A SEI Modeling Approach Distinguishing between Capacity and Power Fade

In this paper we introduce a pseudo two-dimensional (P2D) model for a common lithium-nickel-cobalt-manganese-oxide versus graphite (NCM/graphite) cell with solid electrolyte interphase (SEI) growth as the dominating capacity fade mechanism on the anode and active material dissolution as the main aging mechanism on the cathode. The SEI implementation considers a growth due to non-ideal insulation properties during calendar as well as cyclic aging and a re-formation after cyclic cracking of the layer during graphite expansion. Additionally, our approach distinguishes between an electronic ( σ SEI ) and an ionic ( κ SEI ) conductivity of the SEI. This approach introduces the possibility to adapt the model to capacity as well as power fade. Simulation data show good agreement with an experimental aging study for NCM/graphite cells at different temperatures introduced in literature. © The Author(s) 2017. Published by ECS. This is an open access article distributed under the terms of the Creative Commons Attribution

Lithium-ion batteries are one of the most promising candidates for energy storage in future stationary storage systems and electric vehicles. [1][2][3] Enormous research efforts have been conducted to get a thorough understanding of the system "lithium-ion cell" and to further develop it for higher energy and power density, higher safety standards as well as longer cycle life. 4 The aging behavior of lithium-ion batteries has been a focus issue of battery research since the introduction of lithium-ion cells by Sony in 1991. 5 Reviews by Agubra et al., 6,7 Arora et al., 8 Aurbach et al., 9,10 Birkl et al., 11 Broussely et al., 12 Verma et al. 13 and Vetter et al. 14 are just a few examples of the extensive literature regarding aging behavior. Commonly accepted and experimentally verified aging phenomena as mentioned in the previously cited literature are electrolyte decomposition leading to solid electrolyte interphase (SEI) and cathode electrolyte interphase (CEI) growth, solvent co-intercalation, gas evolution with subsequent cracking of particles, a decrease of accessible surface area and porosity due to SEI growth, contact loss of active material particles due to volume changes during cycling, binder decomposition, current collector corrosion, metallic lithium plating and transition-metal dissolution from the cathode.
The listed aging mechanisms can be assigned to three different categories that are a loss of lithium-ions (LLI), an impedance increase and a loss of active material (LAM). 12,[15][16][17][18] The LLI is synonymous to a decrease in the amount of cyclable lithium-ions as they are trapped in a passivating film on either of the electrodes or in plated metallic lithium. Due to the growth of the passivating layers and/or the formation of rock-salt in the cathode (residue of the cathode active material after transition-metal dissolution), kinetic transport of lithium-ions through those inactive areas is limited and results in an impedance rise. An LAM can be caused by the dissolution of transition-metal-ions from the cathode bulk material, changes in the electrode composition and/or changes in crystal structure of the active material which all diminish the amount of host structure for lithium-ion intercalation. Also mechanical strain during de-/intercalation can contribute to LAM as particles from both electrodes can crack and get electronically separated from the bulk material.
For investigating or describing the behavior of lithium-ion cells, different model categories can be implemented and those can be classified into first-principle, electrochemical engineered and empirical models. 19,20 Atomistic models based molecular dynamics (MD) 21,22 and density functional theory (DFT) [23][24][25] try to recreate molecular behavior on an atomic scale. As they use fundamental physics-based approaches for atom-atom interactions, these models are also called first-principle models. 21 This category of models becomes of greater importance in future research efforts for better understanding interfacial chemistry as it can predict species in the interphases that might be hidden or changed due to poor empirical characterization. The drawback of first-principle models is that they cannot properly handle cycling of intercalation electrodes as the consideration of the bulk structure is necessary which cannot be represented with a traditional surface thermodynamics approach. 26,27 MD and DFT consider clean surfaces and influences of close subsurface layers, so they do not consider structural changes in the electrodes during cycling.
A step closer to modeling complete cell behavior are electrochemical engineered models that are often also known as physicochemical models. Within this class, surface and molecule processes are modeled in a phenomenological manner but the particle and electrode domain are described mechanistically. Based on electrochemical kinetics and transport equations they can simulate cell characteristics and intercalation as well as side reactions. 19,20 The best-known electrochemistrybased models are the pseudo two-dimensional (P2D) model developed by Newman and co-workers [28][29][30] and the single particle model (SPM) first introduced by Zhang et al. 31 The often proved accuracy and agreement with experimental data of the P2D model originate from its basic implementation of porous electrode theory as well as concentrated solution theory. 28,32 Up to today, the P2D model represents the most precise and -though computationally costly -most popular model in lithium-ion battery research. 28 The SPM represents a simplification of the P2D model in order to decrease computational time. As the spatial representation of the liquid phase are neglected and transport phenomena are just considered in one single representative particle, the SPM lacks the accuracy of the P2D model but still shows good agreement with experimental data. 20,31,33 Empirical models are based on implementing behavioral trends from past experiments and predicting future states such as stateof-charge (SOC) and state-of-health (SOH) from there. The best known models of that category are equivalent circuit models and neural network models. 20,34 As they are relatively simple to implement and computationally fast, empirical models are frequently found in literature. [34][35][36][37][38][39][40][41][42] However, their application is limited as they can only describe a previously seen and implemented behavior, so an adaption to another cell or even chemistry needs a completely new database. 19,20 Previous literature described several degradation mechanisms on anode as well as cathode in a P2D model. Ashwin et al. 43,44 investigated the porosity change in the negative electrode due to SEI growth under different cycle and temperature conditions. Fu et al. 45 ascribed capacity fade to SEI growth as well as active material degradation and found an extra deposit layer on the anode near the separator. Lawder et al. 46 studied the influence of different driving cycle profiles on the capacity fade of electric vehicle batteries and ascribed the total capacity fade to SEI growth. The effects of gas evolution due to SEI growth were modeled by Rashid et al. 47 On the cathode side, Cai et al. 48 implemented an SOC independent manganese disproportionation which increased the cathodic resistance and lead to a change both in porosity as well as particle radius. A combination of SEI growth and cathode dissolution in a lithium-cobalt-oxide (LCO) cell was shown by Lam et al. 49 and optimal discharge parameters were derived. Another very extensive model that included manganese dissolution from a lithium-manganese-oxide cathode and the effects of manganese-ions incorporated into the anodic SEI was presented by Lin et al. 50 In this paper we introduce a P2D model for a common NCM/graphite cell with SEI growth as the dominating capacity fade mechanism on the graphite anode and active material dissolution as the main aging mechanism on the cathode. The SEI implementation considers a growth due to imperfections in its insulating properties as well as new SEI formation due to cracking of the layer during graphite expansion when cycling the cell. The novelty of our approach is that we include two separate conductivities within the SEI for lithium-ions (κ SEI ) and electrons (σ SEI ) leading to distinct overpotentials driving the main and side reaction. Simulation data is compared to experimental studies on NCM/graphite cells performed by Ecker et al. 51

Model Development
To analyze the behavior of the most important aging mechanisms and their impact on capacity fade, we implemented a P2D physicochemical model for a common NCM/graphite cell using COMSOL Multiphysics 5.2a. As the basic equations of the P2D model have been extensively shown in literature, [28][29][30] a brief overview of the model and all used parameters (see Table AI) are given in the Appendix. The basic assumptions of the implemented aging mechanisms in the presented model are introduced and discussed subsequently.
Implementation of SEI growth.-For the implementation of SEI growth we introduce a new approach that distinguishes between the transport of two species through the SEI -lithium-ions on the one hand and electrons on the other hand (refer to Figure 1a). This is in accordance with the assumption that the SEI possesses two ideal properties -a maximum conductivity for lithium-ions and an insulating conductivity for electrons. 13,52 We are aware that literature [53][54][55] still debates whether new SEI is formed at the SEI/electrolyte or the graphite/SEI interface. With our approach, we assume that new SEI is formed at the SEI/electrolyte interface. In the case of an SEI formation at the graphite/SEI interface solvent particles would need to be the second species migrating through the SEI besides lithium-ions. As our P2D model treats the SEI as an interface phenomenon influencing charge-transfer, both cases would lead to the same cell behavior, so we stick to electron migration through the SEI for SEI formation.
Our new approach results in a different ohmic drop (i R) for the driving overpotential of the main intercalation reaction at the negative electrode η neg and the SEI forming side reaction η SEI .
Both resistances R neg and R SEI are dependent on the SEI's initial thickness δ 0,SEI , the thickness increase δ SEI and the respective conductivity. The initial thickness is assumed to be 20 nm which is considered a fully formed SEI 56 and the thickness increase is due to the non-ideal insulating properties as well as an SEI re-formation after cracking.
R SEI = δ 0,SEI + δ SEI σ SEI [3] As no measurements of the SEI's electronic conductivity σ SEI are known, 56 we assume σ SEI to be 10 −8 S m −1 which is considered an insulating behavior. 57 In contrast, the ionic conductivity κ SEI is presumed to be 10 −2 S m −1 which is approximately the conductivity of a liquid lithium-ion battery electrolyte. 57 With the introduced overpotentials we implemented a growth due to imperfections in the SEI's insulating properties by Butler-Volmer kinetics with an anodic charge-transfer coefficient α a,SEI = 0.05 and a cathodic charge-transfer coefficient α c,SEI = 0.95. This implementation is close to the often used cathodic Tafel expression 8,15 but considers also dissolution reactions during cycling. 58 The index n in i SEI,n symbolizes LiF and Li 2 CO 3 as we included the two most important SEI products in the model. 59,60 F, R and T represent Faraday's constant, the universal gas constant and the absolute temperature, respectively. Specific values for i 0,SEI will be given in the Results and discussion section by Equation 11.
Additionally, we implemented an SEI re-formation after cracking due to graphite expansion during intercalation of lithium-ions. 61,62 As we only assume a new formation without dissolution by cracking, the anodic part in the Butler-Volmer equation is omitted and the overpotential η crack considers no i R-drop.
The current density computation considers a cracking function dependent on intercalation degree x that is depicted in Figure 2 which is the gradient of a graphite expansion curve as previously introduced by Laresgoiti et al. 62 Furthermore, we included an empirical factor ineg i C/100 to scale the cracking for utilization at different intercalation current densities. For simplicity, we assume that only Li 2 CO 3 is formed in the cracks. The overall increase in SEI thickness δ SEI after integrating the respective current densities is calculated with the molar masses M i and densities ρ i by The lithium-ions consumed in the three SEI forming charge quantities are subtracted from the total amount of cyclable lithium-ions and represent the LLI in the model.
We assume that the known aging behavior of a lithium-ion battery cannot be represented completely by a mere implementation of SEI growth. In literature, models with SEI growth as their only capacity fade mechanism do not show the typical non-linear aging behavior -i.e. the sudden decrease -in usable capacity after several hundred cycles. 53,55,59,61,63 In these models, this non-linear aging behavior can be emulated by a high power fade, though, which shortens charging and discharging due to high overpotentials that decrease the usable capacity. 43,61 Measurements in literature ascribe this non-linear aging behavior to lithium plating 64,65 as well as to degradation mechanisms on the cathode. 5,48,50 For the here introduced model we chose to implement a cathode dissolution reaction as the responsible mechanism for the non-linear aging behavior. As we lack any information on that topic from the chosen experimental data, including a mechanism on the positive electrode seems sensible in regard of the possible interactions between the two mechanisms (SEI growth and lithium plating) at the negative electrode.
We are aware that dissolved transition-metal ions from the cathode have been reported to be incorporated in the anodic SEI and have altering effects on its properties. [66][67][68] Including those effects will be part of future investigations, as they are not crucial for the general improvement of the introduced SEI model representation by two conductivities.
Cathode dissolution reaction.-The implemented dissolution reaction, is reducing the solid phase volume fraction ε s, pos depending on the cathode's intercalation degree. Hence, the reduction of ε s, pos corresponds to the LAM in our model. Acid attack by HF is one of the dominating causes for active material dissolution at the cathode 50,69,70 and is implemented as an irreversible kinetics expression in the positive electrode domain. As HF evolution is promoted at potentials above 4.0 V, 69 this potential is used as the equilibrium potential E Eq,diss . The volume fraction of the cathode active material is continuously calculated by the integrated dissolution current density Q diss of HF dissolving transition-metals from the active material. ε s, pos = ε s, pos,0 − Q diss c s,max, pos · l pos · F [10] In conclusion to the introduction of the side reaction modeling approach, an overview of all parameters for the previously shown side reactions can be found in Table I.

Results and Discussion
Determination of SEI formation exchange current density.-As mentioned before, we used the experimental data for a NCM/graphite cell from Ecker et al. 51 to test our model and fit relevant parameters. By simulating our model in a calendar aging mode (i.e. no applied external current density), we are able to determine the exchange current density of SEI formation. Electrons for forming the SEI are provided by the anode in calendar as well as cyclic aging. Lithium-ions for the reaction are taken from the electrolyte but to keep the charge balance valid in calender aging, lithium-ions also have to deintercalate from the negative electrode whereas during cyclic aging those lithium-ions are deintercalated from the positive electrode. With the provided data for 35, 40 and 50 • C at 50 % SOC, we determined the SEI formation exchange current density i 0,SEI depending on temperature T in an Arrhenius-like behavior as we expect a negligible influence of the cathodic dissolution reaction at this SOC.
The exchange current density for SEI formation calculates to 3.6, 6.1 and 17.1 × 10 −10 A m −2 for 35, 40 and 50 • C respectively and is in agreement with an exchange current density smaller than 1 × 10 −7 A m −2 as proposed by Fu et al. 45 The agreement of experimental and simulation data can be seen in Figure 3.
Quantitatively LiF and Li 2 CO 3 are formed at the same rate in the calendric regime of the introduced model. This is to be expected as the same reduction potential and an overall side reaction exchange current density i 0,SEI is assumed. The incorporation of the two main degradation products is still advisable as it influences the thickness prediction of the SEI by the different molar volumes of LiF and Li 2 CO 3 (V m,LiF = 9.8 × 10 −6 m 3 mol −1 ; V m,Li 2 CO 3 = 3.5 × 10 −5 m 3 mol −1 ).  current-constant voltage (CC-CV) charge), we observe a higher capacity fade due to SEI growth than during calendar aging. One might expect that this increase in capacity fade is solely due to the cracking and re-formation of the SEI which is not occurring during calendar aging. However, as also shown by Purewal et al. 72 the increase in SEI growth is mainly due to the differing overpotentials during cycling and the cracking of the SEI accounts for only a small amount of the total SEI formed.

Capacity fade based on SEI growth during cyclic and calendar
The overall SEI growth close to the separator and close to the current collector as well as the overall capacity fade as shown in Figure 4 follows a √ t-behavior. In contrast to Lin et al., 50 this behavior is not modeled by an exponential decay pre-factor limiting the exchange current density but is based on a different utilized range in the i SEI,n -η SEI -curve determining the kinetics of SEI growth. As the kinetics dependency has an exponential shape and the overpotential changes due to the increasing R SEI , the SEI formation current decreases until a state is reached where resistance increase and current density decrease keep the overpotential effectively steady. Due to that quasi-steady state, SEI formation never stops for reasons of the SEI being insulating enough but changes to a linear growth behavior.  Non-linear aging behavior due to cathode dissolution.-As the SEI is not stopping to grow due to kinetic limitations as discussed in the previous section, another effect has to serve as a limiting condition. Our simulations show that the "stabilization" of SEI growth is influenced by the degradation of the positive electrode. This fact -which seems contradictory to what one would expect -is caused by a straightforward circumstance. As cathode degradation outpaces the amount of lost cyclable lithium-ions contributing to SEI growth, less and less lithium-ions are moved from the anode to the cathode during discharge. 5,50 This effect leads to shorter charging times and, therefore, shorter times during which SEI can grow which results in a decrease of SEI growth in each cycle 75 (see Figure 5). The same would hold true with lithium plating as a source of LLI and the consequent decrease of cyclable lithium-ions. Figure 5 compares the decrease of SEI growth over 1000 cycles close to the current collector with and without an implemented cathode dissolution reaction. The difference in SEI thickness after 1000 cycles is about 10 nm. This thickness difference seems to be very small in comparison to the difference in the corresponding overall capacity fade as depicted in Figure 6 (blue and red line). The reason for the behavior of the model with cathode dissolution is a prolonged Figure 6. Comparison of experimental data taken from Ecker et al. 51 and data from the proposed model for cyclic aging as a result of SEI formation and cathode dissolution. The light blue color covers the range of the three measurements by Ecker et al. 51 Additionally, the red line shows the capacity fade behavior of the model when disabling the cathode dissolution reaction.  CV phase during charging due to the side reactions. The CV phase keeps the SEI reaction below the reduction potential and the cathode dissolution reaction above the oxidation potential, so the current does not drop below the stopping criterion of C/20 due to the side reactions. Therefore, the longer CV phase counterbalances the shorter charging time for the intercalation reaction and does not limit the SEI growth as much as expected. Figure 6 compares the simulated non-linear behavior in usable capacity with experimental results by Ecker et al. 51 The non-linearity in usable capacity occurs as soon as the LAM in the cathode becomes larger than the LLI. Whereas the decrease in the beginning of the capacity fade and the position of the transition zone from linear to non-linear aging behavior are in good agreement, the slope after the transition zone is underestimated by the model. This could be caused by the exclusion of implementing lithium plating as a second source of LLI and will, therefore, be a task for future work. Figure 7 depicts consequences of the capacity fade on the shift within the stoichiometry -which is the intercalation degree -at the end-of-charge (EOC) and end-of-discharge (EOD). Besides the initial conditions of a non-aged cell, the values of an aged cell before and after the transition to non-linear behavior in the capacity curve of Figure 6 are shown. As expected, the stoichiometry of the anode at the EOC decreases due to LLI (shift from blue to red in Figure 7b). In contrast, the stoichiometry of the cathode at the EOC stays (almost) the same as the anode stays in a stage-1 potential plateau and the EOC is defined by the cutoff-voltage of the cell at 4.2 V which is the difference between anode and cathode potential.
At the EOD, we see that the cathode stoichiometry increases (shift from blue to red in Figure 7a) as the LAM is higher than the LLI and percentagewise more lithium-ions intercalate in a smaller cathode active material volume. When the cathode stoichiometry at EOD is reaching 1, the anode stoichiometry also increases as the discharge is terminated before all lithium-ions are deintercalated from the anode. Therefore, at this point we see a change from an anode limitation to a cathode limitation of the cell. Those shifts and the half-cell behavior are also in good agreement with measurements and conclusions reported by Kleiner et al. 76 for an NCA/graphite cell. Figure 8a is the voltage discharge curve of the simulated cell prior and after aging at 100 % SOC and 50 • C compared to data reported in the paper of Ecker et al. 51 As can be seen, the cell shows a capacity fade -recognizable by shorter discharge time -but no significant power fade as voltage levels are almost equal. Figure 8b shows the exemplary behavior of a cell with a uniform conductivity κ SEI of 1 × 10 −7 S m −1 for lithium-ions and electrons within the SEI in the order of often used values in literature. 47,77 The plot shows that we get a totally different power behavior as a result and, therefore, prediction of available energy with a model that does not distinguish between the conductivity of electrons and lithium-ions in the SEI -although we calculate the same capacity fade.

Capacity and power fade behavior with new model.-Depicted in
With our modeling approach we are able to differentiate between capacity fade and power fade, both resulting in an energy loss during aging. We thereby get the possibility to gain new insights into SEI properties for different cell systems and material combinations in future work.

Conclusions
In this paper we introduced a new approach for modeling aging behavior that distinguishes between electronic (σ SEI ) and ionic (κ SEI ) conductivity of the SEI. By this approach we do not only represent the SEI in a way that is more accurate but we can also differentiate between capacity and power fade which is inextricably connected in a single conductivity approach.
The model shows good agreement with experimental data from Ecker et al. 51 as not only an SEI growth due to non-ideal insulation properties and re-formation after cyclic cracking but also a cathode dissolution reaction is implemented. With this cathodic aging mechanism, the transition to non-linear behavior in retrievable capacity can be explained.
Future work will add further aging mechanisms on both electrodes, like e.g. lithium plating and a cathode electrolyte interphase (CEI) formation, to the existing model to get a more thorough understanding of the interactions between the different mechanisms.
and charge balance throughout the electrode domain. The current within the liquid phase is described by the current density i l and potential l , while the pore wall flux at the electrode-electrolyte interface is named j n . R describes the universal gas constant, F the Faraday's constant and T the local absolute temperature. Within the separator domain the equations simplify to and To couple solid and liquid phase, Butler-Volmer kinetics are assumed for the pore wall flux including the lithium-ion concentration at the particle's surface c s and the overpotential where s corresponds to the solid phase potential. Effective transport parameters are used to account for tortuosity in the homogenized P2D model by scaling material parameters with MacMullin's number -a function of Transport number t + 0.38 79 porosity ε l and tortuosity τ 78 To describe the electrolyte's characteristics properly, a concentration dependence is implemented for conductivity, diffusivity and mean molar activity coefficient of the electrolyte. These are taken from fittings to measurements 79 while presuming a constant transport number. The applied diffusion coefficients are estimated from various literature sources. 29,30,80,81 The equilibrium potential is taken from literature 81 as well as the maximum concentration of lithium within active material particles. 71,81 Additional parameters such as reaction rate constants 80,81 are assumed based on references from literature.
The chosen parameters for the above introduced model -measured or taken from literature -are summarized in Table AI.