Modeling Investigation of the Local Electrochemistry in Lithium-O 2 Batteries : A Kinetic Monte Carlo Approach

In this paper we present a mesoscopic model of the transport and electrochemical processes inside a Lithium-O2 battery cathode pore. The model dynamically resolves both Oxygen Reduction Reaction (ORR) thin film and solution phase mechanisms together with the transport of O2, Li+ and LiO2 in the electrolyte. It is supported on an extension to three dimensions of our Kinetic Monte Carlo (KMC) Electrochemical Variable Step Size Method (E-VSSM) recently published by our group in [M. A. Quiroga and A. A. Franco, J. Electrochem. Soc., 162, E73 (2015)]. The model allows predicting porosity evolution as a function of multiple operational, physical and geometrical parameters including the pore size and inlet/outlet channel size, O2 and Li+ concentration, the property of the solvent as well as the applied overpotential. The investigation of the impact of these different aspects reveals that at the mesoscale level, the overall ORR kinetics and the discharge mechanism strongly depend on a balance between the geometrical features of the pore and the transport as well as the electrochemical properties of the system. © The Author(s) 2015. Published by ECS. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 License (CC BY, http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse of the work in any medium, provided the original work is properly cited. [DOI: 10.1149/2.0841602jes] All rights reserved.

The escalating demand for energy and the depletion of fossil fuel resources create an urge to find alternative methods to convert the available energy on Earth into useful energy.2][3] However, the intermittent nature of renewable energy harvest as well as the hourly fluctuation of energy consumption highlight the importance of developing advanced energy storage systems. 4,5Lithium Ion Batteries (LIBs) have already dominated the market of electronics, but for other applications such as electric vehicles, their further enhancement in energy density is still requested.
The Li-O 2 battery, especially the non-aqueous type, attracted much attention in the last decade due to its high theoretical specific energy density.In spite of tremendous efforts, the performance of nonaqueous Li-O 2 batteries in the state-of-art is still far from expectations in view of its unsatisfactory discharge capacity, high overpotential and severe parasitic reactions.All the above deficiencies are partially, if not all, due to the insulating and insoluble nature of Li 2 O 2 formed during discharge.
Johnson et al. 6 reported a two-step discharge process with a dual mechanism as shown in the following reactions: where the subscript "sol" stands for solution phase, while the star sign stands for species that are adsorbed on the electrode surface .The first step is the reduction of O 2 with the existence of Li + to form LiO 2 ion pairs (Reaction 1).Then, the LiO 2 ion pairs can either move into the electrolyte and be disproportionated, generating Li 2 O 2 of toroidal shape and O 2 (Reaction 2a), or they can stay on the electrode surface and be reduced electrochemically, forming a passivation layer (Reaction 2b).The former path is the "solution phase mechanism" but the origin of the LiO 2 solubility, whether from the high donor number solvent 6 or impurities such as water, 7 is still under debate.Relatively, the latter path is the "thin-film mechanism."Li 2 O 2 formed from these two mechanisms adopts different morphologies which are expected to impact the overall cell performance in different ways (Figure 1): 8 r as an insulator, the Li 2 O 2 passivation layer on the electrode surface will hinder the process of charge transfer, either through electron tunneling or polaron charge transport, 9 and cause the "sudden death" of the cell during discharge far before reaching the theoretical capacity. 10 In contrast, Li 2 O 2 toroid particles which are formed in solution have less contact with the electrode surface leading to lower passivation effects; r being insoluble in organic electrolyte, Li 2 O 2 will accumulate inside the porous electrode during discharge, decreasing the porosity and eventually causing clogging.As a consequence, the discharge capacity as well as the cell power density will be limited by the sluggish transport of oxygen and Li + . 11,12Compared to the flat thinfilm, the toroidal Li 2 O 2 particles, located in the electrolyte, may have stronger influences; r the overpotential during charge shows a dependence on the morphology of Li 2 O 2 and higher overpotential is needed to decompose the toroidal particle compared to the thin film on account of the weaker wiring. 13,14However, as carbon electrodes are mostly employed in the state-of-art cells, the rising of overpotential would aggravate carbon corrosion and generate CO 2 as byproduct, which will form stable and To explore and break through the limits of Li-O 2 battery, a better understanding of competitive discharge mechanisms should be developed in both means of experimentations and modeling.
Emerging as a powerful tool for the analysis of a large diversity of experimental observations, 15,16 computational modeling offers a new way to investigate the influence of the Li-O 2 cell operation conditions onto the overall cell performance and to examine the validity of the assumptions in various theories.On one hand, atomistic and molecular simulations, mainly based on Density Functional Theory (DFT) calculations, are used to investigate microscopic properties including the electronic structure properties and conductivity of discharge products such as Li 2 O 2 , 17,18 charge transport mechanisms 9,[19][20][21] and elementary reaction steps on a very local scale (a few ångströms). 22,23][26] While continuum battery cell models have been developed for decades and have been helpful in analyzing several relevant mechanisms, 27 some important phenomena are still overlooked in their mathematical descriptions.In particular, the representation of the electrodes is usually oversimplified which leads to misinterpretation of experimental data.Indeed, transport properties in the electrolyte, such as diffusion and electromigration, have a large impact on the rate performances of the materials.These transport properties can be influenced by factors as the composition of the electrolyte, the geometry of the pores and the configuration of the pore network.In these continuum models, isotropic mass transport is assumed, which can be away from the reality in the case of complex porous materials since the representation of the heterogeneity and diffusion in three dimensions in this case is important.Such complex porous materials are currently considered in the next generation of Li-O 2 cells, therefore the considerations on anisotropic effects and refined microstructural features become very important.
The multi-scale modeling approaches typically refer to methods aiming to connect mathematical descriptions of mechanisms taking place on different spatial scales. 28,29Multi-scale models aim, by construction, to significantly reduce empirical assumptions beyond what can be done in simple multi-physics models.They have a hierarchical structure, meaning that solution variables defined in a lower hierarchy domain have finer spatial resolution than those solved in a higher hierarchy domain. 30ecently, our team proposed multi-scale models of Li-O 2 cells capturing, for the first time in reported literature, the impact of the detailed cathode microstructure (pore size distribution) 16,31 and the competition between the thin film and solution phase mechanisms on the overall cell performance. 32These models are supported on a continuum description through all the length scales and the pores are assumed to have spherical shape.Nevertheless, the continuum approach may meet its limit when the investigation scales down to the mesolevel where the collision frequency of species with boundaries (pores and/or channel walls) is comparable with the intermolecular collision frequency.Therefore, other approaches, such like Molecular Dynamics and the KMC method, should be employed to build the model at this length scale.Besides, in reality pores can have different shapes and coordination (number of interconnections with the neighbors) and it is highly interesting to investigate how these properties may impact the morphology of the formed Li 2 O 2 .
As a first step towards this goal, in this paper we report a KMC model describing the transport and reaction processes inside a pore of a Li-O 2 battery cathode.The model dynamically resolves both Oxygen Reduction Reaction (ORR) thin-film and solution-phase mechanisms together with displacements of O 2 , Li + and LiO 2 in the electrolyte.
This paper is organized as follows.Firstly details about our modeling methodology are provided, followed by discussions on simulation results.Then we conclude and report the future directions of our work.

Methodology
General assumptions.-Themodel presented in this paper is based on the following assumptions: 1. parasitic reactions, such as electrolyte decomposition and carbon degradation reactions, are not described; 2. isothermal conditions are applied as the system is assumed to be in contact with a thermal reservoir with fixed temperature and the temperature fluctuation can be neglected; 3. the electrochemical double layer (EDL) effects are neglected; 33 4. the overpotential remains constant during simulation.
Despite the simplifications, the modular character of this model leaves the flexibility to include more details (e.g, EDL effects, parasitic reaction and dynamic operation conditions) in future work.
Pore and grid geometries.-Theelectrochemistry is investigated in nanoporous media as a representative of electrode made from porous graphene 34 or gold nano-particles 35 (Figure 2).As shown in Figure 3, the geometry of the modeled pore is assumed to be a spherical cavity with two cylindrical channels linking with the sources of O 2 and Li + , respectively.As input parameters, the radius of the pore (r p ), the radius of the channel (r c ) and the length of the channel (l c ) are set in our model within the range of 5 to 15 nm.
Cubic meshes are used to build the three-dimensional grid network of the system and the mesh size is set to be 5 Å, which is similar to the size of solvated Li + in DMSO (Figure 3). 36Only the four species that get involved in the electrochemical or chemical reactions (cf.Equations 2a and 2b) i.e.Li + , O 2 , LiO 2 and Li 2 O 2 , are considered explicitly.Each Li + or O 2 occupies one grid unit, while each LiO 2 occupies two adjacent grid units.For Li 2 O 2 , the occupied volume is always three grid units but its conformation can be linear or angular (see Figure 4).It is worth noting that even though we did not consider the electrolyte molecules directly, their impact on the particles motion and reactions is captured implicitly by the related kinetic and diffusion parameters.
Initial and boundary conditions.-Thereference electrolyte system adopted in this work is 1M LiPF 6 in DMSO, but the proposed investigation method can also be applied to other electrolyte systems.The O 2 has a saturated concentration of 1.6 × 10 −3 M, the Li + concentration is, in all cases much higher (∼10 2 to 10 4 times) than the O 2 concentration; consequently, during the discharge process, the spatiotemporal variation of the Li + concentration is negligible.
Moreover , under the assumption that the pore is connected to an infinite O 2 reservoir and that the backflow (from the pore to the reservoir of O 2 ) is negligible, the flux density at the channel-source boundary is kept constant according to where j O 2 is the O 2 injection flux density, C O 2 is the oxygen concentration in the reservoir and v is the average molecular velocity of the O 2 molecules.The molecular velocity of O 2 is defined as Equation 5 is the Einstein equation of random walk, where v j is the O 2 jumping frequency in a single direction, z is the size of the mesh and D is the diffusion coefficient.Combining Equations 3, 4 and 5, we obtain Displacement events.-All the considered species are assumed to be mobile except Li 2 O 2 regarding the fact that the final product is either a thin film coating on the electrode material surface or a relative large solid particle in the solution.The displacements can be generated by both translation and rotation, with a step size equaling to the mesh size.The translational kinetics are expressed in form of the jumping frequency (ν j ) as shown in Equation 5and the diffusion coefficient can be estimated from the Stokes-Einstein equation where k is the Boltzman constant, T is the temperature, μ is the viscosity of the electrolyte and r is the hydrodynamic radius of the concerned species.
The calculated diffusion coefficients of Li + and O 2 are around 5 × 10 −10 m 2 s −1 , which are similar to previously reported values. 37or LiO 2 , anisotropic diffusion is assumed with coefficients of 5 × 10 −10 and 2.5 × 10 −10 m 2 s −1 for short and long side, respectively.For the same reason, rotation, as another type of displacement, is considered only for LiO 2 and the rotation frequency (ν r ) is estimated to be 5 × 10 9 s −1 by taking LiNO 3 and organic molecules as references. 38,39ectrochemical reaction events.-Reactions 1 and 2b are electrochemical reactions and their reaction kinetics are usually described by the Bulter-Volmer relationship: where k o is the heterogeneous rate constant, β is the charge transfer coefficient, η is the Bulter-Volmer overpotential.It is worth mentioning that the above equations aim at emphasizing the kinetic dependence on overpotential and all other effects from systematic parameters were embedded into k o .Consequently, the value of k o for the same reaction varies from system to system. 40,41ooking down to the mesoscale, the reaction rate constant of an electrochemical reaction can be written as 42 where ν c is the normalized frequency of the reactant particles colliding with the pore wall surface and P o is the probability of electron transfer upon a single collision at open circuit potential.Combining Equations 8 and 9 we get [10]  where P e is the probability of electron transfer upon a single collision at certain overpotential.The value of k o for both Reactions 1 and 2 have already been reported by several groups, 35,43 but we are still not convinced to employ these values directly to extract P o due to the following reasons: 1. k o depends strongly on the surface property (chemical composition, e.g.type of functional groups) of the electrode, which information is usually missing in the literature; 2. rather than replicate certain experimental results, the aim of this work is to have a better understanding of how the microstructure affects the performance.
Therefore, as a preliminary approach, we roughly assumed that the electron transfer probability is at the same order as the jump frequency of species in the electrolyte.
Moreover, when there is Li 2 O 2 formation, tunneling effects were taken into consideration to describe the insulating nature of the product: P = P e P t [11]   where P t is the electron transmission probability which is a function of the thin-film thickness and the electron energy 44 where V o is the energy level of barrier, E is the total energy of electron, δ is the thickness of thin film, h is the Dirac constant.However, for simplicity reasons, a step function is employed: Chemical reaction events.-Thedisproportionation Reaction 2a is a chemical reaction and it could happen when two LiO 2 encounter in the solution.A similar expression of the kinetic rate could be found for the disproportionation reaction k = ν c P e [14]   here, the P e is the probability of effective collision to generate Li 2 O 2 .
Besides, the free LiO 2 would tend to stick to the Li 2 O 2 particle due to the adsorption effects.A sticking coefficient ϒ is used to capture these effects and the study on this parameter will be elaborated in the future work.
KMC algorithm.-Theapproach adopted to simulating the displacement and reaction events within the pore is based on an extension to three dimensions of our Electrochemical Variable Step Size Method (E-VSSM) recently published by us in the context of Polymer Electrolyte Membrane Fuel Cells (PEMFCs). 1This approach can then explicitly resolve the influence of different operational, physical and geometrical parameters onto the competitive thin film and solution phase ORR, Li 2 O 2 particle growth morphology, pore clogging, etc.The KMC approach, also called Gillespie algorithm, 45 is highly relevant to the present study that focuses on a stochastic system with a large number of molecules (∼2000 for the reference case).For time scales larger than the microsecond the diffusion and chemical reaction are unfeasible to be performed with other techniques such as Molecular Dynamics or DFT. 46As Gillespie 45 stated, the method is useful to describe the chemical species mixture with a specific amount of pair chemical reactions.The algorithm may be applied to any chemical system, however it is particularly relevant for those where the fluctuations and degree of correlation do not allow us to apply deterministic methodologies.
Despite that KMC has been already applied to LIBs 47 and PEM-FCs, as far of our knowledge, this is the first time that the KMC method is applied to a three dimensional domain such a nanopore in Li-O 2 cells.As a first attempt to describe the dynamics and the electrochemical reactions in a three dimensional non-cubic domain, the aim of the current paper is to describe and to discuss both the capabilities and the limitations of this tool.The algorithm behind our KMC method was introduced in our previous PEMFC work. 1 The whole flow chart of the overall code adopted in this paper is provided in Figure 5.After the code building the pore geometry, a random-based algorithm distributes the species inside the pore according to the desired initial condition.Then all the possible events as well as the corresponding rates are counted and injected into a decision algorithm, obtaining a new configuration of the system, which serves as the starting point for next counting-deciding iteration.A statistic weight, which depends on the related kinetic rates expressed in frequency unit, is assigned to each event; then, the E-VSSM selects the event j to be executed according to: j i=1 q i ≤ X 1 .qT ≤ j+1 i=1 q i [15]   where q i refers to the statistic weight of event i, q T refers to the sum of all involved rates and X 1 is a random number comprised between (0,1].In this way, the algorithm manages to assign a still feasible chance to be selected to those events with very low associated rates.Apart from events execution, the KMC method also gives a stochastics-based evolution of the time step according to t = − ln (X 2 ) q T [16]   where X 2 is another random number comprised within [0,1].One can summarize this idea by stating that as higher is total rate (it means more possible execution event with the higher associated rate) smaller is the expected time step.Once the decision is taken by the algorithm, it is followed by the execution of the selected event and the clock is updated until the finalizing criteria is achieved, for instance, in the present paper, the iteration stops when the volume fraction reaches the steady state.
In Table I, we summarize all the parameters and constants that we used in our simulations.

Results
With the aim of investigating how different scenarios impact the discharge processes in mesopores, we selected different systems to study the effects of pore geometry, electrolyte properties and overpotential.
To have a basis for comparison, a reference case was established and the values of related parameters are reported in Table I.Five parallel simulations of the reference case were conducted to examine the reproducibility of the method.Outputs of the simulations are the volume fractions of Li 2 O 2 with respect to the volume of the entire system, which stands for the specific volumetric capacities.As shown in Figure 6, good reproducibility of parallel simulations was achieved with a relative dispersity as small as 2%, coming naturally from the random nature of the KMC method.Hence, in the following, for each case under investigation, only the average results of five parallel simulations are reported.
As shown in Figure 6, the evolution curve of Li 2 O 2 volume fraction could be divided into two stages.The first stage (stage I), with a sharp slope, mainly corresponds to reactions that take place before the inlet channel being clogged or fully passivated; while the second stage (stage II), with a gentle slope (sometimes even flat), corresponds to reactions that happened when the inlet channel was clogged or fully passivated.In the reference case, it is found that almost all of the Li 2 O 2 were formed during stage I in the channel, clogging the O 2 transport pathway, leading to a huge disuse of the active surface area as well as the free space in the pore and ending up with a discharge capacity much lower than the theoretical value.This result is in agreement with the experimental observation reported by Landa-Medrano et al. 48oreover, other information can also be extracted from this calculation: for instance, the slope of the curve reflects the apparent reaction  rate of Li 2 O 2 formation, while the maximum volume fraction reflects the volumetric specific capacity of the system at the end of discharge.
Geometrical parameters.-Theimpacts of geometric parameters, i.e. pore radius and channel radius, were investigated.Four systems with pore radius of 10 or 15 nm, combining channels with radius of 7.5 or 5 nm were studied in this part and outputs of simulations, related to the amount of Li 2 O 2 formed in each case, are reported in Figure 7.It is shown that the performance depends mainly on the channel radius.
When the channel is enlarged, two reverse effects on reaction rate are expected.On one hand, the frequency of collision between particles and channel surface would be brought down due to the decrease of surface/volume ratio, leading to a decrease of apparent reaction rate; on the other hand, the O 2 flux for a given flux density would be enhanced as the intersection area was increased.Simulations were conducted by taking both effects into consideration.As shown in Figure 7a, the correlation between the reaction rate and channel radius turns out to be positive, indicating that the benefit from the growth of O 2 population counteracted and even overwhelmed the loss from the decrease of collision frequency.
The enlargement of the channel radius could also improve the formation of Li 2 O 2 and there are several factors that could be ascribed to that enlargement.Firstly, a larger channel showed a higher capacity to accommodate Li 2 O 2 and lower risk to be clogged; secondly, a larger channel enhances the possibility for O 2 to go through; lastly, when the channel radius became larger than the critical tunneling distance (5 nm in our simulation), the channel would be covered by a passivation layer of 5 nm but not be clogged anymore, which leads to a continuous growing at stage II as showed by curves of systems with 7.5 nm channels.
Besides, the pore size also has an effect on the reaction rate, especially after the passivation of the channel surface (Figure 7c).It is found that the curve corresponding to the smaller pore has a steeper slope, which is due to the increase of the collision frequency in a more confined environment.Moreover, the system with pore radius of 10 nm has an advantage in specific volumetric capacity compared to the system with pore radius of 15 nm when the amount of lithium peroxide is converted to the volume fraction with respect to the volume of the entire system.
Electrolyte properties.-Although it does not participate directly in the reaction, the electrolyte has strong impacts on the discharge process by affecting the population of species as well as their mass transport.Several investigated scenarios are presented in the following subsections.Concentration effects.-Inthis subsection, effects of Li + concentration (C Li + ) and O 2 concentration in the reservoir (C O 2 ) on discharge processes were studied.The increase in either of the two parameters resulted in an improvement of the reaction rate (Figures 8 and 9) due to the increase of collisions; whereas the two parameters have different impacts on the discharge capacity.As shown in Figure 8, although the j O 2 in each case was different, it turned out that almost all the O 2 was consumed in the channel before entering into the pore, resulting in a similar discharge capacity with a slight improvement when the C O 2 increased.However, higher discharge capacity was obtained with lower C Li + , mainly due to the promotion of O 2 transport.Coexisting in the system, Li + ions were hindering the O 2 transport by steric effect resulted from finite volume.Thus lowering the C Li + reduced the hindrance and improved the mobility of O 2. As a consequence, the competitiveness of O 2 transport in the competition with its reduction was improved and the channel clogging was delayed, leading to a higher volumetric capacity accompanied by a more uniform distribution of Li 2 O 2 (Figure 9).Mobility effects.-Themobility of species is also electrolytedependent and the diffusion coefficient is the parameter which captures this property.Under the assumption that Li + , O 2 and the short side of LiO 2 have the same radius in their solvated form, the diffusion coefficient for these species is identical according to the Stokes-Einstein relationship (Equation 7).It is shown in Figure 10 that the reaction rate was raised when we increase the diffusion coefficient.This is due to the enhancement of the collision frequencies, indicating that the limiting factor of apparent reaction rate is mass transport under simulated conditions in the present paper.Moreover, the higher diffusion coefficient also benefits the discharge capacity due to the improvement of O 2 transport.As same as in the case of Li + dilution, while the mobility is increased, O 2 is more likely to diffuse through the channel rather than being anchored at the surface to form Li 2 O 2 , contributing to the capacity improvement and more uniform distribution of Li 2 O 2 as showed by the snapshots in Figure 10.
Overpotential.-It is widely reported that there is a close correlation between discharge capacity and overpotential.As discussed previously in the methodology part, it is through the electron transfer probability that the overpotential influences the discharge process.The higher the overpotential is, the higher the probability for an electrochemical reaction to occur upon a single collision.We thus explored the effects of overpotential by tuning the electron transfer probability (P e ) on the basis of the reference case.
The evolution of overall Li 2 O 2 volume fraction in those cases is plotted in Figure 11a.It turns out that increasing P e , which refers to raising the overpotential in experiment, leads to the speeding up of the reaction rate.However, the favoring of electrochemical reactions   crippled further the O 2 transport, advancing the channel clogging and ending up with low discharge capacity.In addition, raising the overpotential is also unfavorable for the solution-phase mechanism, as the LiO 2 is prone to be further reduced rather than to move in the solution.Consequently, the ratio of Li 2 O 2 formed from the solutionphase mechanism drops significantly as shown in Figure 11c.

Conclusions and Perspectives
In this paper we have presented a modeling approach to resolve the ORR, O 2 and Li + transport inside a pore within a Li-O 2 battery cathode.The approach is based on a KMC algorithm resolving electrochemical, transport and nucleation processes in three dimensions.The influence of several parameters onto the reaction rate, discharge capacity, product localization and relative weight between the thin film and solution phase mechanisms has been investigated.
The simulation results straightforwardly showed that the discharge capacity of Li-O 2 batteries with mesoporous cathode is mainly from the part that is closer to the O 2 inlet.Therefore, the discharge capacity relies more on the geometry surface area of the electrode other than the carbon loading, which corresponds to the experiment observation by Landa-Medrano et al. 48Besides, in the confined system, apart from the electron transfer, the kinetic rate also strongly depends on the size of the system.A smaller system yields a larger kinetic rate due to the increase of collision frequency.However, a smaller system would result in lower capacity due to the faster channel/pore clogging.In order to optimize the performance, both of the above effects should be balanced in the electrode design.Moreover, under the simulated conditions, it is found that the competition between electrochemical reactions and mass transport determines the capacity as well as the discharge pathway.Operations that improve the mobility of species, for instance, dilution of Li + and increase of diffusion coefficient, lead to a more effective discharge and higher capacity.It is worth noting that similar effects of salt concentration on discharge capacity in Li-O 2 battery have been reported with experimental data without explanation, 29 and we believe that our model could provide insights into these observations.
As far of our knowledge, in this paper we presented the first threedimensional KMC model describing the ORR, O 2 and lithium ions transport inside a pore.Some assumptions were done in order to provide a first flavor of our method capabilities.Nevertheless, the first aim of this paper is to present a methodology able to tackle the complexity of LiO 2 and Li 2 O 2 mechanism growth in a time range sufficiently long to describe the significant events during the pore-filling.We underline that this approach can be extended to the investigation of competitive reactions and transport mechanisms in other electrochemical systems, such as lithium sulfur batteries: this will be the focus of one of our incoming investigations.

Figure 1 .
Figure 1.Schematic illustration of a Li-O 2 cell at the end of discharge.) unless CC License in place (see abstract).ecsdl.org/site/terms_useaddress.Redistribution subject to ECS terms of use (see 80.13.180.142Downloaded on 2015-12-08 to IP

Figure 2 .
Figure 2. Schematics of a) the nanoporous cathode structure and b) different reactions considered in the present model.

Figure 3 .
Figure 3. (a) 3D visualization of the simulated pore-channel system; (b) 2D projection of the simulated pore-channel system with the cut-view of the grid structure indicated.

Figure 4 .
Figure 4.The blue square depicts the lithium ion, the green square lithium superoxide and red square the lithium peroxide.(a) Initial configuration; (b) first possibility: the lithium ion on the top of lithium superoxide reacts and the code forces the thin film formation; (c) second possibility: the lithium ion on the left reacts and the code forces again the thin film formation on the interface.

Figure 5 .
Figure 5. Flow chart of the simulation code used in this paper.

Figure 6 .
Figure 6.(a) Evolution of Li 2 O 2 volume fraction with time for reference case; (b) snapshots with initial '1', intermediate '2' and final '3' geometries of the pore for a typical case.

Figure 7 .
Figure 7. (a) Evolution of Li 2 O 2 formation in systems with pores of 10 or 15 nm and channels of 5 or 7.5 nm from beginning; (b) snapshots with the final geometry for all the simulated cases; (c) Evolution of Li 2 O 2 formation all the systems after channel passivation.

Figure 8 .
Figure 8. Simulation results of Li 2 O 2 volume fraction dynamics in systems where the O 2 concentrations in reservoir are 1.6, 3.2 and 8.0 mM and images of each system at the end of discharge; (b) snapshots with the final geometry for all the simulated cases.

Figure 9 .
Figure 9. Simulation results of Li 2 O 2 volume fraction dynamics in systems where the Li + concentration are 1, 0.5, 0.2 and 0.1 M, respectively and the images of each system at the end of discharge; (b) snapshots with the final geometry for all the simulated cases.

Figure 10 .
Figure 10.Li 2 O 2 volume fraction evolution in time for three different Li + diffusion coefficients; (b) snapshots with the final geometry for all simulated cases.

Figure 11 .
Figure 11.(a) Li 2 O 2 volume fraction evolution in time for three different reaction probabilities; (b) snapshots with the final geometry for all the simulated cases; (c) Li 2 O 2 volume fraction in solution phase evolution in time.