# A four parameter model for the solid-electrolyte interphase to predict
battery aging during operation

Lars von Kolzenberg These authors contributed equally to this work German
Aerospace Center, Pfaffenwaldring 38-40, 70569 Stuttgart, Germany  Helmholtz
Institute Ulm, Helmholtzstraße 11, 89081 Ulm, Germany  Jochen Stadler = Ulm
University, Albert-Einstein-Allee 47, 89081 Ulm, Germany  Mercedes-Benz AG,
Mercedesstr. 120, 70372 Stuttgart, Germany  Johannes Fath  Mercedes-Benz AG,
Mercedesstr. 120, 70372 Stuttgart, Germany  Madeleine Ecker  Mercedes-Benz AG,
Mercedesstr. 120, 70372 Stuttgart, Germany  Birger Horstmann  German Aerospace
Center, Pfaffenwaldring 38-40, 70569 Stuttgart, Germany  Helmholtz Institute
Ulm, Helmholtzstraße 11, 89081 Ulm, Germany  Ulm University, Albert-Einstein-
Allee 47, 89081 Ulm, Germany  Arnulf Latz Corresponding Author:
<EMAIL_ADDRESS>German Aerospace Center, Pfaffenwaldring 38-40, 70569
Stuttgart, Germany  Helmholtz Institute Ulm, Helmholtzstraße 11, 89081 Ulm,
Germany  Ulm University, Albert-Einstein-Allee 47, 89081 Ulm, Germany

###### Abstract

Accurately predicting aging of lithium-ion batteries would help to prolong
their lifespan, but remains a challenge owing to the complexity and
interrelation of different aging mechanisms. As a result, aging prediction
often relies on empirical or data-driven approaches, which obtain their
performance from analyzing large datasets. However, these datasets are
expensive to generate and the models are agnostic of the underlying physics
and thus difficult to extrapolate to new conditions. In this article, a
physical model is used to predict capacity fade caused by solid-electrolyte
interphase (SEI) growth in 62 automotive cells, aged with 28 different
protocols. Three protocols parametrize the time, current and temperature
dependence of the model, the state of charge dependence results from the
anode’s open circuit voltage curve. The model validation with the remaining 25
protocols shows a high predictivity with a root-mean squared error of
$1.28\%$. A case study with the so-validated model shows that the operating
window, i.e. maximum and minimum state of charge, has the largest impact on
SEI growth, while the influence of the applied current is almost negligible.
Thereby the presented model is a promising approach to better understand,
quantify and predict aging of lithium-ion batteries.

##  1 Introduction

Lithium-ion batteries are the current benchmark technology for mobile energy
storage because they combine high energy density and longevity. Nevertheless,
different aging phenomena continuously decrease the usable capacity and limit
the battery’s lifetime. This is a major challenge for battery electric
vehicles (BEV), which require a battery lifetime of about 10 years. To ensure
a certain remaining capacity in this timespan, a detailed understanding and
quantification of the capacity fade in lithium-ion batteries is imperative.

Two main approaches exist to predict capacity fade in lithium-ion batteries.
On the one hand, (semi-)empirical approaches based on simplified physical
equations or data driven methods generate a precise aging quantification from
large datasets 1, 2, 3, 4, 5, 6. However, obtaining the necessary datasets is
costly and the resulting models can not easily be extrapolated or adapted to
new cell chemistries. On the other hand, physics-based approaches model the
underlying aging phenomena and infer equations to describe aging based on
operating conditions 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17.

Most physical battery aging models rely on resolving solid-electrolyte
interphase (SEI) growth as the major cause of continuous capacity fade 18, 19,
20, 21. The SEI is a thin layer on the anode, which emerges in the initial
battery cycle from electrochemical reactions of electrolyte molecules,
electrons and lithium ions 22, 23, 24, 25, 26, 27, 28, 29. In subsequent
cycles, the SEI shields electrolyte molecules from electrons, but continues to
grow due to a leak current through the SEI.

During storage, the square-root-of-time ($\sqrt{t}$) dependent capacity fade
points to a self-limiting kinetic behind this continuous growth process 7, 8,
10, 15, 30. Diffusion of localized electrons, e.g. as neutral lithium
interstitial atoms 31, 32, emerged as most likely growth limiting mechanism as
it best captures the experimentally observed voltage dependence 33, 34, 15.

During operation, recent experiments of Attia et al. 35 reveal that SEI growth
accelerates with increasing charging current and decreases with increasing
discharging current. We orient us on these findings in our recent work 17 and
extend the model of Single et al. 15 for the effect of battery operation. The
resulting model shows good accordance with both, storage and operation
capacity fade measurements.

In this paper, we use the long-term limit of our model from ref. 17 to
describe capacity fade of 62 automotive grade pouch cells aged with 28
different protocols 36. Using differential voltage analysis (DVA), we first
separate capacity fade into the three aging modes loss of active material on
positive and negative electrode and loss of active lithium (LL), which we
account to SEI growth 37, 38. Then, three protocols and the anode open circuit
voltage (OCV) curve parametrize the time, current, temperature and state of
charge (SoC) dependence of the model, while the remaining 25 protocols
validate our model predictions. Finally, post-mortem experiments with
experimental coin cells built from harvested electrode materials provide
further complementary validation for the model’s assumptions.

We present the experiments and the implemented model detailedly in the
following Sections 2 and 3. In Section 4, we present the experimental and
theoretical results. Finally, Section 5 summarizes the main findings of the
work and shows future applications and extensions.

##  2 Experimental

In the following section, we outline our experimental methods. Subsection 2.1
concisely presents the cell specifications and the test matrix. Figure 1 shows
our methodology to separate capacity fade into different aging modes and
isolate the influence of LL. During operation, we determine LL with DVA,
presented in Subsection 2.2. Post mortem, we analyse heterogeneities using DVA
as described in Subsection 2.3.

###  2.1 Aging test

The aging test comprises 62 automotive grade lithium ion pouch cells with a
nominal capacity of $43\text{\,}\mathrm{A}\text{\,}\mathrm{h}$ and nominal
voltage of $3.63\text{\,}\mathrm{V}$. The cells contain a graphite anode and a
blend cathode consisting of $\ce{Li(Ni_{0.6}Mn_{0.2}Co_{0.2})O_{2}}$ and
$\ce{Li(Ni_{1}/3Mn_{1}/3Co_{1}/3)O_{2}}$. The aging procedure is detailedly
described in ref. 36 and the aging conditions are listed in Table SI-1.

In short, the cycling protocol consists of a charging sequence with a constant
power $\text{P}_{\text{CH}}$ and subsequent constant voltage until reaching
$\text{SoC}_{\text{max}}$. Afterwards, a dynamic EV profile discharges the
cell to $\text{SoC}_{\text{min}}$ with maximum currents of up to
$200\text{\,}\mathrm{A}$ , followed by repetition of charge sustaining hybrid-
EV profiles until the specified $\text{EV}_{\text{ratio}}$, which is the ratio
of energy throughput stemming from EV and hybrid-EV driving, is met.Real
driving profiles are the basis for EV and hybrid-EV profiles which roughly
compare to the widely used WLTP profiles. To reveal the influence of these
factors on aging, we use a central composite design of experiments. This
approach varies the influencing factors around a common center point at
$T=$30\text{\,}\mathrm{\SIUnitSymbolCelsius}$$,
$\text{SoC}_{\text{max}}=90\%$, $\text{SoC}_{\text{min}}=28\%$,
$\text{EV}_{\text{ratio}}=60\%$ and
$\text{P}_{\text{CH}}=$136\text{\,}\mathrm{W}$$. Additionally we perform one
calendar aging test at $T=$30\text{\,}\mathrm{\SIUnitSymbolCelsius}$$ and
$\text{SoC}_{\text{max}}=48\%$. Twofold measurement of each aging protocol and
eightfold measurement of the center point results in 28 different conditions
of the 62 cells, detailedly described in ref. 36 and listed in Table SI-1.

To track the capacity fade we conduct reference parameter tests (RPT) at
$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$ after every two weeks of cycling.
These tests consist of three 1C/1C cycles and one 1C/0.1C cycle between
$4.2\text{\,}\mathrm{V}$ and $2.5\text{\,}\mathrm{V}$ with CCCV charging with
a cut-off at 0.05C and CC discharging. Additionally we measure
$30\text{\,}\mathrm{s}$ discharge pulses with a current of
$200\text{\,}\mathrm{A}$ at $100\text{\,}\mathrm{\char 37\relax}$,
$75\text{\,}\mathrm{\char 37\relax}$, $50\text{\,}\mathrm{\char 37\relax}$ and
$25\text{\,}\mathrm{\char 37\relax}$ SoC. For this work we focus on the C/10
capacity fade as exemplarily shown in Figure 1a). In the context of the aging
test a C-rate of 1C corresponds to the current that charges the cell within
one hour based on a nominal capacity of
$43\text{\,}\mathrm{A}\text{\,}\mathrm{h}$. In our analysis in Subsection 4.4
we base the C-rate on the measured mean BoL capacity of
$45.89\text{\,}\mathrm{A}\text{\,}\mathrm{h}$. The measured capacity fade
consists of an irreversible part and a reversible part originating from
lateral flow of lithium into the anode overhang 39. To retrieve the bound
charge in the anode overhang and thus isolate the irreversible part, we remove
one cell for each aging condition from the aging test after about two years
and discharge to $0\%$ SoC. Subsequently we perform another RPT after two to
three months prior to post mortem analysis.

Figure 1: Evaluation of LL and homogeneity of lithium distribution. a) State
of health determined by C/10 capacity check. b) The algorithm simulates aged
dU/dQ measurements by shifting the electrode balancing as a result of LL and
stinting the electrode potentials caused by loss of active material. The
figure shows $10\%$ LL and $10\%$ active material loss on cathode side (AMLp).
c) A superposition of simulated dU/dQ curves with a linear distribution of LL
can reproduce the peak broadening in aged dU/dQ curves caused by inhomogeneous
LL. d) Capacity fade due to lithium loss (crosses) and fitted distribution
indicating inhomogeneities (orange area). d-g) During post mortem analysis
experimental cells (f) are built with coins from harvested electrodes to
determine local capacity loss within a layer (e). C/100 discharge curves are
further used for DVA analysis (g) to validate local LL after the end of test
(d).

###  2.2 Isolating active lithium loss

We quantify the contribution of active material and active lithium loss on the
overall capacity fade with the DVA fitting algorithm described by Fath et al.
40 based on the C/10 discharge. The algorithm relies on changes in the cell’s
voltage curve $U_{\text{0,cell}}=U_{\text{0,cath}}-U_{\text{0,an}}$ relative
to the begin of life (BoL), see Figure 1b). Loss of active material on the
anode and cathode side stints the electrode potential curves
$U_{\text{0,cath}}$ and $U_{\text{0,an}}$, respectively. In contrast, loss of
lithium inventory, e.g. stemming from SEI growth or lithium plating, shifts
the anode’s vs. the cathode’s potential curve 37, 38, 40.

The algorithm varies the BoL potential curves accordingly to fit the aged
$\text{d}U/\text{d}Q$ curves and consists of two parts exemplarily shown in
Figure 1c). First, the algorithm varies the LL and the loss of active material
on anode and cathode side until the peak positions match, see I in Figure 1c).
Second, the algorithm fits the peak broadening observed in II-III in Figure
1c) as inhomogeneous aging by dividing the cell into seven parallelly
connected cell segments. Each segment shows a particular LL according to a
linear distribution around the previously determined mean LL 41, 42. From this
distribution, the overall DVA results as weighted sum of the cell segments
inverse $(\text{d}U/\text{d}Q)^{-1}$ 42, 40, 43. Both steps are repeated until
the root mean square error (RMSE) between fitted and measured
$\text{d}U/\text{d}Q$ curve is minimal. Minimum and maximum of the fitted LL
then indicate the homogeneity of lithium distribution inside the cell.

Fitting all available C/10 curves, we obtain a distribution of lithium loss
(LL) across the cell throughout the entire course of aging, shown in Figure 1
d). Here, the orange crosses indicate the capacity fade resulting from LL and
the orange area shows the inhomogeneity of lithium loss based on the
previously fitted distribution. As we observe significant inhomogeneity after
around two years, we also conducted post mortem experiments to validate our
assumptions.

###  2.3 Post mortem analysis

As first step of our post mortem analysis, we uniformly discharge the cells
with C/3 CCCV and C/50 cutoff to $3\text{\,}\mathrm{V}$. Then, we open the
cells in a glovebox with argon atmosphere and $\ce{H_{2}O}$ and $\ce{O_{2}}$
levels below $0.1\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ following the
procedure described by Sieg et al. 42. To remove residues of conducting salt,
we wash anode and cathode layers from the middle of the stack in a bath of
dimethyl carbonate (DMC) for at least three minutes. Afterwards, we remove
active material from one side of the electrode and cut coins with
$18\text{\,}\mathrm{mm}$ diameter (see Fig 1 d). From coins that are opposing
in the original pouch cell configuration, we assemble experimental cells using
the PAT-Cell Setup from EL-Cell with the FS-5P core cells according to Figure
1 e). The cells contain $95\text{\,}\mathrm{\SIUnitSymbolMicro l}$ of a DMC/EC
electrolyte, a $220\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick double
layered PE-fibre / PP-membrane separator, a metallic lithium reference
electrode, and aluminum and copper plungers as current collectors. Lastly,
each cell undergoes a formation procedure with CC cycles between
$4.2\text{\,}\mathrm{V}$ and $2.5\text{\,}\mathrm{V}$ consisting of 2x $\pm$
$0.7\text{\,}\mathrm{mA}$, 2x $\pm$ $1.4\text{\,}\mathrm{mA}$, 2x $\pm$
$2.3\text{\,}\mathrm{mA}$, 1x $\pm$ $0.7\text{\,}\mathrm{mA}$.

We electrochemically test the so-formed cells at
$25\text{\,}\mathrm{\SIUnitSymbolCelsius}$ with a BaSyTec CTS system.
Discharge capacity and potential curves result from charging three times to
4.2V CCCV with $5\text{\,}\mathrm{mA}$ and $0.07\text{\,}\mathrm{mA}$ cutoff
and subsequently CC discharging with $0.07\text{\,}\mathrm{mA}$ to
$2.5\text{\,}\mathrm{V}$, corresponding to a C-rate of $\text{C}/100$ at BoL.

##  3 Theory

In this subsection, we present our mathematical model to describe capacity
fade during battery operation caused by SEI growth. We start with a half-cell
cycling model followed by an SEI growth model. Concludingly, we summarize the
implemented system of equations.

###  3.1 Half-Cell Cycling

The applied intercalation current $J_{\text{int}}$ changes the half-cell
charge $Q_{\text{int}}$ over time according to differential Equation 1

$\frac{\text{d}Q_{\text{int}}}{\text{d}t}=-J_{\text{int}}.$ (1)

Here, we use the IUPAC convention and define the intercalation current as
negative and the deintercalation current as positive. We obtain the
overpotential $\eta_{\text{int}}$ from the intercalation current
$J_{\text{int}}$ based on a symmetric Butler-Volmer approach

$\eta_{\text{int}}=\frac{2RT}{F}\sinh^{-1}\left(\frac{J_{\text{int}}}{2J_{\text{int,0}}}\right)$
(2)

with Faraday’s constant $F$, the universal gas constant $R$ and the
temperature $T$ in Kelvin. The intercalation exchange current
$J_{\text{int,0}}=J_{\text{int,0,0}}\sqrt{\tilde{c}}$ depends on the reaction
rate $J_{\text{int,0,0}}$ and the SoC
$\tilde{c}=Q_{\text{int}}/Q_{\text{max}}$ with the maximum capacity
$Q_{\text{max}}$ 44. In the following we state equations to describe capacity
fade resulting from SEI growth.

###  3.2 SEI Growth

We rely on our model developed in ref. 17 to describe SEI growth during
battery operation. From the multitude of possible reactions, we assume that
the SEI predominantly forms from the reaction of lithium ions $\ce{Li}^{+}$,
electrons $\ce{e^{-}}$ and electrolyte molecules, e.g. ethylene carbonate
$\ce{EC}$, according to reaction 3

$2\ce{Li^{+}}+2\ce{e^{-}}+2\ce{EC}\rightarrow\ce{Li_{2}EDC}+\ce{R}.$ (3)

Here, $\ce{Li_{2}EDC}$ is the commonly observed SEI component lithium ethylene
dicarbonate and $\ce{R}$ is a gaseous residue. Assuming that lost lithium
inventory $Q_{\text{LL}}$ is completely bound in the SEI with thickness
$L_{\text{SEI}}$, we linearly link both quantities with

$Q_{\text{LL}}=\frac{\nu_{\text{SEI}}AF}{\bar{V}_{\text{SEI}}}L_{\text{SEI}}.$
(4)

Here, $\nu_{\text{SEI}}$ is the stoichiometric coefficient of lithium ions in
the SEI formation reaction 3, $\bar{V}_{\text{SEI}}$ is the mean molar volume
of the resulting SEI components and $A$ denotes the active electrode surface.
Over time, the SEI grows in thickness according to differential Equation 5

$\frac{\text{d}L_{\text{SEI}}}{\text{d}t}=-\frac{\bar{V}_{\text{SEI}}}{\nu_{\text{SEI}}F}j_{\text{SEI}}.$
(5)

The SEI formation current density $j_{\text{SEI}}$ results from the diffusion
of localized electrons, e.g. as lithium interstitial atoms
$\ce{Li^{0}}=\ce{Li^{+}}+\ce{e^{-}}$, from the electrode to the electrolyte
31, 15, 17. We describe this process with the long-term limit of our model
developed in ref. 17,

$j_{\text{SEI}}=-\frac{c_{\text{e}^{\text{-}},0}D_{\text{e}^{\text{-}}}F}{L_{\text{SEI}}}e^{-\frac{F\eta_{\text{SEI}}}{RT}}.$
(6)

Here, $c_{\text{e}^{\text{-}},0}$ and $D_{\text{e}^{\text{-}}}$ are the
equilibrium concentration and the diffusivity of lithium atoms inside the SEI.
The diffusivity $D_{\text{e}^{\text{-}}}$ depends on the temperature according
to an Arrhenius kinetic

$D_{\text{e}^{\text{-}}}=D_{{\text{e}^{\text{-}},0}}e^{-\frac{E_{\text{A}}}{RT}}$
(7)

with the reference diffusivity $D_{\text{e}^{\text{-}},0}$ and the activation
energy $E_{\text{A}}$. The overpotential $\eta_{\text{SEI}}$ for the formation
of these lithium atoms depends on the OCV of the anode $U_{0}$ (see Figure
SI-1), the intercalation overpotential $\eta_{\text{int}}$, and the reference
chemical potential of lithium atoms $\mu_{\text{e}^{\text{-}},0}$, which we
gauge to 0, according to

$\eta_{\text{SEI}}=U_{0}+\eta_{\text{int}}.$ (8)

###  3.3 Model Summary

In the following, we summarize our set of equations to determine active
lithium loss due to SEI growth. We start with the differential equation to
determine the SoC from the applied current profile

$\frac{\text{d}\tilde{c}}{\text{d}t}=-\frac{J_{\text{int}}(t)}{Q_{\text{max}}}.$
(9)

From the SoC $\tilde{c}$ and the current profile $J_{\text{int}}(t)$, we then
calculate the SEI formation overpotential

$\eta_{\text{SEI}}=U_{0}(\tilde{c})+\frac{2RT}{F}\sinh^{-1}\left(\frac{J_{\text{int}}}{2J_{\text{int,0,0}}\sqrt{\tilde{c}}}\right).$
(10)

Followingly, we determine the SEI thickness evolution with the semi-analytical
solution of Equation 5 and Equation 6, Equation 11

$L_{\text{SEI}}=\sqrt{2\frac{c_{\text{e}^{\text{-}},0}D_{\text{e}^{\text{-}},0}\bar{V}_{\text{SEI}}}{\nu_{\text{SEI}}}\int_{0}^{t}e^{-\frac{F}{RT}\left(\eta_{\text{SEI}}+\frac{E_{\text{A}}}{F}\right)}\text{d}t^{\prime}+L_{\text{SEI},0}^{2}}.$
(11)

Lastly, the active lithium loss LL results from the SEI thickness via

$\text{LL}=\frac{\nu_{\text{SEI}}AF}{Q_{\text{max}}\bar{V}_{\text{SEI}}}\left(L_{\text{SEI}}-L_{\text{SEI},0}\right).$
(12)

Overall, the model relies on four fitting parameters to describe SEI growth.
The diffusivity $D_{\text{e}^{\text{-}},0}$ and the initial SEI thickness
$L_{\text{SEI},0}$ quantify the calendaric SEI growth rate. The exchange
current density $J_{\text{int,0,0}}$ affects the resulting overpotential and
thereby describes the current dependence of SEI growth. Lastly, the activation
energy $E_{\text{A}}$ specifies the temperature dependence of SEI growth.
Implementing the model equations 9-11 as numerical integration with e.g.
MATLABs trapz yields results for yearlong cycling protocols within seconds.

##  4 Results and Validation

In this section, we present the experimental and theoretical results obtained
from the precedingly discussed experiments and model. We start by pointing out
the dominant influence of lithium loss on capacity fade which we analyze with
the DVA. Afterwards we compare the measured lithium loss of the test cells to
our theoretical predictions and thereby parametrize and validate our model.
Subsequently, a post mortem analysis shows the validity of our experimental
and theoretical assumptions. Concludingly, we show the sensitivity of our SEI
growth model to operating patterns and identify detrimental operating
conditions.

###  4.1 Identifying the cause of capacity fade

Using the DVA fitting algorithm as presented in Subsection 2.2 we quantify the
contribution of the aging modes loss of active lithium (LL) and active
material losses on anode side (AMLn) and on cathode side (AMLp) to overall
capacity loss.

Figure 2: a) Measured and fitted DVA with increasing testing time from bottom
to top. The peaks do not significantly shift which b) Measured capacity fade
QLoss in comparison to fitted loss of active lithium LL as well as active
material losses on anode side (AMLn) and cathode side (AMLp). The orange area
indicates inhomogeneity of lithium loss.

Figure 2a) exemplarily shows measured and fitted dU/dQ curves at different
times for the reference cell 39. In the measured curves we observe only slight
changes in the peak positions, pointing to negligible active material losses
38. In the beginning, the peaks become sharper, which indicates a more
homogeneous lithium distribution 42, possibly due to the anode overhang
effect45, 39. After about 300 days the peaks broaden again, pointing towards a
more heterogeneous lithium distribution due to heterogeneous aging, as we also
observe in our post mortem analysis conducted after ending of test, see
Section 4.3.

The results of the fitting algorithm in Figure 1b) support these observations
and further show that AMLp exists but is not limiting. The fitted LL shows
good accordance with the measured capacity loss except for a slight offset at
BoL, which we attribute to a slight mismatch in initial balancing due to
inhomogeneities.

We emphasize that overall capacity loss is not equal to the sum of LL, AMLp
and AMLn. Instead, capacity loss stems from limitations at the lower or upper
cutoff voltage, which in turn result from changes in electrode balancing due
to the different aging modes. For example, the aging simulation with AMLp and
LL in Figure 1b) shows, that LL shifts the steep anode potential and thus
limits the lower cutoff potential, while AMLp is not significantly limiting
the lower cutoff potential as it only slightly shifts the shallow cathode
potential.

We observe similar trends with LL being the dominant aging factor for all
cells. This allows us to focus our modeling approach on the main aging
mechanism SEI growth as root-cause of LL and thus to keep the model simple.
However, harsher aging conditions or different cell chemistries can induce
more severe active material losses, where our simple approach might fall
short.

###  4.2 Predicting lithium loss

####  4.2.1 Parametrization

As a first step, we use storage data to parametrize the initial SEI thickness
$L_{\text{SEI},0}$ and the diffusivity
$D_{\text{e}^{\text{-}}}(T=$30\text{\,}\mathrm{\SIUnitSymbolCelsius}$)$, see
Equation 7. In this case, we solve Equation 11 analytically, because the SEI
overpotential $\eta_{\text{SEI}}$ is constant in time. This leads to the well-
known $\sqrt{t}$-dependence of SEI growth during battery storage 15

$L_{\text{SEI}}=\sqrt{2\frac{c_{\text{e}^{\text{-}},0}D_{\text{e}^{\text{-}}}(T)\bar{V}_{\text{SEI}}}{\nu_{\text{SEI}}}e^{-\frac{F}{RT}U_{0}}t+L_{\text{SEI},0}^{2}}.$
(13)

Figure 3 compares the remaining active lithium inventory predicted by our
model with experimental data.

Figure 3: Comparison of experimental (orange crosses) and simulated (blue
line) capacity fade during storage at
$T=$30\text{\,}\mathrm{\SIUnitSymbolCelsius}$$ with $50\%$ SoC.

We see that the analytically derived $\sqrt{t}$ SEI growth law, Equation 13,
agrees excellently with the experimentally obtained capacity fade curves. In
the following, we thus rely on the so-determined values for $L_{\text{SEI},0}$
and $D_{\text{e}^{\text{-}}}(T=$30\text{\,}\mathrm{\SIUnitSymbolCelsius}$)$
and proceed to determine the two remaining fitting parameters.

Next, we parametrize the influence of charging current on SEI growth with the
exchange current $J_{\text{int},0,0}$. For parametrization, we choose cell 39
from the reference cells 39-47, which were cycled at the same temperature as
the storage cell and showed low variance among the eight fold measurement.

Figure 4: a) SEI growth simulation for the current profile of the reference
cell 39 (see Table SI-1), which was continuously repeated in the aging
protocol. The blue curve shows the SEI growth during the cycle based on an
initial SEI thickness $L_{\text{SEI},0}=$3.9\text{\,}\mathrm{nm}$$, the orange
curve shows the SoC. Comparison of experimental (orange crosses) and simulated
(blue line) fade of active lithium inventory for b) the reference cell 39
cycled with the protocol listed in Table SI-1 at
$T=$30\text{\,}\mathrm{\SIUnitSymbolCelsius}$$ and c) the high temperature
cell 19 which were cycled with a distinct current profile, see Table SI-1 at
$T=$50\text{\,}\mathrm{\SIUnitSymbolCelsius}$$. The orange areas indicate
heterogeneity within the cell as measured by DVA.

Before comparing our simulations with experiments, we illustrate the simulated
SoC profile and the resulting SEI growth in Figure 4 a). The orange line
indicates the SoC of the battery during one cycle profile, which was then
repeated several times. For the center cells, the SoC was mostly kept around
30% with a charging/discharging sequence to 90% in between. This protocol
largely influences the SEI growth, which is shown in the blue curve. We
observe that SEI growth proceeds slowly in the 30% SoC phases. During charging
to 90%, the SEI growth is fastest, followed by the storage at 90%. This
accelerated growth results from the influence of intercalation overpotential
$\eta_{\text{int}}$ and OCV $U_{0}$ on the SEI overpotential
$\eta_{\text{SEI}}$ according to Equation 8, which is in line with previous
capacity fade measurements 33, 34, 35.

Figure 4b) compares the average lithium loss in the center point cells with
the simulated capacity fade. The distribution of LL (orange area) exhibits a
hourglass shape. In the first 150 days lithium distribution becomes more
homogeneous, possibly due to the anode overhang effect. After about 300 days
lithium distribution becomes increasingly inhomogeneous due to heterogeneous
aging, as we also observe in our post mortem analysis conducted after ending
of test, see Section 4.3.

Overall, our mathematical SEI model agrees well with the experiments,
especially in the homogeneous area between 200 and 400 days. However, between
0 and 100 days, we observe a deviation of experiment and simulation, primarily
because the experimental active lithium inventory starts below 100%. We
attribute this offset to experimental uncertainties, as the BoL electrode
potentials that were used as basis for DVA fitting were measured well after
test start. Hence the BoL cell can stem from a different production batch and
thus show different initial lithium distribution resulting e.g. from the
reversible anode overhang effect 45, 39. The kinks in the simulation profile
result from the modular build-up of our simulation consisting of storage,
cycle-profile and RPT phases. As each of these phases shows a different aging
behaviour, we observe kinks between the profiles. After 800 days the active
lithium inventory remains nearly constant in our simulation, because the
battery is stored at a low SoC.

The intercalation reaction rate $J_{\text{int},0,0}$ is not exclusive to our
SEI model, but also a parameter in the commonly used Doyle-Fuller-Newman
battery model 46. This enables a comparison of our value
$j_{\text{int},0,0}=J_{\text{int},0,0}/A=$0.09\text{\,}\mathrm{A}\text{\,}{\mathrm{m}}^{-2}$$
with values found in literature. Sauer and coworkers 47, 48 derived values
ranging from $0.7\text{\,}\mathrm{A}\text{\,}{\mathrm{m}}^{-2}$ to
$7\text{\,}\mathrm{A}\text{\,}{\mathrm{m}}^{-2}$ from electrochemical
impedance spectroscopy (EIS). However, the determination of
$j_{\text{int},0,0}$ is ambiguous as the electrode active surface area $A$ is
not easily attainable. As a consequence, Ng et al. 49 determine the exchange
current of a $50\text{\,}\mathrm{A}\text{\,}\mathrm{h}$ cell with
galvanostatic intermittent titration technique (GITT) to be around
$10\text{\,}\mathrm{A}$, which is comparable to our value of
$J_{\text{int},0,0}=$21\text{\,}\mathrm{A}$$ for a
$46\text{\,}\mathrm{A}\text{\,}\mathrm{h}$ cell.

Building up on this parametrization, we next determine the dependence of SEI
growth on temperature by fitting the diffusivity
$D_{\text{e}^{\text{-}}}(T=$50\text{\,}\mathrm{\SIUnitSymbolCelsius}$)$ to the
high temperature cells cycled at
$T=$50\text{\,}\mathrm{\SIUnitSymbolCelsius}$$. The parameters
$D_{\text{e}^{\text{-}},0}$ as well as $E_{\text{A}}$ then result from the
Arrhenius Equation 7 and the diffusivity
$D_{\text{e}^{\text{-}}}(T=$30\text{\,}\mathrm{\SIUnitSymbolCelsius}$)$, which
we previously determined from storage data, see Figure 3. In Figure 4c), we
again observe the hourglass shape of heterogeneity with a minimum around
200-400 days. Compared to the previous cell, the heterogeneity is even more
pronounced in the long term and spans up to 10% at around 800 days. Our model
shows good accordance to the average experimental values, but deviates in the
long term. As we also observe the highest heterogeneity in this case, we
account the long term deviation between simulation and experiment to
additional thermal effects, which are not captured by our simple SEI growth
model. These effects could comprise thermal SEI degradation, lithium plating
or electrolyte dry out.

####  4.2.2 Validation

In this section, we validate our aging model with the remaining 25 protocols,
which were not used for parametrization.

Figure 5: Scatterplot of the simulated active lithium loss
$\text{LL}_{\text{sim}}$ vs. the experimentally measured active lithium loss
$\text{LL}_{\text{exp}}$

Figure 5 summarizes the validation results in a scatterplot, which compares
the simulated and the experimentally measured lithium loss. The complete
results of our aging simulations are listed in Figure SI-4. The points are
colored according to the operating temperature of the cells, which ranged from
$10\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to
$40\text{\,}\mathrm{\SIUnitSymbolCelsius}$. In general, we observe a good
accordance of simulation and experiments, which also reflects in the low root-
mean-square error of $\text{RMSE}=1.28\%$. This underlines the outstanding
global predicitivity of our model, which we obtained already after
parametrizing the model with only three aging protocols.

However, we observe a systematic deviation of our simulated results from the
experiments for temperatures below
$T=$20\text{\,}\mathrm{\SIUnitSymbolCelsius}$$ as well as high aging states
with more than 10% lithium loss. We attribute this deviation to increased
lithium plating at low temperatures as well as the large heterogeneity
observed for these cells. Also the mentioned offset of measured LL at
beginning of test amplifies the deviation. Followingly, we further validate
these assumptions with post mortem analysis.

###  4.3 Post Mortem Validation

In Figure 6, we exemplarily illustrate the model predictions for three cells
cycled with different protocols at different temperatures and compare them to
the measured local LL from experimental cells prepared as part of the post
mortem analysis.

Figure 6: a-c) Photographs of anode electrode layer taken from the stack and
respective local capacities of experimental cells in colored relation to BoL.
Photographs reveal a moderate aging over the entire electrode for Cell 39 and
severe aging and plating mainly focused to the center of the electrode for
Cells 37 and 51. d-f) Measured and simulated loss of lithium inventory for the
three exemplary cells. Orange indicates the experimental results consisting of
an average active lithium inventory (cross) and inhomogeneity (orange area).
The blue line shows the simulated loss of active lithium caused by SEI-growth.
Markers show the active lithium loss fitted to the local three electrode
cells.

Visually, we already suspect an accelerated degradation focused to the center
of the electrode from the anode photographs of the three cells in Figure 6 due
to the intensified blue coloring. Additionally cell 37 and 51 also exhibit
shiny metallic grey spots that we attribute to lithium plating. Experimental
cell measurement of particular anode areas allow to further evaluate
heterogeneous capacity fading. For this purpose, we separate the electrode
area into edge ($\square$), mid($\triangle$) and center ($\bigcirc$) as
indicated in Figure 1. Local LL of the respective areas are shown as points at
the end of life in the bottom part plots of Figure 6.

We observe that especially the center shows the highest LL and thus drives
inhomogeneity of lithium distribution for all three cells. In contrast, the
edge and mid areas age more homogeneously and our model nicely accords to this
experimental LL for the moderately aged cells shown in Figure 6 d) and e).
However, we see a large deviation to the fitted mean LL for the more harshly
aged cell 51, Figure 6 f). We attribute this model deviation to increased
aging due to lithium plating in this case, as is also evident from the
photograph in Figure 6 c). To further increase our model accuracy for these
cases, inhomogeneous SEI growth and lithium plating should be considered in
future works.

###  4.4 Identifying detrimental conditions

In this section, we conduct a sensitivity analysis to unravel particularly
detrimental operating patterns. Subsequently, we study the impact of
different, simplified real-life motivated user profiles on aging in a case
study.

####  4.4.1 Sensitivity analysis

Figure 7: a) Sensitivity analysis of the influence of current, SoC and
temperature on the SEI growth rate. The blue cross indicates the center point
$J_{\text{int}}=0$, $T=$35\text{\,}\mathrm{\SIUnitSymbolCelsius}$$,
$\text{SoC}=50\%$ around which we vary one factor at a time. Orange indicates
the influence of the SoC, yellow the influence of the charging/discharging
current and purple the influence of the temperature. The dashed yellow line
shows the normalized degradation rate
$\bar{J}_{\text{SEI}}=J_{\text{SEI}}\cdot
t_{\text{cycle}}/$1\text{\,}\mathrm{h}$$. b) Sensitivity analysis of the depth
of discharge influence (maximum SoC - minimum SoC) on the active lithium loss
after cycling with 1C for two years. The blue curve shows the active lithium
loss for DoD variation around a mean SoC=50% whereas the red curve shows for
the effect of DoD variation from a maximum SoC=100%.

Figure 7a) shows the influence of current $J_{\text{int}}$, SoC $\tilde{c}$
and temperature $T$ on aging compared to a central point, which is stored
($J_{\text{int}}=0$) with $\text{SoC}=50\%$ at
$T=$35\text{\,}\mathrm{\SIUnitSymbolCelsius}$$. We observe that the influence
of electrochemistry, i.e. applied current and SoC exceeds the influence of
temperature. In particular, we observe the highest SEI current
$J_{\text{SEI}}$ for charging at a rate of 1C. This influence arises from the
intercalation overpotential $\eta_{\text{int}}$, which increases the SEI
overpotential $\eta_{\text{SEI}}$ (Equation 8) and thereby accelerates SEI
growth. During discharging, in contrast, increasing the C-Rate has the adverse
effect and decelerates the SEI growth until it is nearly suppressed at
discharging with 1C. This is in good accordance with recent experiments of
Attia et al. 35, who showed this asymmetry in SEI growth between charging and
discharging and thus motivated our model extension in reference 17.

However, the shown influence of applied current on degradation is misleading,
because the applied current also affects the cycle time. To emphasize this
relationship, we also plot the normalized degradation rate
$\bar{J}_{\text{SEI}}=J_{\text{SEI}}\cdot
t_{\text{cycle}}/$1\text{\,}\mathrm{h}$/$ as dashed yellow line in Figure 7a).
This rate compares the mean cumulated SEI growth within one cycle. We observe
three main effects: First, the influence of charging current on degradation is
not as severe as we would expect solely from the yellow graph. Second, the
degradation per cycle diverges for $J_{\text{int}}\rightarrow 0$, because the
long cycle times increase the calendaric aging. Third, the opposing effects of
current and cycle time on the degradation rate lead to an optimum at around
C/3. To further analyse the complex interrelation of charging current and
cycle time, we conduct a case study with different charging profiles in
Subsection 4.4.2.

Another major contribution to SEI growth arises from the state of charge. We
observe in Figure 7a) that the SEI current closely follows the anode OCV-curve
(see Figure SI-1), in line with the experimental results of Keil et al. 33, 34
and the model of Single et al. 15. This dependence causes cells stored at 60%
SoC to age nearly four times as fast as cells stored at 50% SoC, while storage
at 80% leads to a similar capacity fade as storage at 60%.

This trend also reflects in the depth of discharge
$\text{DoD}=\text{SoC}_{\text{max}}-\text{SoC}_{\text{min}}$, which we analyze
in the following. Figure 7b) plots the remaining active lithium inventory
after cycling for two years with $\text{J}_{\text{int}}=\pm 1\text{C}$,
depending on the DoD. We simulate two different approaches; The blue curve
shows a DoD variation around a mean SoC=50%, while the orange curve varies the
DoD with a fixed maximum SoC=100%.

The blue curve shows that increasing the DoD around a mean SoC causes
accelerated aging up to a plateau of 91% active lithium inventory at 50% DoD.
This dependence results from the SoC-dependence of aging, see Figure 7a). In
the regime of 0-50% DoD, the battery experiences increasing share of SoCs
larger than 60%, which according to Figure 7a) increases aging. In the regime
of 50-100% this trend is balanced by the decreased aging for SoCs below 20%
leading to the observable plateau. In comparable experiments, Ecker et al. 50
and Hoog et al. 51 observed an approximately linear dependency of capacity
degradation with increasing DoD when cycling NMC/graphite cells around a mean
SoC of 50%. Contradictory Sarasketa-Zabala et al. 52 showed a more complex
dependency for LFP/graphite cells with the highest capacity fade between 10%
and 50% DoD and lower capacity fade at very high and very small DoD, again
while cycling around a mean SoC of 50%.

The orange curve shows that for cycling around a fixed maximum SoC of 100%,
increasing the DoD results in slower aging. Again, this trend originates from
the influence of SoC on aging, depicted in Figure 7a). A systematic DoD
variation from 100% maximum SoC is not reported as often in literature, but
results from Rechkemmer et al. 53 for $\ce{LiMn_{2}O}$ hint to accelerated
degradation at high DoD which is in contrast to our predictions based on anode
SoC. Also Laresgoiti et al. 54 derive an exponential acceleration of SEI-
growth with increasing DoD from their SEI cracking approach as the particles
undergo greater volume change. Benavente-Araoz et al. 55 on the other hand
observed a significantly higher degradation for cycling between 65-95% SoC
compared to 20-95 % which is qualitatively in line with the SoC dependence as
predicted in our model.

Summing up the depth of discharge dependence of aging predicted by our model
compared to literature 50, 51, 52, 53, 54, 55, we at most partially reproduce
the experimentally observed trends. Accordingly, the influence of DoD on aging
arises not only from a shift of the anode OCV, but also from additional
effects. For example, the DoD affects particle volume changes and thus may
cause SEI fracture or loss of electrical contact of individual particles. To
capture these effects and increase our model predictivity in this respect,
extending the present simplistic SEI model for mechanics seems promising 56,
57.

####  4.4.2 Case Study

We now apply our model to study battery aging in different, real-life
motivated battery user profiles. We consider three electric vehicle
enthusiasts: standard driver Stan (blue), range anxious Robert (red) and aging
optimizer Alfred (yellow), who travel $50\text{\,}\mathrm{km}$ to work back
and forth every working day. All three drive steadily at
$50\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$ and for simplicity
discharge their EV with a range of $500\text{\,}\mathrm{km}$ constantly at
C/10 for one hour at a constant temperature of
$20\text{\,}\mathrm{\SIUnitSymbolCelsius}$.

Figure 8: Aging results for different exemplary battery use cases at a
temperature $T=$20\text{\,}\mathrm{\SIUnitSymbolCelsius}$$. a) State of charge
over the course of a working week. During the working days, the battery is
discharged twofold with C/10 for 10% SoC in each case. Different colors
indicate different charging behaviours. Blue: Standard behaviour with charging
once the SoC goes beyond 40% SoC. Red: Range anxiety with charging once the
SoC is below 100%. Yellow: Aging optimization with charging once the SoC is
below 20% SoC and recharging up to 50% SoC. The standard charging protocol
charges with C/8, C/3 is colored brighter and 1C the brightest. b) Battery
aging after applying the user profile of a) continuously for three years.

Figure 8a) shows the SoC during each working week for the three drivers. Stan
starts his working week with a fully charged battery and charges at home once
the SoC is below $40\%$. Robert also starts his week with a fully charged
battery, but charges every time his battery falls below $100\%$. Alfred
carefully read the papers of Keil et al. 33, 34 and thus starts at $50\%$ and
recharges only if the SoC is below $20\%$. At the end of each working week,
all three recharge their EV to their initial SoC and don’t move their EVs over
the weekend. They all try different charging rates, starting from the standard
charging rate C/8 shown in the darkest color, over the brighter C/3, up to the
brightest 1C. In the course of one week, all three drive the same range and
thus also have the same charge throughput.

In Figure 8b), we show the loss of cyclable lithium after three years for each
driver. We clearly see that range anxious Robert’s EV shows the highest
capacity loss of $10.4\%$, closely followed by standard driver Stan with
$9.2\%$. In contrast, aging optimizer Alfred only loses $3.6\%$ of his EV’s
starting capacity. The different charging rates, colored in brighter colors,
hardly affect the battery aging and cause only an increase about $0.2\%$ to
$9.4\%$ from C/8 to 1C for standard driver Stan.

We best comprehend the different aging characteristics with the sensitivity
analysis of our model shown in Figure 7a). Range anxious Robert and standard
driver Stan operate their batteries mostly at high SoCs, contrary to aging
optimizer Alfred. Following the red curve in Figure 7a), we see that Robert
and Stan are thus mostly on the highest graphite stage, which exhibits nearly
the fourfold degradation rate compared to the second graphite stage on which
Alfred mostly operates his battery. This is the main cause of accelerated
aging that we see in Figure 8b).

Surprisingly, the charging current plays only a minor role, although according
to the yellow line in Figure 7a) it should have the largest impact on the
degradation rate. However, because the charging time shortens with increasing
current, this dependence is misleading as we also showed with the dashed
yellow line in Figure 7a). The presented user-profiles further enrich this
analysis and show that degradation depends rather on the charge throughput
than the applied charging rate. However, our simplistic model only considers
SEI growth as aging mechanism. For lithium plating we would expect a major
effect of fast charging rates on aging.

Summarizing the findings of our experimentally validated aging model, we
predict that battery aging depends mainly on the operating window, i.e.
maximum and minimum SoC. Charging current and temperature, in contrast, have
only a minor influence.

However, our predictions are based on a homogeneous SEI growth model with some
inherent shortcomings. First of all, DVA and post mortem analysis reveal
heterogeneous aging within the cells. Implementing our model in three-
dimensional thermal and electrochemical battery models will help to resolve
heterogeneous SEI growth 58, 59, 60. Furthermore, we observe excessive lithium
plating in the cell center, which results presumably from thermal hotspots 61.
Extending our model for a plating kinetic will not only help to predict these
heterogeneities, but also increase our model predictivity for low
temperatures. In the long-term, the locally strong aging will eventually cause
”sudden death” of specific cell areas due to electrolyte dry-out. Coupling our
SEI-growth model to the percolation model of Kupper et al. 62 is a promising
approach to capture this effect. Lastly, DVA reveals an OCV-curve change over
time, which will also affect the aging kinetics. Future works can rely on our
SEI growth model and refine it for these secondary aging modes to further
increase the model predictivity.

##  5 Conclusion

We applied an electrochemical SEI growth model 17 to predict active lithium
loss in 62 automotive grade batteries cycled with 28 different protocols 36.
The simplicity of our model allows us to to parametrize the state of charge,
time, current and temperature dependence of the model with only three
protocols and the open circuit voltage curve of the anode. The validation with
the remaining 25 protocols shows remarkable accordance of predicted and
measured active lithium loss with a global root-mean-squared error of 1.28%.
Thus, our methodology reduces the set of experiments to parametrize a global
aging prediction.

For the first time, our so-validated model quantitatively predicts how
operating conditions affect battery lifetime. Along three exemplary use cases,
we show that the battery operating window, i.e. minimum and maximum state of
charge, mainly drives aging. In contrast, temperature and current play only a
minor role. These insights help in deriving battery design and usage
recommendations to prolong lifetime.

Future works can further refine and extend our model for several effects. In
particular, post mortem analysis of aged cells with differential voltage
analysis reveal lithium plating and heterogeneous aging inside the cells.
Extending the model for lithium plating is straightforward, because we assume
lithium atom mediated SEI growth. To resolve heterogeneous aging, implementing
our model in three-dimensional cell simulations is a promising approach 58,
59, 60.

## Acknowledgements

Lars von Kolzenberg gratefully acknowledges funding and support by the German
Research Foundation (DFG) within the research training group SiMET under the
project number 281041241/GRK2218. The support of the bwHPC initiative through
the use of the JUSTUS HPC facility at Ulm University is acknowledged. This
work contributes to the research performed at CELEST (Center for
Electrochemical Energy Storage Ulm-Karlsruhe). We also thankfully acknowledge
Mercedes-Benz AG for funding of the extensive cyclic aging test. We thank Timm
Konstantin Groch for his support in the post mortem analysis.

## Conflict of interest

The authors declare no conflict of interest.

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