Towards Ultra Low Cobalt Cathodes: A High Fidelity Phase Search Incorporating Uncertainty Quantification of Li-Ni-Mn-Co Oxides

Towards Ultra Low Cobalt Cathodes: A High Fidelity Phase Search Incorporating Uncertainty Quantification of Li-Ni-Mn-Co Oxides

Abstract

Layered Li(Ni,Mn,Co,)O (NMC) presents an intriguing ternary alloy design space for the optimization of performance as a cathode material in Li-ion batteries. In the case of NMC, however, only a select few proportions of transition metal cations have been attempted and even fewer show promise. Recently, due to cost and resource limitations of Co, high Ni-containing NMC alloys have gained enormous attention. Here, we present a high fidelity computational search of the ternary phase diagram with an emphasis on high-Ni containing compositional phases. This is done through the use of density functional theory training data fed into a reduced order model Hamiltonian that accounts for effective electronic and spin interactions of neighboring transition metal atoms at various lengths in a background of fixed lithium and oxygen atoms. This model can then be solved to include finite temperature thermodynamics into a convex hull analysis. We also provide a method to propagate the uncertainty at every level of the analysis to the final prediction of thermodynamically favorable compositional phases thus providing a quantitative measure of confidence for each prediction made. Due to the complexity of the three component system, as well as the intrinsic error of density functional theory, we argue that this propagation of uncertainty, particularly the uncertainty due to exchange-correlation functional choice is necessary to have reliable and interpretable results. With our final result, we recover the prediction of already known phases such as LiNiMnCoO (111) and LiNiMnCoO (811) in exact proportion while finding other proportions very close to the experimentally claimed LiNiMnCoO (622) and LiNiMnCoO (532) phases, and overall predict a total of 37 phases with reasonable confidence and 69 more phases with a lower level of confidence. Through our analysis, we also can identify the phases with the highest average operational voltage at a given Co composition. Our method presents a framework that can be extended to searches for other high Ni cathode materials by substituting other transition metal atoms into the lattice such as aluminum and magnesium, which have already shown promise.

pacs:

I Introduction

The first rechargeable Li-ion battery cathode material, layered LiCoO,Mizushima et al. (1980) revolutionized the world of portable electronics. The high cost and low operational capacity of this battery material was improved with the addition of Ni and Mn to create LiNiMnCoO (NMC). This was first successfully done by Lui et al.Liu et al. (1999) and later the most popular phase, x=y=1/3 (NMC111) was synthesized by Ohzuku et al.Ohzuku and Makimura (2001) Since then a small collection of phases have been used in various applications with some of the most promising candidates being x=0.5, y=0.3 (NMC532) Wang et al. (2003); Wu et al. (2015); Li et al. (2017), and x=0.8,y=0.1 (NMC811) Choi and Manthiram (2006); Kim et al. (2006). Additionally, NMC cathode materials are now present in a majority of electric vehicle battery chemistries Blomgren (2017).

Many studies have explored trends of operational performance as a function of Ni, Mn, and Co content. Julien et al. demonstrated that with increasing Co content, the specific capacity is increasedJulien et al. (2016). Manthiram et al on the other hand showed that capacity increased with increasing Ni content Manthiram et al. (2016). The discrepancy of these studies is that they each probed a different one dimensional subset of the full two dimensional phase space. Another consideration is the thermal stability of the cathode. Recent work has shown that thermal stability decreases with increasing Ni contentBak et al. (2014). In this study, however, no phases were tested for Ni content between 60 and 80 and only one possible combination of Co and Mn content was used for each Ni content. Only with a full understanding of the entire compositional phase space, can trends be understood and possibly broken. What therefore is needed, is an extensive computational search of the ternary phase diagram of NMC cathodes. Previous computational works have provided reasonable guesses for various phases with no exploration for the stability with respect to other cation orderings at the same composition Wu et al. (2015); Liang et al. (2017); Xu et al. (2017). Given the large number of possible atomic arrangements at a fixed composition, this represents a significant computational challenge. A full understanding of the computational phase space, however, also requires the comparison of the lowest energies of each fixed composition with other nearby composition in phase space, furthering the complexity of the problem. Additionally, due to the closeness in energies of phases both the same in composition and with slightly different composition, a careful, high precision calculations must be performed. Within this work, we specifically address the need for new low Co content cathode materials as Co production will present major price volatility and supply concerns in the near future.Olivetti et al. (2017)

Density functional theory (DFT) alongside thermodynamic modeling has been successfully used as a method of understanding and predicting the stable phases of many binary alloys van de Walle and Ceder (2002); Ruban and Abrikosov (2008). Density functional theory calculations train a computationally efficient model and with the use of statistical simulations using the model Hamiltonian, the thermodynamically stable phases with respect to composition can be predicted.

One of the largest factors in the accuracy of density functional theory calculations is the choice of the exchange correlation functional. Different exchange correlation functionals are shown to have drastically different levels of accuracy for different material classes with certain functionals performing better for certain applications Peverati and Truhlar (2014); Mardirossian and Head-Gordon (2017). Recently, a class of functionals known as Bayesian Error Estimation Functionals have been developed to incorporate a collection of exchange correlation functionals and therefore provide an estimate of uncertainty related to the calculation of exchange correlation energetics. Previous work has applied this uncertainty estimation to the prediction of magnetic ground states, bulk properties, and activity of catalysts Houchins and Viswanathan (2017); Ahmad and Viswanathan (2016); Krishnamurthy et al. (2018); Deshpande et al. (2016).

Given the complexity of the problem, it is necessary to fully understand the uncertainty from systematic errors in the DFT training data as well as random errors from the statistical nature of Monte Carlo simulations. The energy differences between various compositions can be on the order of the uncertainty, and therefore we must cleverly propagate the uncertainty from every step to the final prediction of the thermodynamically stable phases. We focus most heavily on understanding the uncertainty derived from density functional theory itself. Reduced order models such as cluster expansion can be fit to arbitrary accuracy given enough terms, and the statistical error of Monte Carlo can be controlled given a sufficient number of sweeps in your simulation as well as repeated simulations. The systematic error in density functional theory, however, is impossible to control as there is no way to find an exchange correlation functional for an arbitrary system with arbitrary accuracy. We therefore attempt to understand the sensitivity of our results to variation of the exchange correlation function. Additionally, we attempt to show that without considering uncertainty propagation, this work could not be done with trustable results.

Figure 1: A top view of the lattice structure. The cation site are shown pink, while the oxygen sites are red and the lithium atoms are not pictured. The black triangle shows the idealized triangle lattice the cations form.

i.1 Calculation Details

Ii Methods

In this work, we use a total of 84 density functional theory calculations for the formation energy of various NMC compositions, focusing on high Ni-content structures, to train a reduced order model in order to more efficiently evaluate the formation energy with only a minor decrease in accuracy compared to input. The reduced order model is then solved in a high fidelity search of the composition space using Monte Carlo simulations that incorporate finite temperature effects including entropy contributions to the Gibbs free energy. From these simulations, the change in Gibbs free energy from pure metal oxide end member states to a NMC phase at a given composition is fed into a convex hull analysis. Finally we are able to predict all thermodynamically stable NMC phases. Here we describe briefly each step of this process providing the full details in the Supplementary Materials.

ii.1 Training Data

For all of the 84 density functional calculations used as training data, the Bayesian Error Estimation Functional with van der Waals (BEEF-vdW) Wellendorff et al. (2012) was used to treat the exchange-correlation energy at the level of the general gradient approximation. Recent work has shown that systematic error related to oxide materials can be reduced for energy difference if the proper reference state is used Christensen et al. (2015a). This idea was further extended to show that the error in DFT predicted formation enthalpies of two similiar reactions are correlated. That is the differnce in formation enthalpies is constant and independent of exchange correlation function allowing the difference in these formation enthalpies to be compared accuratatly without the use of a Hubbard U correction Christensen et al. (2016). We therefore do not include the Hubbard correction within our calculations.

The BEEF-vdW functional has the ability to estimate the error of density functional theory with respect to experiment. This exchange correlation functional incorporates an estimation of uncertainty through the generation of an ensemble of energetic predictions. The ensemble of energetic predictions is generated by feeding the electron density of a fully self consistent calculation of the empirically fit functional from BEEF-vdW to an ensemble of exchange correlation functionals. These functionals then provide an ensemble of non-self consistent energetic predictions. The spread of this ensemble has be pretuned to recreate the error of the DFT calculation with respect to the experimental data it was trained on. In this way, BEEF-vdW links the precision of the ensemble of predictions, to the accuracy of the prediction of the self consistent calculations. With this empirical error estimation, we can provide a quantification of uncertainty within each DFT calculation and propagate that uncertainty to the model parameters that are trained with respect to the DFT input. Beyond the result that the spread of energies estimates the uncertainty of the calculation, the ensemble of functionals provides a way to probe exchange-correlation space to understand the confidence of a prediction at the level of the generalized gradient approximations. This aspect can be used to ask if another exchange-correlation functional was used, would the prediction qualitatively change. The spread of the BEEF functional estimates has been been shown to bound the prediction of other general gradient approximation functionals for mechanical properties, magnetic ground states, and reaction enthalpies for hydrocarbons Ahmad and Viswanathan (2016); Houchins and Viswanathan (2017); Christensen et al. (2015b, 2016). As there is no way to ensure accuracy or even the estimation of accuracy, we argue this aspect of uncertainty quantification strengthens the interperability of DFT prediction and predictions made by models trained on DFT.

ii.2 Reduced Order Model

Once the DFT data is generated, a reduced order model for the formation enthalpy is chosen. The change in volume of the lattice from pure states to the mixed state simulated by DFT is on the order of 1. Therefore at atmospheric pressure, which is well below the accuracy of DFT and the convergence of the calculations here. The goal of this reduced order model is to recreate the change in enthalpy with respect to the end members of homogeneous lithium metal oxides, rather than the enthalpy from constituent elements. Therefore, our model focuses on the effective interactions of the changing compositions and arrangement of the transition metal ions. The energetic effects of these constant background lithium and oxygen atoms with each other and a particular cation should be largely subtracted away as these interaction are assumed to be independent of composition. By treating the lithium and oxygen atoms as a constant background, we create a model lattice containing the relevant cations and only consider the energetics and interactions related to the proportions and orderings of the transition metal ions. This type of consideration of only specific interaction terms while leaving other atoms as a constant background term has been used sucessfully in other cluster expansion studies of transition metal oxide materials Zhou et al. (2006); Malik et al. (2009); Lee et al. (2017). We assume a triangle lattice where at each site on the lattice, one of the three metal cations (Ni, Mn, Co) is present as seen in Figure 1. The model used here, as with other cluster expansion models, contains energetic terms for the species that occupies a lattice site, and N-body interactions over various length scales. The final model is given by:

(1)

Where i and j are sums over the lattice sites, is a sum over nearest neighbor interactions, is a sum over next-nearest neighbor interactions, and x and y are sums over the three possible species (Ni,Mn,Co). The J,J terms represent the electron nearest neighbor and next nearest neighbor interactions respectively while the K terms are the energies related to magnetic interactions. The values of each of these coefficients can be seen in Table 1.

Iii Entropy

To understand the stability of phases at finite temperature, a comparison of the change in Gibbs free energy rather than enthalpy is needed.

For this we must understand the entropy of the systems. In order to calculate this we estimate the entropy change to be purely configurational and assume that any entropy due to internal degrees of freedom of the lattice remain relatively constant between all compositions due to the similarities in lattices. To a first approximation ignoring spin, the configurational entropy is given by that of a three component system that ideally mixes which after the application of Sterling’s approximation, is given by

This assumes that all of the possible states contribute to the entropy. In reality many fewer states are degenerate and therefore this is a large overestimation of the configurational entropy. This overestimation of entropy is large enough to dominate the enthalpic differences between two neighboring compositions and leads to a convex hull that suggests that any composition is thermodynamically stable. In order to mitigate this, we calculate the entropy more exactly by calculating the entropy as

where is the probability of a given state at a given temperature. This probability can be calculated using the Boltzmann factor if the energy of every state could be calculated. Due to the large number of possible states, calculating this for all (N=81, with 3403 unique sets of for a 9x9 lattice) possible states is computationally unfeasible. We turn to Montropolis Monte Carlo sampling and estimate this probability as the sample probability of the energy state within the Monte Carlo simulation sampling.

This method leads to a entropic term in the Gibbs energy this is orders of magnitude smaller than that of the ideal mixing. As a result the enthalpy plays a more equal part, along with the entropy in determining the phases on the convex hull. The change in Gibbs Energy for all compositions simulated can be seen in Figure 2.

Figure 2: Formation energy heat map of all NMC combinations tested.

Iv Phase diagram and Results

Once the change in Gibbs energy is calculated, we perform a hull analysis in an attempt to find the lowest-lying compositional states and determine what compositional states will appear as pure states within a battery. To do this, the points were fed into a 3D convex hull algorithm and all points on the hull with formation energy less than or equal to 0 were plotted as points on a ternary phase diagram.

Figure 3: Predicted thermodynamically favorable compositional phases of NMC are shown as blue diamonds. For reference, a collection of experimentally claimed phases from Ref. 10 are shown in red. A large disagreement between the the experimentally claimed phases and predicted phases should be noted, especially for the NMC111 phase.

As can be seen in Figure 3, the predicted points in blue poorly match the red points of experimentally used NMC phases only agreeing with a subset. Perceivable issues with phase diagram prediction methods like the one used here is the size of absolute error with respect to the energy difference of two adjacent points. For a given DFT test calculations, the error in the prediction of energy could be as large as on the order of 10 meV while the energy difference of two points adjacent in compositional space could be on the order of meV. It is therefore hard to guarantee that the energetic ordering of points in reality match that of the ordering with pure DFT and the reduced order model. The product of this error with statistical error in the Monte Carlo simulations causes the poor disagreement in the predicted ternary phase diagram shown in Figure 3. It is therefore necessary to understand the uncertainty and create a well defined method to propagate this uncertainty onto the convex hull and predicted phase diagram and order to understand the prediction confidence of this result.

V Uncertainty Quantification

Although density functional theory is largely accepted as a computationally viable way to calculate the energy of a system from first principles, the topic of uncertainty quantification remain an issue. Attempts have been made to understand the uncertainty by testing predictions over large data sets and trying to understand the errors in statistical or regression models Pernot and Savin (2018); De Waele et al. (2016); Lejaeghere et al. (2016). Other attempts have tried to incorporate the uncertainty into the generation of a new exchange correlation functionals Aldegunde et al. (2016); Petzold et al. (2012); Proppe et al. (2016); Wellendorff et al. (2012, 2014); Simm and Reiher (2016). The exchange-correlation functional used within this work, BEEF-vdW, allows for the estimation of error that has been empirically fit to experiment. Along with a main exchange-correlation functional that is used to self consistently converge the electron density and derive the energy, BEEF-vdW carries with it 2000 functionals with slight perturbations from the main functional as described earlier. The spread of these 2000 functionals has been pre-tuned to recreate a non-self consistent spread of energies that matches the error of the main functionals energetic prediction with respect to experimental training data. Previous work has utilized this spread of functionals to quantify the uncertainty at the level of GGA in the predciton of magnetic statesHouchins and Viswanathan (2017), bulk propertiesAhmad and Viswanathan (2016), surface Pourbaix diagramsVinogradova et al. (2017); Sumaria et al. (2018), scaling relations in oxygen reductionYan et al. (2017); Deshpande et al. (2016), formation enthalpies of reactions Christensen et al. (2015b), and the prediction efficiency of DFT derived descriptors.Krishnamurthy et al. (2018).

v.1 Prediction confidence

Recently, the BEEF-vdW functional was was used to quantify the prediction confidence of magnetic ordering of a material using spin polarized DFT at the GGA level. The confidence of the prediction can be understood through a c-value that defined as the ratio of functionals that give the same prediction of energetic favorabililty of a given state as the main BEEF-vdW functional Houchins and Viswanathan (2017). We extend this principle here to quantify the confidence of our parameter fits by calculation the c-value that a given interaction is antiferromagnetic versus ferromagnetic, or favorable versus not.

Value (meV) c-value
J 8.6 1.00
J 185.0 1.00
J -0.2 0.56
J -141.0 1.00
J 19.7 0.67
J -26.5 1.00
J -32.8 1.00
J -11.4 0.63
J -23.1 1.00
K -12.1 1.00
K 47.4 1.00
K 49.0 1.00
K 16.3 1.00
K -5.1 0.58
K -26.3 1.00
Table 1: The values in meV for the parameters of the model fit with normal regression.

For each DFT training calculation, an ensemble of energies is generated and the reduced order model is trained. We can then define a c-value, shown in Table 1, for the value of a training parameter as the normalized number of times that a parameter has the same sign as the parameter from the model trained from the best fit exchange correlation functional. This gives us a confidence of the physical interperability of a term within the model trained on DFT data. If a large ratio of models predict the same qualitative interaction between two species, then there is a high confidence that the interaction is independent of exchange correlation functional and is likely to be the qualitatively correct interaction. This quantification of confidence is independent of the reduced order model performance in recreating the training data but rather propagates the sensitivity of functional choice to the final model parameters. From this quantification of uncertainty in the model parameters, we can then understand the level of confidence in predictions of the qualitative arangement of cations in the lowest energy structure as we discuss in Section VI.2.

As the c-value is ultimately coupled to the number of input calculations and the variety of interactions sampled within the data, it could be used to assist in the choice of what additional data to add. It is to be noted that this will lead to structure choices that could be distinctly different than that obtained purely based on cluster expansion. If a c-value of an interaction is low or even below 0.5, indicating that the majority of functionals disagree with the main prediction, then further input data targeting that interaction should be used. With a sufficient number of training points and a model that is arbitrarily accurate, the c-values should converge to reflect purely the approximate uncertainty in the underlying DFT data.

Figure 4: An example path in compositional phase space. The convex hull generated from the values given by the Monte Carlo simulations of the model from the self consistent DFT calculation is shown is black. At every point, an error bar of one standard deviation in the change of Gibbs free energy over all 2000 models is given. As can be seen, the error bars are much larger than the difference in energies of two adjacent points.

This provides qualitative understanding of confidence at the DFT and an additional way to select training data. The ultimate goal, however, is to propagate this uncertainty to the final predicted phase diagram as well as incorporate the statistical error within Monte Carlo. To further understand the challenge of propagating error through a hull analysis we look at a single line within the ternary phase space. Figure 4 shows the predicted change in Gibbs energy, fitted convex hull, and error bars of LiNiMnCoO. The error bars shown are calculated as the standard deviation of the predicted change in Gibbs energy over the 2000 Monte Carlo simulations each with a distinct model from the BEEF-vdW ensemble. A similar plot in Figure 5 over the same portion of phase space show a collection of 50 of the full 2000 hulls for further elucidation of the uncertainty associated with the final hull analysis.

Figure 5: A plot of the the first 50 convex hulls from the 2000 total hulls.

We extend the concept of c-value to convex hull analysis by repeating our whole precedure of fiting the reduced order model, performing Monte Carlo to simulate thousands of compositions, and feeding the results to a convex hull analysis for all 2000 ensembles. We then count the number of times each specific composition appears on the convex hull over all of the 2000 convex hulls ultimately generated. We then plot this normalized predominance in Figure 6. This results in much better agreement with the experimentally seen phases as well as shows high confidence for the prediction of new compositional phases.

Figure 6: A plot of the predominance of each composition on the 2000 different convex hulls fit at at a temperature of 297K. The darker the circles, the more often the phase appeared on the final hull generated by a single exchange correlation functionals. Experimental phases from Ref. 10 are shown in red.

v.2 Analysis of Predicted Phases

The model used within this work assumes a fixed layered structure and hence, we limit our discussion to phases below a certain threshold Mn content. Beyond this level, the material will exhibit a layered to spinel structural transformation that is seen in the LiMnO end member. Given recent work on the structural phase diagram of the Li-Ni-Mn-Co oxide pseudoquaternary system, we set this cutoff to be 40 Mn contentBrown et al. (2015). We also restrict our analysis to predicted phases that have a low Co content in order to reduce the overall cost of the predicted material. We show in Table 2 the 30 phases with the highest c-value that contain all three cations species and satisfy the criteria of less than 40 Mn for structural stability during cycling and less than 40 Co for cost. Notice that the c-value for the NMC 111 phase, one of the most predominate phases in literature, is 0.21. We use this as a calibration for what level of confidence within our model has a plausible chance of occurring in reality. The absence of the NMC11 phase in the prediction from only a single exchange correlation functional in Figure 3, is one example of the importance of propagation of uncertainty in this work. Overall, we predict 37 phases that satisfy the design criteria with c-value above 0.2 which we consider to be of reasonable probability. There are 69 more phases satisfying our constraints with a c-value from 0.1 to 0.2. The full set of 303 predicted phases, including compositions outside of our restricted design space, can be seen in Table S1 in the Supplementary Materials.

As mentioned before, the 811 phase is a promising candidate for a high Ni-content cathode and we recover this phase with high confidence (c=0.73). Other experimentally claimed phases such as 622, 532 are not seen in those exact proportions with signifigant confidence. In the case of 622, the phase was not predicted in those specific proportions for any of the 2000 functionals. While the phase NMC 14 5 5, which is very close in composition, was not only predicted in the single functional hull, but appeared in the final result with a c-value of 0.25 and therefore is a possibly the exact composition of the proposed 622 phase. Similarly in the case of 532, which has a c-value of 0.058, is more likely to be NMC321, a phase predicted with a c-value of 0.3665 or NMC 12 7 5 as predicted by othersXu et al. (2017), which has a c-value of 0.215. The level of prediction confidence of the 532, however, does not completely exclude it from possibility.

Aside from gaining a better understanding of what the exact proportions of experimentally used phases, of particular interest are phases that have likely not been experimentally explored yet. Most specifically those Ni content at or above 75%. Interesting candidates include 16 1 3, 16 3 1, 10 1 1, 912, and 921 which present an interesting collection of phases as a starting point for experimental verification and testing.

By intentionally designing the input data to heavily include high Ni content phases, the final hull predictions are expected to be more reliable in the high Ni region. This may explain the general trend that there are more points with higher c-value in the high Ni region.

Ni Mn Co c-value
Exact
Proportions
Base
0.800 0.050 0.150 0.77 16 1 3 20
0.800 0.100 0.100 0.73 8 1 1 10
0.800 0.150 0.050 0.62 16 3 1 20
0.833 0.083 0.083 0.52 10 1 1 12
0.750 0.083 0.167 0.48 9 1 2 12
0.708 0.083 0.208 0.48 17 5 2 24
0.708 0.042 0.250 0.47 17 1 6 24
0.500 0.250 0.250 0.46 2 1 1 4
0.850 0.050 0.100 0.46 17 1 2 20
0.875 0.042 0.083 0.41 21 1 2 24
0.750 0.167 0.083 0.37 9 2 1 12
0.500 0.333 0.167 0.37 3 2 1 6
0.750 0.208 0.042 0.36 18 5 1 24
0.667 0.250 0.083 0.36 8 3 1 12
0.625 0.250 0.125 0.36 5 2 1 8
0.583 0.250 0.167 0.34 7 3 2 12
0.750 0.125 0.125 0.34 6 1 1 8
0.722 0.083 0.194 0.32 26 3 7 36
0.722 0.111 0.167 0.31 13 2 3 18
0.708 0.125 0.167 0.26 17 3 4 24
0.500 0.083 0.417 0.26 6 1 5 12
0.583 0.208 0.208 0.25 14 5 5 24
0.667 0.042 0.292 0.25 16 1 7 24
0.667 0.111 0.222 0.24 6 1 2 9
0.750 0.042 0.208 0.24 18 1 5 24
0.500 0.375 0.125 0.23 4 3 1 8
0.667 0.083 0.250 0.23 8 1 3 12
0.500 0.292 0.208 0.22 12 7 5 24
0.333 0.333 0.333 0.21 1 1 1 3
0.722 0.056 0.222 0.21 13 1 4 18
Table 2: Predicted phases and their corresponding c-values. The exact proportions are given in the most reduced form as N M C

Vi Analysis

The prediction of the exact compositions of NMC phases will allow more accurately prediction of the properties and degradation of these cathode materials. Recent work measuring the oxygen release during cycling of NMC based Li ion batteries have shown conflicting understandings of the relationship between degradation related to oxygen release and the specific content of the NMC. Further work using the specifically predicted phases here may allow better understanding of oxygen release as it relates to not only composition but cation ordering. The specific composition also allows us to accurately predict important properties related to the performance of these materials as a cathode in a Li-ion battery.

vi.1 Voltage

One of the easiest predictions that can be made once the Gibbs Free energy is known is the average voltage of the predicted NMC cathode with respect to Li/Li given by the Nernst Equation.

The change in Gibbs energy is given by

If we assume that the delithiated phase has the same Gibbs energy as the delithiated end members as is discussed in the Supplementary Materials, we can calculate the average voltage of LiNiMnCoO as

A plot of the predicted average voltage over the entire phase can be seen in Figure 7 and a table of the predicted voltage of common NMC phases is presented in Table 3. The prediction is compared to the predictions from AutoLion-1D Kalupson et al. (2013), which is well benchmarked to experiment and allows for the simulation of an extremely slow discharge to better match the perfectly ideal average voltage predicted by this work.

Phase AL1D (V) Model (V)
111 3.83 3.88
532 3.82 3.89
811 3.83 3.86
622 3.80 3.92
001 3.98 4.10
Table 3: The average voltage of various cathodes with respect to Li/Li. To give an idea of the difference in the DFT predicted voltage and AL1D, the predicted average voltage of pure LiCoO is also given. In general, the DFT predicted voltages are about 0.1eV or less higher than those predicted by AL1D.
Figure 7: The predicted average voltage for all compositions calculated.

One of the largest design problems in NMC is currently related to the price of Co, therefore we explore the maximum average voltage at every Co content in Figure 9. From this plot we see an overwhelming trend of decreasing average voltage with decreasing Co content. We have also reploted Figure 6 with the addition of points in black showing the phases predicted to have the highest average voltage at that Co content. Overlaid with the c-value, we can then identify particularly promising phases at low Co content. For a Co content of about 10% the 831 phase with a predicted voltage of 3.88 V or the 921 phase with a predicted voltage of 3.88 V present a small improvements over the current 811 phase with a predicted voltage of 3.86. At approximately the amount of Co present in 111, we predict the 16 1 7 phase to have a voltage of 3.96 V compared to 3.88 V for 111. Overall, we expect the possibility of small improvements in the average voltage with future work needed to understand the possibility of improvement in the open circuit voltage at every state of charge.

Figure 8: An overlay of the composition with the maximium voltage at a given content with the c-value hull. This allows for a match up optimal voltage with phases that have a high confidence of thermodynamic stability. Experimental phases from Ref. 10 are shown in red.
Figure 9: A plot of the maximum predicted voltage as a function of Co content.

vi.2 Cation Ordering

Another important result we can get from this work is a prediction of the specific ordering of the cations. To do this, we use Monte Carlo to stochastically solve for what the absolute minimum enthalpy is for a given composition using our reduced order model. We have done this for a collection of the interesting phases that were discussed and the results are shown in Figure 10. The structures shown are the minimum energy structures at the specific concentration over all of the supercells tested. The qualitative results of these structures can be interpreted using the information of the interactions from Table 1 and the confidence of these interpretations can be understood through the c-value of each interaction parameter. We note a tendency for the formation of stripes of Ni ions next to stripes of Mn and Co ions. Despite a repulsive Ni-Ni nearest neighbor interaction, the Ni ions prefer this arrangement due to a larger magnetic interaction that can create an overall negative energy interaction. This nearest neighbor interaction is then relatively stable compared to Ni-Ni next nearest neighbor interactions due to the attractiveness of heterogeneous next nearest neighbor interactions involving Ni. In the phases with higher Mn content than Co as is seen in Figure 10c and 10d, the Mn is seen clustering slightly within the stripe despite Mn-Mn nearest neigbor interactions not having an overall favorable interaction as seen in Ni. This clustering is a result of the tendency to maximize the number of Mn-Co next nearest neighbor interactions. However, when possible as seen in Figure 10 (a,b) Mn and Co will arrange to have no interactions with each other as their interaction with Ni is more favorable.

{overpic}

[width=1.0trim=0 0cm 0 0cm,clip]811.png (a){overpic}[width=1.0trim=0 0cm 0 -0.2cm,clip]1275.png (c)

{overpic}

[width= 1.0trim=0 0cm 0 0cm,clip]921.png (b){overpic}[width=1.0trim=0 0cm 0 -0.2cm,clip]321.png (d)

Figure 10: Top view of the specific arrangement of cations predicted in (a) 811, (b) 921, (c) 12 7 5 and (d) 321. The oxygen are small red, Ni is green, Mn is purple, and Co is pink. Li is not shown here as it would be above and below the transition metal plane.

Vii Conclusion

We have presented a comprehensive exploration of the NMC phase space with the use of a reduced order model trained on DFT input. The addition of uncertainty quantification related to the largest and least controllable source of error, the exchange correlation functional choice, leads to a more interperatable and reliable reduced order model and final prediction of compositional phases. From the understanding gained from uncertainty quantification of the reduced order model parameters we can understand the qualitative patterns of cation ordering and the level of certainty of this ordering. And from the propagation of error to the final convex hull analysis, we were able to have a more comprehensive list of predicted phases and assign a confidence to these predictions. A collection of promising phases exist in the high Ni content region that satisfy both the constraint of low Mn content to avoid a structural transition to spinel during operation, as well the constraint of low Co content derived from economic concerns. Of the phases predicted, we select some particularly promising candidates such as 831 and 921 for low Co content, and 16 1 7, as an alternative with the similar Co content to 111.

Future work would include using the cation orderings found here to understand surface effects in existing and newly predicted NMC phases. We believe a better atomistic picture of the cation ordering may help to solve the mystery of how the composition of the cathode affects the production of O and CO gas during the operation of batteries containing various NMC phases. Further exploration of the operational properties of the newly predicted phases may also lead to marginal improvements over existing phases even when coupled with external design constraints such as low Co content with the most interesting properties to study being the open circuit voltage and operational capacity. Additionally, the extension of this work to include other doping elements such as Al, and Mg may provide a larger design space to tune the properties of low Co cathode materials.

Acknowledgements.
G. H. and V. V. acknowledge support from the Scott Institute for Energy Innovation at Carnegie Mellon. G. H. would like to also thank Shashank Sripad for assistance with AL1D calculations.

Appendix A Supplementary Materials

a.1 Delithated Case

The same method of preforming a set of density functional theory calculations as training data for a reduced order model was used in the case of a fully delitiated cathode Ni Mn CoO which in the case of the Ni and Co end members is experimentally known to be in the O1 phase. This training data was fed into the same functional form of the reduced order model used for the lithiated case, and the fit model was simulated using Metropolis Monte Carlo. It was found, however that there are no mixed phases with a negative change in Gibbs energy. The resulting change in Gibbs Energy is very small and is therefore expected to the be kinetically stabilized especially if during operation, the cathode is not charged to the fully delithiated state.

a.2 Computational Details

Spin polarized density functional theory using the Generalized Gradient Approximation for the exchange-correlation energy were used to train the model for formation energy. The DFT training data was performed with various supercells of the O-3 structure of LiMO where M=Ni,Mn,Co as is the experimentally seen phase for LiNiO and LiCoO Ebner et al. (1994); Mizushima et al. (1980), and is a metastable phase for LiMnO Armstrong and Bruce (1996); Vitins and West (1997).

A 3x1, 2x2, 3x2, and 3x4 rhombohedral supercell as well as a triangular supercell were used. For the 3x1, and triangular all unique combinations of cation ordering within a single layer were used where the same cation ordering was repeated within the three layers. For the 2x2 and 3x2 supercells a random selection of 33 unique cation ordering each were used. Two possible 3x4 cation orderings for the a structure near the 811 phase (actual proportions of 10 1 1) were used to increase the model accuracy in the high-Ni region of the phase space. One extra calculation of the 532 phase in a 5x2 supercell from Wu el al.Wu et al. (2015) was also include to assess the proposed cation ordering of a NMC532 phase. The out of plane interactions of the cations are ignored as justified later in the Reduced Order Model section. The initial spin state of the DFT training data was randomly chosen to be positive or negative to create a random sampling of spin interaction.

All density functional calculations were performed using the Gradient Projector Augmented Wavefunction (GPAW)Mortensen et al. (2005) approach with a grid spacing of 0.16 Åand [12,12,2] k-points per conventional cell, with Monkhorst-Pack grid. Both the grid spacing and the k-point mesh were converged to eV with respect to the predicted formation energy of test LiNMC mixture. Within each DFT calculation with these grid-point and k-point specifications, the densities were converged to 0.0001 electrons, and the energies were converged to eV . The specific lattice parameters of each cation composition and ordering was found by varying the lattice parameters in the a and b directions equally with strain of and where are the experimental lattice parameters of LiNiO, while fixing the c direction. The volumes and energies of at each point were fit to a jellium equation of state Alchagirov et al. (2001) and the ideal lattice parameters were extracted. This process was then repeated by fixing the a and b constants and varying the c constants then fitting to find the ideal constant in the way as before. The resulting structure at the optimized lattice parameters was then relaxed to a maximum force of eV/ Å.

a.3 Selection of Reduced order Model

Of particular relevance to our system are the nearest neighbor electronic interactions, next-nearest neighbor electronic interactions, and nearest neighbor spin interactions. We also investigate the addition of a three body equilateral cluster term. Early in the work, the effect of out of plane interactions was investigated and structures with differing out of plane interactions differed in energy on the order of eV or less which is at the scale of numerical convergence and therefore is ignored. The number of interactions in the final model was determined by calculating the cross validation score

The model chosen, gave the smallest cross validation score of 2.9 meV. With the addition of the triangle cluster term, the cross validation score rose to 3.6 meV, while dropping the next nearest neighbor interaction weakened the cross validation score to 3.3 meV.

It should be noted that the inclusion of all terms in the model would lead to a rank deficient system of equation when trying to fit the interaction coefficients through a system of linear equations. This is because there is an external constraints of all the lattice sites being filled that creates a linear dependence on the number of cations with the number of nearest neighbor interactions. Because of this, we choose to artificially set the within our model to choose a particular solution to the model. This choice is made due to the fact that the formation enthalpy is being fit by the model and the compositional details of a given state should be almost completely subtracted away, only leaving interaction energies. There is a similar linear dependence between the number of next nearest neigbor interactions and the number of each ion. Because of this we choose to set . What is left in the model after removing the occupation terms is the relative favorability of a given interaction compared to all other interactions. The model was fit using multivariate normal regression where the design matrix of number of interactions and the formation energies were normalized to one stoichiometric unit of LiMO. This prevents overweighting the larger supercells. The magnitude of the spins of each cation is assumed to be constant for all composition of Ni, Mn, and Co with the same level of lithiation. This assumption allows for the reduced order model to be solved more easily since allowing the the magnetic moments of each atom to vary continuously and unbounded during Monte Carlo simulations would cause the spin interaction term to be unbounded. There is also no reason to suggest that any of the cations exhibit a set of high-spin and low-spin states given a fixed state of Li content.

a.4 Monte Carlo Simulations

Once the model is trained, the enthalpy of formation at finite temperature is simulated using a Metropolis Monte Carlo algorithm. For each composition of transition metals, a random initial cation and spin ordering is assumed. Then randomly the spin of a single cation or the position of two cations is swapped. After each swap, the condition for acceptance is given by estimating the probability of the swap happening spontaneously.

Where is the energy change of the swap. If the energy change is negative then the swap is always kept. If the energy change is positive, it is kept if the probability is less than than a randomly generated number between 0 and 1. This algorithm allows for the thermal population of states and given a sufficiently long simulation, will accurately predict the probability of thermal population.

Each simulation is thermalized by running the calculation for 500 sweeps (steps per lattice site) and then the calculation is sampled every sweep for 1000 more sweeps. The enthalpy is then calculated to be the average enthalpy of all of the sweeps. We run 9x9, 8x8, 8x5, 6x5, 6x6, and 6x4 supercells for a total of over 7,000 points on the ternary phase diagram.

a.5 Presentation of All Compositions Tested

Here we present our full findings over all of the compositions tested. Figure S1 shows the distribution of c-values for all compositions tested. To show detail for the phases with a significant c-value, the compositions with c-value below 0.05 are not shown. There are a total 1930 compositions with nonzero c-value below 0.05, and 5365 phase with a zero c-value not pictured. From the total of 7609 compositions tested, we extract 303 phases with a c-value greater than or equal to 0.05. That is that 100 or more functionals agree that the phase is on the convex hull.

Figure S1: A histogram of the number of phases with each c-value.

The full listing of these phases can be seen in Table S1 below. The information is presented in the same way as Table 2 but there is no exclusion based on the Co and Mn composition restrictions presented above. It should be noted that while the Co constraint was due to economic and material availability reasons, the Mn constraint we based off of structural instability reasoning and should still be kept in mind. For completeness, we also show phases that do not contain all three cations.

Ni Mn Co c-value
Exact
Proportions
Base
0.000 0.988 0.012 1.00 0 80 1 81
0.012 0.000 0.988 1.00 1 0 80 81
0.012 0.988 0.000 1.00 1 80 0 81
0.000 0.012 0.988 1.00 0 1 80 81
0.167 0.000 0.833 0.99 1 0 5 6
0.333 0.667 0.000 0.99 1 2 0 3
0.167 0.833 0.000 0.98 1 5 0 6
0.500 0.500 0.000 0.98 1 1 0 2
0.333 0.000 0.667 0.96 1 0 2 3
0.750 0.000 0.250 0.95 3 0 1 4
0.750 0.250 0.000 0.95 3 1 0 4
0.250 0.750 0.000 0.91 1 3 0 4
0.667 0.333 0.000 0.83 2 1 0 3
0.988 0.012 0.000 0.81 80 1 0 81
0.988 0.000 0.012 0.81 80 0 1 81
0.250 0.000 0.750 0.81 1 0 3 4
0.500 0.000 0.500 0.80 1 0 1 2
0.833 0.000 0.167 0.76 5 0 1 6
0.800 0.050 0.150 0.76 16 1 3 20
0.800 0.100 0.100 0.73 8 1 1 10
0.111 0.000 0.889 0.73 1 0 8 9
0.833 0.167 0.000 0.66 5 1 0 6
0.583 0.000 0.417 0.64 7 0 5 12
0.222 0.000 0.778 0.63 2 0 7 9
0.800 0.150 0.050 0.62 16 3 1 20
0.900 0.000 0.100 0.58 9 0 1 10
0.028 0.000 0.972 0.54 1 0 35 36
0.200 0.800 0.000 0.53 1 4 0 5
0.833 0.083 0.083 0.52 10 1 1 12
0.417 0.000 0.583 0.50 5 0 7 12
0.750 0.083 0.167 0.48 9 1 2 12
0.708 0.083 0.208 0.48 17 2 5 24
0.984 0.016 0.000 0.47 63 1 0 64
0.708 0.042 0.250 0.47 17 1 6 24
0.500 0.250 0.250 0.46 2 1 1 4
0.850 0.050 0.100 0.46 17 1 2 20
0.417 0.583 0.000 0.45 5 7 0 12
0.000 0.333 0.667 0.44 0 1 2 3
0.000 0.167 0.833 0.44 0 1 5 6
0.917 0.083 0.000 0.43 11 1 0 12
0.900 0.100 0.000 0.41 9 1 0 10
0.875 0.042 0.083 0.41 21 1 2 24
0.000 0.222 0.778 0.38 0 2 7 9
0.750 0.167 0.083 0.37 9 2 1 12
0.500 0.333 0.167 0.37 3 2 1 6
0.000 0.667 0.333 0.36 0 2 1 3
0.750 0.208 0.042 0.36 18 5 1 24
0.667 0.250 0.083 0.36 8 3 1 12
0.625 0.250 0.125 0.36 5 2 1 8
0.984 0.000 0.016 0.35 63 0 1 64
0.583 0.250 0.167 0.34 7 3 2 12
0.750 0.125 0.125 0.34 6 1 1 8
0.111 0.889 0.000 0.34 1 8 0 9
0.500 0.417 0.083 0.33 6 5 1 12
0.722 0.083 0.194 0.32 26 3 7 36
0.012 0.012 0.975 0.32 1 1 79 81
Table S1: Predicted phases and their corresponding c-values. The exact proportions are given in the most reduced form as N M C
Ni Mn Co c-value
Exact
Proportions
Base
0.917 0.000 0.083 0.31 11 0 1 12
0.722 0.111 0.167 0.31 13 2 3 18
0.000 0.500 0.500 0.30 0 1 1 2
0.667 0.000 0.333 0.27 2 0 1 3
0.100 0.000 0.900 0.27 1 0 9 10
0.708 0.125 0.167 0.26 17 3 4 24
0.000 0.100 0.900 0.26 0 1 9 10
0.500 0.083 0.417 0.26 6 1 5 12
0.083 0.833 0.083 0.26 1 10 1 12
0.583 0.208 0.208 0.25 14 5 5 24
0.056 0.000 0.944 0.25 1 0 17 18
0.667 0.042 0.292 0.25 16 1 7 24
0.800 0.200 0.000 0.25 4 1 0 5
0.583 0.417 0.000 0.25 7 5 0 12
0.667 0.111 0.222 0.24 6 1 2 9
0.708 0.292 0.000 0.24 17 7 0 24
0.750 0.042 0.208 0.24 18 1 5 24
0.234 0.766 0.000 0.24 15 49 0 64
0.000 0.833 0.167 0.24 0 5 1 6
0.222 0.778 0.000 0.23 2 7 0 9
0.542 0.458 0.000 0.23 13 11 0 24
0.500 0.375 0.125 0.23 4 3 1 8
0.625 0.375 0.000 0.23 5 3 0 8
0.667 0.083 0.250 0.23 8 1 3 12
0.025 0.975 0.000 0.22 2 79 0 81
0.417 0.056 0.528 0.22 15 2 19 36
0.333 0.028 0.639 0.22 12 1 23 36
0.200 0.000 0.800 0.22 1 0 4 5
0.250 0.333 0.417 0.22 3 4 5 12
0.000 0.111 0.889 0.21 0 1 8 9
0.500 0.292 0.208 0.21 12 7 5 24
0.333 0.333 0.333 0.21 1 1 1 3
0.722 0.056 0.222 0.21 13 1 4 18
0.975 0.025 0.000 0.21 39 1 0 40
0.950 0.050 0.000 0.20 19 1 0 20
0.583 0.292 0.125 0.20 14 7 3 24
0.028 0.972 0.000 0.20 1 35 0 36
0.583 0.333 0.083 0.20 7 4 1 12
0.012 0.975 0.012 0.20 1 79 1 81
0.000 0.889 0.111 0.20 0 8 1 9
0.167 0.417 0.417 0.20 2 5 5 12
0.167 0.167 0.667 0.19 1 1 4 6
0.583 0.375 0.042 0.19 14 9 1 24
0.967 0.033 0.000 0.19 29 1 0 30
0.250 0.167 0.583 0.19 3 2 7 12
0.000 0.778 0.222 0.19 0 7 2 9
0.333 0.583 0.083 0.19 4 7 1 12
0.333 0.500 0.167 0.19 2 3 1 6
0.444 0.000 0.556 0.19 4 0 5 9
0.167 0.667 0.167 0.19 1 4 1 6
0.750 0.222 0.028 0.19 27 8 1 36
0.694 0.028 0.278 0.19 25 1 10 36
0.167 0.250 0.583 0.18 2 3 7 12
0.444 0.056 0.500 0.18 8 1 9 18
0.300 0.000 0.700 0.18 3 0 7 10
0.100 0.500 0.400 0.18 1 5 4 10
0.083 0.583 0.333 0.18 1 7 4 12
0.500 0.458 0.042 0.18 12 11 1 24
0.875 0.083 0.042 0.18 21 2 1 24
0.750 0.016 0.234 0.17 48 1 15 64
Ni Mn Co c-value
Exact
Proportions
Base
0.000 0.400 0.600 0.17 0 2 3 5
0.542 0.167 0.292 0.17 13 4 7 24
0.900 0.033 0.067 0.17 27 1 2 30
0.972 0.028 0.000 0.17 35 1 0 36
0.975 0.000 0.025 0.17 39 0 1 40
0.389 0.056 0.556 0.17 7 1 10 18
0.167 0.333 0.500 0.17 1 2 3 6
0.625 0.292 0.083 0.17 15 7 2 24
0.958 0.042 0.000 0.16 23 1 0 24
0.250 0.500 0.250 0.16 1 2 1 4
0.667 0.125 0.208 0.16 16 3 5 24
0.000 0.600 0.400 0.16 0 3 2 5
0.025 0.025 0.951 0.16 2 2 77 81
0.950 0.000 0.050 0.16 19 0 1 20
0.222 0.750 0.028 0.16 8 27 1 36
0.125 0.625 0.250 0.16 1 5 2 8
0.528 0.083 0.389 0.15 19 3 14 36
0.889 0.111 0.000 0.15 8 1 0 9
0.750 0.028 0.222 0.15 27 1 8 36
0.667 0.167 0.167 0.15 4 1 1 6
0.000 0.972 0.028 0.15 0 35 1 36
0.417 0.500 0.083 0.15 5 6 1 12
0.625 0.333 0.042 0.15 15 8 1 24
0.833 0.125 0.042 0.15 20 3 1 24
0.667 0.222 0.111 0.15 6 2 1 9
0.150 0.050 0.800 0.15 3 1 16 20
0.050 0.100 0.850 0.14 1 2 17 20
0.025 0.012 0.963 0.14 2 1 78 81
0.972 0.000 0.028 0.14 35 0 1 36
0.000 0.056 0.944 0.14 0 1 17 18
0.900 0.050 0.050 0.14 18 1 1 20
0.111 0.028 0.861 0.14 4 1 31 36
0.542 0.333 0.125 0.14 13 8 3 24
0.250 0.250 0.500 0.13 1 1 2 4
0.750 0.194 0.056 0.13 27 7 2 36
0.542 0.375 0.083 0.13 13 9 2 24
0.222 0.694 0.083 0.13 8 25 3 36
0.333 0.556 0.111 0.13 3 5 1 9
0.472 0.056 0.472 0.13 17 2 17 36
0.750 0.111 0.139 0.12 27 4 5 36
0.722 0.139 0.139 0.12 26 5 5 36
0.850 0.100 0.050 0.12 17 2 1 20
0.389 0.611 0.000 0.12 7 11 0 18
0.208 0.667 0.125 0.12 5 16 3 24
0.333 0.250 0.417 0.12 4 3 5 12
0.833 0.042 0.125 0.12 20 1 3 24
0.375 0.083 0.542 0.12 9 2 13 24
0.333 0.167 0.500 0.12 2 1 3 6
0.556 0.111 0.333 0.12 5 1 3 9
0.042 0.125 0.833 0.12 1 3 20 24
0.167 0.625 0.208 0.12 4 15 5 24
0.417 0.333 0.250 0.11 5 4 3 12
0.306 0.028 0.667 0.11 11 1 24 36
0.100 0.100 0.800 0.11 1 1 8 10
0.722 0.278 0.000 0.11 13 5 0 18
0.472 0.083 0.444 0.11 17 3 16 36
0.500 0.056 0.444 0.11 9 1 8 18
0.778 0.111 0.111 0.11 7 1 1 9
0.867 0.100 0.033 0.10 26 3 1 30
0.967 0.000 0.033 0.10 29 0 1 30
Ni Mn Co c-value
Exact
Proportions
Base
0.500 0.278 0.222 0.10 9 5 4 18
0.375 0.042 0.583 0.10 9 1 14 24
0.933 0.000 0.067 0.10 14 0 1 15
0.250 0.722 0.028 0.10 9 26 1 36
0.000 0.016 0.984 0.10 0 1 63 64
0.333 0.042 0.625 0.10 8 1 15 24
0.194 0.667 0.139 0.10 7 24 5 36
0.208 0.583 0.208 0.10 5 14 5 24
0.792 0.208 0.000 0.10 19 5 0 24
0.800 0.000 0.200 0.10 4 0 1 5
0.042 0.083 0.875 0.09 1 2 21 24
0.734 0.266 0.000 0.09 47 17 0 64
0.222 0.667 0.111 0.09 2 6 1 9
0.766 0.234 0.000 0.09 49 15 0 64
0.958 0.000 0.042 0.09 23 0 1 24
0.028 0.111 0.861 0.09 1 4 31 36
0.542 0.250 0.208 0.09 13 6 5 24
0.361 0.056 0.583 0.09 13 2 21 36
0.016 0.000 0.984 0.09 1 0 63 64
0.600 0.350 0.050 0.09 12 7 1 20
0.750 0.234 0.016 0.09 48 15 1 64
0.083 0.083 0.833 0.09 1 1 10 12
0.333 0.222 0.444 0.09 3 2 4 9
0.625 0.083 0.292 0.09 15 2 7 24
0.778 0.222 0.000 0.09 7 2 0 9
0.050 0.200 0.750 0.09 1 4 15 20
0.083 0.000 0.917 0.08 1 0 11 12
0.125 0.042 0.833 0.08 3 1 20 24
0.333 0.444 0.222 0.08 3 4 2 9
0.208 0.625 0.167 0.08 5 15 4 24
0.300 0.300 0.400 0.08 3 3 4 10
0.625 0.125 0.250 0.08 5 1 2 8
0.000 0.750 0.250 0.08 0 3 1 4
0.208 0.542 0.250 0.08 5 13 6 24
0.000 0.028 0.972 0.08 0 1 35 36
0.867 0.033 0.100 0.08 26 1 3 30
0.900 0.067 0.033 0.08 27 2 1 30
0.200 0.200 0.600 0.08 1 1 3 5
0.194 0.333 0.472 0.08 7 12 17 36
0.028 0.139 0.833 0.08 1 5 30 36
0.667 0.292 0.042 0.08 16 7 1 24
0.000 0.083 0.917 0.08 0 1 11 12
0.167 0.583 0.250 0.08 2 7 3 12
0.500 0.400 0.100 0.08 5 4 1 10
0.033 0.100 0.867 0.08 1 3 26 30
0.250 0.625 0.125 0.07 2 5 1 8
0.444 0.083 0.472 0.07 16 3 17 36
0.028 0.167 0.806 0.07 1 6 29 36
0.167 0.750 0.083 0.07 2 9 1 12
0.133 0.033 0.833 0.07 4 1 25 30
0.542 0.292 0.167 0.07 13 7 4 24
0.333 0.083 0.583 0.07 4 1 7 12
0.333 0.111 0.556 0.07 3 1 5 9
0.528 0.472 0.000 0.07 19 17 0 36
0.333 0.139 0.528 0.07 12 5 19 36
0.050 0.150 0.800 0.07 1 3 16 20
0.050 0.250 0.700 0.07 1 5 14 20
0.083 0.056 0.861 0.07 3 2 31 36
0.067 0.133 0.800 0.07 1 2 12 15
0.194 0.750 0.056 0.07 7 27 2 36
Ni Mn Co c-value
Exact
Proportions
Base
0.417 0.167 0.417 0.07 5 2 5 12
0.611 0.222 0.167 0.07 11 4 3 18
0.266 0.734 0.000 0.07 17 47 0 64
0.300 0.500 0.200 0.07 3 5 2 10
0.167 0.306 0.528 0.07 6 11 19 36
0.222 0.639 0.139 0.07 8 23 5 36
0.933 0.067 0.000 0.07 14 1 0 15
0.083 0.028 0.889 0.07 3 1 32 36
0.083 0.167 0.750 0.07 1 2 9 12
0.400 0.000 0.600 0.07 2 0 3 5
0.750 0.139 0.111 0.07 27 5 4 36
0.550 0.350 0.100 0.07 11 7 2 20
0.944 0.000 0.056 0.07 17 0 1 18
0.833 0.133 0.033 0.06 25 4 1 30
0.333 0.194 0.472 0.06 12 7 17 36
0.350 0.550 0.100 0.06 7 11 2 20
0.361 0.167 0.472 0.06 13 6 17 36
0.750 0.056 0.194 0.06 27 2 7 36
0.734 0.078 0.188 0.06 47 5 12 64
0.500 0.111 0.389 0.06 9 2 7 18
0.975 0.000 0.025 0.06 79 0 2 81
0.800 0.133 0.067 0.06 12 2 1 15
0.167 0.042 0.792 0.06 4 1 19 24
0.250 0.600 0.150 0.06 5 12 3 20
0.139 0.028 0.833 0.06 5 1 30 36
0.042 0.250 0.708 0.06 1 6 17 24
0.667 0.028 0.306 0.06 24 1 11 36
0.625 0.208 0.167 0.06 15 5 4 24
0.975 0.025 0.000 0.06 79 2 0 81
0.734 0.094 0.172 0.06 47 6 11 64
0.306 0.139 0.556 0.06 11 5 20 36
0.250 0.417 0.333 0.06 3 5 4 12
0.278 0.111 0.611 0.06 5 2 11 18
0.067 0.100 0.833 0.06 2 3 25 30
0.250 0.583 0.167 0.06 3 7 2 12
0.500 0.300 0.200 0.06 5 3 2 10
0.806 0.194 0.000 0.06 29 7 0 36
0.100 0.150 0.750 0.06 2 3 15 20
0.111 0.056 0.833 0.06 2 1 15 18
0.792 0.083 0.125 0.06 19 2 3 24
0.333 0.542 0.125 0.06 8 13 3 24
0.167 0.361 0.472 0.06 6 13 17 36
0.333 0.056 0.611 0.06 6 1 11 18
0.600 0.300 0.100 0.06 6 3 1 10
0.083 0.750 0.167 0.06 1 9 2 12
0.542 0.417 0.042 0.06 13 10 1 24
0.556 0.083 0.361 0.05 20 3 13 36
0.025 0.716 0.259 0.05 2 58 21 81
0.125 0.583 0.292 0.05 3 14 7 24
0.042 0.167 0.792 0.05 1 4 19 24
0.167 0.500 0.333 0.05 1 3 2 6
0.306 0.167 0.528 0.05 11 6 19 36
0.969 0.000 0.031 0.05 31 0 1 32
0.734 0.125 0.141 0.05 47 8 9 64
0.458 0.125 0.417 0.05 11 3 10 24
0.694 0.306 0.000 0.05 25 11 0 36
0.250 0.550 0.200 0.05 5 11 4 20
0.056 0.056 0.889 0.05 1 1 16 18
0.278 0.028 0.694 0.05 10 1 25 36
0.917 0.042 0.042 0.05 22 1 1 24
Ni Mn Co c-value
Exact
Proportions
Base
0.750 0.031 0.219 0.05 24 1 7 32
0.734 0.250 0.016 0.05 47 16 1 64
0.031 0.031 0.938 0.05 1 1 30 32
0.583 0.194 0.222 0.05 21 7 8 36
0.083 0.125 0.792 0.05 2 3 19 24
0.734 0.109 0.156 0.05 47 7 10 64
0.867 0.067 0.067 0.05 13 1 1 15

References

  1. K. Mizushima, P. C. Jones, P. J. Wiseman,  and J. B. Goodenough, Mat. Res. Bull 15, 783 (1980).
  2. Z. Liu, A. Yu,  and J. Y. Lee, J. Power Sources 8182, 416 (1999).
  3. T. Ohzuku and Y. Makimura, Chem. Mater. 30, 642 (2001).
  4. G. Wang, S. Bewlay, J. Yao, Y. Chen, Z. Guo, H. Liu,  and S. Dou, J. Power Sources 119-121, 189 (2003).
  5. Z. Wu, S. Ji, J. Zheng, Z. Hu, S. Xiao, Y. Wei, Z. Zhuo, Y. Lin, W. Yang, K. Xu, K. Amine,  and F. Pan, Nano Lett.  (2015).
  6. J. Li, H. Li, W. Stone, R. Weber, S. Hy,  and J. R. Dahn, J. Electrochem. Soc. 164, A3529 (2017).
  7. J. Choi and A. Manthiram, J. Power Sources 162, 667 (2006).
  8. M.-H. Kim, H.-S. Shin, D. Shin,  and Y.-K. Sun, J. Power Sources 159, 1328 (2006).
  9. G. E. Blomgren, J. Electrochem. Soc. 164, A5019 (2017).
  10. C. Julien, A. Mauger, K. Zaghib,  and H. Groult, Materials 9, 595 (2016).
  11. A. Manthiram, J. C. Knight, S. T. Myung, S. M. Oh,  and Y. K. Sun, Adv. Energy Mater. 6, 1501010 (2016).
  12. S.-M. Bak, E. Hu, Y. Zhou, X. Yu, S. D. Senanayake, S.-J. Cho, K.-B. Kim, K. Y. Chung, X.-Q. Yang,  and K.-W. Nam, ACS Appl. Mater. Interfaces 6, 22594 (2014).
  13. C. Liang, R. C. Longo, F. Kong, C. Zhang, Y. Nie, Y. Zheng, J. S. Kim, S. Jeon, S. A. Choi,  and K. Cho, J. Power Sources 340, 217 (2017).
  14. S. Xu, G. Luo, R. Jacobs, S. Fang, M. K. Mahanthappa, R. J. Hamers,  and D. Morgan, ACS Appl. Mater. Interfaces 9, 20545 (2017).
  15. E. A. Olivetti, G. Ceder, G. G. Gaustad,  and X. Fu, Joule 1, 229 (2017).
  16. A. van de Walle and G. Ceder, J. Phase Equilib. 23, 348 (2002).
  17. A. V. Ruban and I. A. Abrikosov, Rep. Prog. Phys. 71, 046501 (2008).
  18. R. Peverati and D. G. Truhlar, Philos. Trans. R. Soc. London, Ser. A 372, 20120476 (2014).
  19. N. Mardirossian and M. Head-Gordon, Mol. Phys. 115, 2315 (2017).
  20. G. Houchins and V. Viswanathan, Phys. Rev. B 96, 134426 (2017).
  21. Z. Ahmad and V. Viswanathan, Phys. Rev. B 94, 064105 (2016).
  22. D. Krishnamurthy, V. Sumaria,  and V. Viswanathan, J. Phys. Chem. Lett. 9, 588 (2018).
  23. S. Deshpande, J. R. Kitchin,  and V. Viswanathan, ACS Catal. 6, 5251 (2016).
  24. J. Wellendorff, K. T. Lundgaard, A. Møgelhøj, V. Petzold, D. D. Landis, J. K. Nørskov, T. Bligaard,  and K. W. Jacobsen, Phys. Rev. B 85, 235149 (2012).
  25. R. Christensen, J. S. Hummelshøj, H. A. Hansen,  and T. Vegge, J. Phys. Chem. C 119, 17596 (2015a).
  26. R. Christensen, H. A. Hansen, C. F. Dickens, J. K. Nørskov,  and T. Vegge, J. Phys. Chem. C 120, 24910 (2016).
  27. R. Christensen, H. A. Hansen,  and T. Vegge, Catal. Sci. Technol. 5, 4946 (2015b).
  28. F. Zhou, T. Maxisch,  and G. Ceder, Phys. Rev. Lett. 97, 155704 (2006).
  29. R. Malik, F. Zhou,  and G. Ceder, Phys. Rev. B 79, 214201 (2009).
  30. E. Lee, H. Iddir,  and R. Benedek, Phys. Rev. B 95, 085134 (2017).
  31. P. Pernot and A. Savin, J. Chem. Phys. 148, 241707 (2018).
  32. S. De Waele, K. Lejaeghere, M. Sluydts,  and S. Cottenier, Phys. Rev. B 94, 235418 (2016).
  33. K. Lejaeghere, L. Vanduyfhuys, T. Verstraelen, V. V. Speybroeck,  and S. Cottenier, Comput. Mater. Sci. 117, 390 (2016).
  34. M. Aldegunde, J. R. Kermode,  and N. Zabaras, J. Comput. Phys. 311, 173 (2016).
  35. V. Petzold, T. Bligaard,  and K. W. Jacobsen, Top. Catal. 55, 402 (2012).
  36. J. Proppe, T. Husch, G. N. Simm,  and M. Reiher, Faraday Discuss. 195, 497 (2016).
  37. J. Wellendorff, K. T. Lundgaard, K. W. Jacobsen,  and T. Bligaard, J. Chem. Phys. 140, 144107 (2014).
  38. G. N. Simm and M. Reiher, J. Chem. Theory Comput.  (2016).
  39. O. Vinogradova, D. Krishnamurthy, V. Pande,  and V. Viswanathan, arXiv  (2017), arXiv:1710.08407 [cond-mat.mtrl-sci] .
  40. V. Sumaria, D. Krishnamurthy,  and V. Viswanathan, arXiv  (2018), arXiv:1804.02766 [cond-mat.mtrl-sci] .
  41. B. Yan, D. Krishnamurthy, C. H. Hendon, S. Deshpande, Y. Surendranath,  and V. Viswanathan, Joule 1, 600 (2017).
  42. C. R. Brown, E. McCalla, C. Watson,  and J. R. Dahn, ACS Comb. Sci.  (2015).
  43. J. Kalupson, G. Luo,  and C. E. Shaffer, in SAE 2013 World Congress Exhibition (SAE International, 2013).
  44. W. Ebner, D. Fouchard,  and L. Xie, Solid State Ionics 69, 238 (1994).
  45. A. R. Armstrong and P. G. Bruce, Nature 381, 499 (1996).
  46. G. Vitins and K. West, J. Electrochem. Soc. 144, 2587 (1997).
  47. J. J. Mortensen, L. B. Hansen,  and K. W. Jacobsen, Phys. Rev. B 71, 035109 (2005).
  48. A. B. Alchagirov, J. P. Perdew, J. C. Boettger, R. C. Albers,  and C. Fiolhais, Phys. Rev. B 63, 224115 (2001).
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