# D2LV: A Data-Driven and Local-Verification Approach for Image Copy Detection

Wenhao Wang  Yifan Sun  Weipu Zhang  Yi Yang
Baidu Research
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Image copy detection is of great importance in real-life social media. In this
paper, a data-driven and local-verification ($D^{2}LV$) approach is proposed
to compete for Image Similarity Challenge: Matching Track at NeurIPS’21. In
$D^{2}LV$, unsupervised pre-training substitutes the commonly-used supervised
one. When training, we design a set of basic and six advanced transformations,
and a simple but effective baseline learns robust representation. During
testing, a global-local and local-global matching strategy is proposed. The
strategy performs local-verification between reference and query images.
Experiments demonstrate that the proposed method is effective. The proposed
approach ranks first out of $1,103$ participants on the Facebook AI Image
Similarity Challenge: Matching Track. The code and trained models are
available here.

##  1 Introduction

The goal of image copy detection is to determine whether a query image is a
modified copy of any images in a reference dataset. It has extensive numbers
of applications, such as checking integrity-related problems in social media.
Although this topic has been researched for decades, and it has been deemed as
a solved problem, most state-of-the-art solutions cannot deliver satisfactory
results under real-life scenarios [8]. There are two main reasons. First,
real-life cases in social media involve billions to trillions of images, which
introduces many “distractor” images to degrade the performance. Second, the
transformations used to edit images are countless. It is very challenging for
an algorithm to be robust to unseen scenarios.

Figure 1: The data-driven and local-verification approach. Unsupervised pre-
training is used to substitute supervised one. During training, we get
different trained models by designing different sets of augmentations. Global-
local and local-global matching strategy is proposed for testing. Figure 2:
The proposed data-driven and local-verification (D2LV) approach. Recent self-
supervised learning methods, BYOL [13] (its Momentum2 Teacher version [19])
and Barlow-Twins [37], are performed for pre-training. During training, we get
different trained models by using the combination of basic augmentations with
different advanced augmentations separately. When testing, global-local and
local-global matching strategy is proposed for the local-verification between
reference and query images.

In this paper, a data-driven and local-verification (D2LV) approach is
proposed to compete for Image Similarity Challenge: Matching Track at
NeurIPS’21 (ISC2021) [8]. This competition builds a benchmark that features a
variety of image transformations to mimic real-life cases in social media. To
mimic a needle-in-haystack setting, both the query and reference set contain a
majority of “distractor” images that do not match. The evaluation metric
adopted is micro Average Precision, which penalizes any detected pair for a
distractor query.

The approach consists of three parts, pre-training, training, and test. In
pre-training, unsupervised pre-training on ImageNet [6] instead of the
commonly-used supervised pre-training is performed. Specifically, we
empirically find that BYOL [13] pre-training (its Momentum2 Teacher version
[19]) and Barlow-Twins [37] pre-training are superior to some other
unsupervised pre-training methods, and a further ensemble of these two is even
better. In training, a strong but simple deep metric learning baseline is
designed by combining both the classification loss and the pairwise loss
(triplet, in particular). Moreover, to make the sample pairs more informative,
a battery of image augmentations is employed to generate training images.
Empirically, we find that different augmentations may disturb each other to
degrade the performance. Therefore, we use different sets of augmentations to
train models separately and then ensemble them. The diversity of augmentations
promotes learning robust representation. During testing, a robust global-local
and local-global matching strategy is proposed. We observe two hard cases: a)
some query images are generated by overlaying a reference image on top of a
distractor image and b) some queries are cropped from the reference and thus
contain only a partial patch of the reference images. In response, both
heuristic and auto-detected bounding boxes are adopted to crop some local
patches for global-local and local-global matching. The illustration of the
proposed approach is shown in Fig. 1.

In summary, the main contributions of this paper are:

  1. 1.

The paper proposes a data-driven and local-verification approach for image
copy detection.

  2. 2.

The proposed D2LV approach handles real-life copy detection scenarios in
social media well and generalizes well to unseen transformations.

  3. 3.

Our approach outperforms the baseline models and achieves the state-of-the-art
in the Image Similarity Challenge: Matching Track at NeurIPS’21.

##  2 Related Work

###  2.1 Copy Detection

Although copy detection plays an important role in social media, the
publication of copy detection is not well known because organizations want to
keep copy detection techniques obscure, and the researchers often consider
that the task is easy [8]. For classical approaches, some methods are used to
extract global [16, 32, 12] and local descriptors [1, 3, 15]. For deep
learning methods, a descriptor, or known as a feature, is extracted by
convolutional neural networks [21, 39]. However, none of them gives a
satisfying performance on challenging, large benchmarks like ISC2021.

###  2.2 Unsupervised Pre-training

The unsupervised pre-trained models are from recent self-supervised learning
methods. They are trained to extract image embedding unsupervisedly. MoCo [9]
builds a dynamic dictionary with a queue and a moving-averaged encoder to
conduct contrastive learning. BYOL [13] and its Momentum2 Teacher version
achieve a new state-of-the-art without negative pairs. Other self-supervised
methods, such as Barlow Twins [37], SimSiam [5], and SwAV [4], also show
promising performance.

For copy detection in ISC2021, we find that unsupervised pre-trained models
show superior performance to their fully-supervised counterparts. That may be
because the category defined in unsupervised pre-training is much more similar
to that in copy detection than in fully-supervised one.

Figure 3: The set of basic augmentations. It includes random resized cropping,
random rotation, random pixelization, random pixels shuffling, random
perspective transformation, random padding, random image underlay, random
color jitter, random blurring, random gray scale, random horizontal flipping,
random Emoji overlay, random text overlay, random image overlay, and resizing.
The first image is the resized original image.

###  2.3 Deep Metric Learning

Learning a discriminative feature is a crucial component for many tasks, such
as image retrieval [25, 17], face verification [28, 33], and object re-
identification [29, 38]. The commonly-used loss functions can be divided into
two classes: pair-based [11, 27, 22] and proxy-based [20, 34, 7] losses.
Circle loss [28] gives a unified formula for two paradigms.

In our solution to ISC2021, we only use two common losses, triplet loss with
hard sample mining and cross-entropy loss with label smoothing [30]. The two
losses are proved to be simple but effective.

##  3 Proposed Method

In this section, we introduce each important component in the proposed D2LV
approach. In the training part, we discuss the built simple but effective
baseline and designed different sets of augmentations. In the test part, we
discuss the local verification and ensemble methods. The whole approach is
shown in Fig. 2.

###  3.1 Unsupervised Pre-training

In the ISC2021, a category is defined at a very “tight” level, images
generated by exact duplication, near-exact duplication, and edited copy are
considered in the same category while two images that belong to the same
instance or object are not in the same category. However, when using fully-
supervised pre-training on the ImageNet, two images share the same instance or
object are in one class. Therefore, the definition of the category is
contradicted between the ISC2021 and ImageNet.

Fortunately, the recent research on self-supervised learning provides a new
direction. It defines every image as a category and uses invariance to data
augmentation as their main training criterion [8]. Although it seems natural
to directly adopt a self-supervised learning method to train a model for copy
detection, we choose to give up this solution for two reasons. First, the
self-supervised learning methods often cost a lot of days to get convergence
results. For instance, on ImageNet, BYOL [13] or its Momentum2 Teacher version
[19] takes about two weeks to pre-train $300$ epochs with ResNet-152 [10]
backbone using $8$ NVIDIA Tesla V100 32GB GPUs. In addition, when using
sophisticated transformations online, the time becomes much longer, which is
unaffordable. Second, even if the resource is unlimited, we do not get a
satisfying performance only trained by self-supervised learning methods, which
may be because that the hyper-parameters are not adjusted carefully.

As a result, we choose to use BYOL [13] (its Momentum2 Teacher version [19])
and Barlow-Twins [37] to get the pre-trained model on ImageNet. The used
augmentations follow the default setting in their original implementations.

###  3.2 Baseline

We design a simple but effective baseline for ISC2021. Its original version is
from DomainMix [35]. The built baseline newly includes GeM [26], WaveBlock
[36], a high-dimension projector, two commonly-used losses, and a warming up
with cosine annealing learning rate. The backbones selected are ResNet-50
[10], ResNet-152 [10], and ResNet50-IBN [24]. For the design of GeM [26] and
WaveBlock [36], please refer to their original papers, and we follow their
default hyper-parameters. The high-dimension projector projects a learned
$2048$-dim feature into $8192$-dim via some linear and non-linear connections.
We empirically find learning from high dimension space is beneficial for
datasets with large-scale categories. The two commonly-used losses are triplet
loss with hard sample mining and cross-entropy loss with label smoothing [30].
The $2048$-dim feature is used for triplet loss, and the $8192$-dim feature is
used for classification. The baseline is trained for $25$ epochs, and the
change of ratio for learning rate is:

$ratio=\left\\{\begin{array}[]{l}0.99\cdot epoch/5+0.01,0\leq epoch<5\\\
1,5\leq epoch<10\\\
0.5\cdot\left({\cos\left({{\textstyle{{epoch-10}\over{25-10}}}\cdot\pi}\right)+1}\right),10\leq
epoch<25\end{array}\right.$ (1)

(a) Super-blur augmentation.

(b) Super-color augmentation.

(c) Super-dark augmentation.

(d) Super-face augmentation.

(e) Super-opaque augmentation.

(f) Super-occlude augmentation.

Figure 4: The six advanced augmentations. They are added into the original set
of basic augmentations separately. The first image is the resized original
image.

###  3.3 Augmentation

In the ISC2021, the data is crucial in our solution. That is why the proposed
approach is named “data-driven”. First, we design a set of basic augmentations
to transform images. Then, we build six advanced augmentations to provide a
supplementary.

The set of basic augmentations includes random resized cropping, random
rotation, random pixelization, random pixels shuffling, random perspective
transformation, random padding, random image underlay, random color jitter,
random blurring, random grayscale, random horizontal flipping, random Emoji
overlay, random text overlay, random image overlay, and resizing. A display of
using all the basic augmentations on one image is shown in Fig. 3. The first
image is the resized original image.

Further, together with the set of basic transformations, we design six
advanced augmentations, super-blur, super-color, super-dark, super-face,
super-opaque, and super-occlude. The six augmentations are added into the set
of basic augmentations separately. The super-blur augmentation uses enhanced
blur augmentation; the super-color augmentation uses enhanced color jitter
augmentation; the super-dark augmentation darkens the images; the super-face
augmentation adds some face images into training; the super-opaque
augmentation overlays one image on another with certain transparency; the
super-occlude augmentation adds more occlusion into original images. These six
advanced augmentations are shown in Fig. 4.

Besides the basic and advanced augmentations, we also employ black-white
augmentation. The augmentation changes all the color images into black and
white style. Some of the sets, “basic”, “basic + super-blur”, “basic + super-
color”, and “basic + super-face”, use this augmentation. As a result, we have
$11$ sets containing different augmentations. A pre-trained model is trained
$11$ times to get $11$ trained models.

###  3.4 Local Verification

During testing, we observe two corner cases: a) some query images are
generated by overlaying a reference image on top of a distractor image. b)
some queries are cropped from the reference images and thus only contain parts
of the reference images. Therefore, we propose a global-local and local-global
matching strategy for testing. For global-local matching, it matches the
global features of reference images with local features of query images. For
local-global matching, it matches the local features of reference images with
global features of query images.

Generating local features of query images. We use heuristic and auto-detected
bounding boxes to crop local patches from query images. The features of
cropped local patches are used as local features of query images. Given a
query image, first, rotating and center cropping are used to generate patches.
The rotating degrees are $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$. The
center cropping crops not only the “exact center” but also the “$1/3$ center”
of an image. The illustration of “exact center” and “$1/3$ center” is shown in
Fig. 5. Together with the original image, now nine patches are generated.
Second, selective search [31] is used to generate proposal regions. The
features of proposal regions are also regarded as local features of query
images. Last, we use YOLOv5 [14] to detect overlay automatically. For
training, we just generate some images with overlay augmentations and
corresponding bounding boxes automatically from the training dataset. Note
that the above-defined “patch” includes the original image and its rotations.

Figure 5: The nine patches generated by rotating and center cropping. The
dotted lines in the original image show the definition of “exact center” and
“$1/3$ center”.

Generating local features of reference images. We only divide reference images
into small patches to match locally. First, given a reference image, it is
divided into four parts evenly and nine parts evenly. Then, we crop the “exact
center” and “$1/3$ center” of a reference image. Together with the original
image, we can extract $19$ features from a reference image. Note that the
above-defined “patch” also includes the original image.

###  3.5 Ensemble

First, we introduce two ensemble criteria, confidence and completeness. Assume
that we have a pair of images, e.g. a patch $A$ from a query image and an
original reference image $B$, and two models, $f$ and $g$. $f\left(A,B\right)$
and $g\left(A,B\right)$ stand for the similarity score between $A$ and $B$
using models $f$ and $g$, respectively. Also, two score thresholds are
$\alpha$ and $\beta$. When using confidence criterion, the ensemble result is:

$score=\begin{cases}max(f\left(A,B\right),g\left(A,B\right)),\ if\
f\left(A,B\right)>\alpha\ and\ g\left(A,B\right)>\beta,&\\\ None,\
others.&\end{cases}\ $ (2)

The “None” means both the scores are discarded, it does not contribute to the
final score of the image pair. Under completeness criterion, the ensemble
result is:

$score=max(f\left(A,B\right),g\left(A,B\right)).$ (3)

Both the confidence and completeness criteria can be extended to multi-models
by

$score=\begin{cases}max(f_{0}\left(A,B\right),f_{1}\left(A,B\right),...,f_{n}\left(A,B\right)),\
if\ f_{i}\left(A,B\right)>\alpha_{i},&\\\ None,\ others,&\end{cases}$ (4)

and

$score=max(f_{0}\left(A,B\right),f_{1}\left(A,B\right),...,f_{n}\left(A,B\right)),$
(5)

where: $i$ is from $0$ to $n$, $f_{0},f_{1},...,f_{n}$ denotes $n$ different
models, and $\alpha_{0},\alpha_{1},...,\alpha_{n}$ denotes $n$ different score
thresholds.

Remember that we have three different backbones, when using the global-local
matching strategy, the confidence criterion is used for ResNet50$\&$ResNet152,
ResNet50$\&$ResNet50-IBN, and ResNet152$\&$ResNet50-IBN. When using the local-
global matching strategy, the confidence criterion is used for
ResNet50$\&$ResNet152$\&$ResNet50-IBN. For any other model ensembles, we use
the completeness criterion.

Further, to ensemble different local-global and global-local scores, we repeat
the completeness criterion by

$score=max\left(\begin{gathered}max(f\left(A_{0},B\right),\
f\left(A_{1},B\right),...,f\left(A_{l},B\right)),\\\
max(f\left(A,B_{0}\right),\
f\left(A,B_{1}\right),...,f\left(A,B_{m}\right))\end{gathered}\right)$ (6)

where: $A_{0},A_{1},...,A_{l}$ denotes $l$ different patches of the query
image, and $B_{0},B_{1},...,B_{m}$ denotes $m$ different patches of the
reference image.

After finishing all models and patches ensemble, the final score represents
the similarity between the pair of images.

##  4 Experiments

###  4.1 Experimental Settings

In the unsupervised pre-training part, we essentially follow the experimental
setting given in Momentum2 Teacher [19] and Barlow-Twins [37]. For Momentum2
Teacher [19], the selected backbone is ResNet152 [10] and ResNet50-IBN [24].
$8$ NVIDIA Tesla V100 32GB GPUs are used. When training ResNet152 [10], the
batch size is $512$, and it is trained for $300$ epochs. The training process
can be finished in about two weeks. When training ResNet50-IBN [24], the batch
size is $1024$, and it is also trained for $300$ epochs. The training process
can be finished in about six days. For Barlow-Twins [37], the selected
backbone is ResNet50 [10]. We do not re-train the model, and just use the
model supplied by the official implement.

In training, we select $100,000$ out of $1000,000$ images from the official
training set. The criterion is selecting one every $10$. Each image is
augmented $19$ times, and thus for training, each ID has $20$ images. We
trained $33$ models in total. Each training takes about/less than one day on
$4$ NVIDIA Tesla V100 32GB GPUs (bigger backbone takes longer and smaller
backbone takes shorter). Each training batch includes $128$ images of $32$
IDs. Adam optimizer [18] is used to optimize the networks. The image size is
$256\times 256$. The model is trained for $25$ epochs, and in each epoch, we
have $8000$ iterations. The basic learning rate is set to $3.5\times 10^{-4}$.

During testing, for global-local matching, we test the $33$ trained models for
three scales, $200\times 200$, $256\times 256$, $320\times 320$, then ensemble
them to get the final results. The time for extracting all query patches’
features using $33$ NVIDIA Tesla V100 32GB GPUs is about six hours. For local-
global matching, we only test the three models trained on the basic
augmentations for one scale, $256\times 256$. The time for extracting all
$19,000,000$ reference patches’ features using $3$ NVIDIA Tesla A100 40GB GPUs
on a DGX-server is less than one day. We apply a PCA dimensionality reduction
to decrease dimension from $8192$ to $1500$ with the code from the competition
official. The $1500$-dim feature is used for final matching. Besides, some
tricks are used to further improve performance. First, when performing local-
global strategy, it is inappropriate to divide reference images containing
faces into patches. Therefore, when a face detector finds face(s) existing in
one reference image, we do not perform the local-global strategy. Second,
compared to original images, the partial image in a pair is more likely to be
“false positive”, and thus we penalize the corresponding score. Third, we
delete generated query patches with too small image size. Fourth, the final
ensemble score uses the average of two maximum scores instead of the only one
as introduced before.

###  4.2 Comparison with State-of-the-Arts

Table 1: Comparison with state-of-the-art methods from the leaderboard in the Phase $2$. Recall@Precision $90$ is a secondary metric provided for information purposes only as an indicator of a model’s recall at a reasonably good precision level, but is not used for ranking purposes. Our results are highlighted in bold. Team | Score
---|---
Micro-average Precision |  Recall@Precision 90
Ours |  0.8329 |  0.7309
separate |  $0.8291$ |  $0.7917$
imgFp |  $0.7682$ |  $0.6715$
forthedream |  $0.7667$ |  $0.7218$
titanshield |  $0.7613$ |  $0.7557$
VisonGroup |  $0.7169$ |  $0.5963$
mmcf |  $0.7107$ |  $0.5986$
… |  … |  …
MultiGrain[2] |  $0.2761$ |  $0.2023$
GIST [23] |  $0.0526$ |  $-$

To prove the superiority of the D2LV, we compare the proposed model with
state-of-the-art methods from the leaderboard in Phase $2$. The comparison
results are shown in Table 1. In the ISC2021, there are $1103$ participants,
and $36$ teams have submitted their final results. Compared with the strongest
official benchmark MultiGrain [2], the proposed D2LV achieves about $56\%$
improvements, which is considerable. Most Micro-average Precisions from other
teams are less than $77\%$, while our approach gives an $83.29\%$ result. It
is interesting to mention that, in Phase $1$, our result is $90.08\%$. There
is only a less than $7\%$ gap between the two leaderboards for us. However,
for some other teams, the gap can be larger than $30\%$. The superiority
performance is attributed to two aspects. First, the sophisticated
augmentations give us more generalizability when facing unseen
transformations. That is called “Data-Driven”. Second, the global-local and
local-global matching strategy provides an exhaustive matching between a pair
of images. That is called “Local-Verification”.

###  4.3 Ablation Studies

The ablation studies are conducted on $25,000$ query images in Phase $1$ to
prove the effectiveness of each component in the D2LV approach. The evaluation
metric is Micro-average Precision. The experimental results are displayed in
Table 2.

Table 2: The ablation study about the proposed D2LV. “Supervised” or “Unsupervised” denotes the supervised pre-training or unsupervised pre-training is used, respectively. “Global-local” denotes using global-local matching strategy, and “Both’ denotes using global-local and local-global matching strategy. “Adv-Aug” denotes using different kinds of advanced augmentations. “Multi+Tricks” denotes using multi-scale testing and other mentioned tricks. The best results are highlighted in bold. Method | Score
---|---
Micro-average Precision |  Recall@Precision 90
Supervised |  $0.68726$ |  $0.54678$
Unsupervised |  $0.70813$ |  $0.62773$
Global-local |  $0.82726$ |  $0.74755$
Both |  $0.83720$ |  $0.75155$
Adv-Aug |  $0.88640$ |  $0.80124$
Multi+Tricks |  0.90035 |  0.81887

Comparison between supervised pre-training and unsupervised pre-training.
First, we discuss the improvement brought by unsupervised pre-training. The
experimental results are denoted as “Supervised” and “Unsupervised” in Table
2, respectively. The training images only use the basic augmentations. The
matching strategy is only global-to-global. We can find that, instead of
supervised pre-training, $2.1\%$ improvement has been obtained by using
unsupervised pre-trained models. This phenomenon may be because the definition
of category in unsupervised pre-training is similar to that in the task of
copy detection.

The improvement from local-verification. In this part, the experimental
results are from the ensemble result of the three backbones. The experimental
results are denoted as “Global-local” and “Both” in Table 2, respectively.
When only using the global-local matching strategy, the performance can be
improved from $70.81\%$ to $82.73\%$. When we perform global-local and local-
global matching strategy, the highest performance, $83.72\%$, is achieved. The
local-verification plays a very important role in the proposed D2LV.

The improvement from advanced augmentations. To be convenient, we only give
the final ensemble experimental results of all the advanced augmentations with
the basic ones. The experimental results are denoted as “Adv-Aug” in Table 2.
Indeed, each advanced augmentation with the basic one can improve performance
separately. Also, the results use local-verification and three backbones. The
sophisticated augmentations contribute a lot to Phase 2 where some unseen
transformations appear.

The improvement from multi-scale testing and some tricks. The experimental
results are denoted as “Multi+Tricks” in Table 2. The performance improved by
multi-scale testing and some tricks is not very significant. It is less than
$1.5\%$ on $25,000$ queries in Phase 1. After finding the huge performance gap
between the two phases, we are not quite sure whether these methods really
contribute to the final results in Phase 2.

##  5 Conclusion

In this paper, we introduce our winning solution to Image Similarity
Challenge: Matching Track at NeurIPS’21. The proposed D2LV approach uses
recent self-supervised learning methods for pre-training instead of
traditional supervised ones. Further, we find that exploring data is of great
importance, and thus many strong augmentations are designed. Also, the
proposed global-local and local-global matching strategy contributes a lot to
the final submission. We hope the proposed solution is beneficial for real-
life applications including content tracing, copyright infringement, and
misinformation.

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SupplementaryInformation [1]organization=Institute of Engineering
Thermodynamics, German Aerospace Center (DLR), addressline=Wilhelm-Runge-
Straße 10, city=89081 Ulm, country=Germany [2]organization=Helmholtz Institute
Ulm (HIU), addressline=Helmholtzstraße 11, city=89081 Ulm, country=Germany
[3]organization=Institute of Electrochemistry, Ulm University,
addressline=Albert-Einstein-Allee 47, city=89081 Ulm, country=Germany

# Growth of the Solid-Electrolyte Interphase: Electron Diffusion versus
Solvent Diffusion

Lukas Köbbing  Arnulf Latz  Birger Horstmann

###### Abstract

The solid-electrolyte interphase (SEI) substantially influences the lifetime
of lithium-ion batteries. Nevertheless, the transport mechanism responsible
for the long-term growth of the SEI remains controversial. This study aims at
discussing the characteristic time and state-of-charge dependence of SEI
growth mediated by electron diffusion versus solvent diffusion. We describe
both transport mechanisms with continuum models and compare them to
experimental results. We show that electron diffusion can explain both the
observed state-of-charge dependence and the time dependence. In contrast, we
demonstrate that solvent diffusion can reproduce either the state-of-charge
dependence or the time dependence of capacity fade. There is no intermediate
regime where solvent diffusion can explain both dependencies simultaneously.
Furthermore, we emphasize the crucial role of anode voltage and state-of-
charge on SEI growth in general. Due to self-discharge, this dependence can
explain deviations from the typical square-root behavior in the time domain.
We conclude that electron diffusion is the relevant process leading to the
state-of-charge dependent SEI growth. Further experiments are needed to
investigate the reason for contributions to the capacity fade that are
independent of the state-of-charge.

##  1 Introduction

The solid-electrolyte interphase (SEI) crucially determines the performance
and degradation of lithium-ion batteries. Since its first observation, Peled
contributed significantly to the theoretical understanding of the SEI: Peled
provided the first theoretical description of the SEI in 1979 [1], considered
its complex structure and composition [2], and constructed an equivalent
circuit model for a mosaic SEI [3]. His recent review from 2017 highlights the
great importance of ongoing SEI research for the improvement of future
batteries [4].

Despite a lot of effort that has been going into the theoretical and
experimental investigation of the SEI for many years, the SEI is still not
fully understood [5, 6, 7]. This results from the complexity of the SEI due to
the variety of organic and inorganic SEI components [8, 9, 10], mixed layered
and mosaic SEI structures [11, 12, 13, 14, 15], a heterogeneous SEI morphology
[16, 17, 18], and the contribution of different SEI formation reactions to the
evolution of the SEI [19, 20, 21, 22, 23].

One major open question is the exact growth mechanism responsible for the
long-term SEI growth. In this paper, we focus on the two most widely used
mechanisms: electron diffusion through localized states [19, 20, 24, 25, 26]
and solvent diffusion [20, 27, 28, 29, 30, 31]. We illustrate both mechanisms
in Fig. 1. The most promising mechanism for describing experimental data for
SEI growth is electron diffusion [7], e.g. in form of neutral lithium
interstitial diffusion [20, 26]. However, the literature also provides
arguments supporting a porous SEI allowing for solvent diffusion [32].
Arguments for solvent diffusion reported in the literature are observed
currents from redox shuttles through the SEI [33, 34] as well as swelling of
the SEI inside the electrolyte compared to a dried SEI [35].

Figure 1: Schematic of the two transport mechanisms investigated in this
study. The upper part depicts the diffusion of electrons from the electrode
through the SEI towards the electrolyte. The lower part illustrates the
diffusion of solvent molecules from the electrolyte through the SEI towards
the electrode.

To compare these two mechanisms, we investigate the SEI growth during open-
circuit battery storage, usually called calendar aging. During calendar aging,
the growth of the SEI is the main issue leading to capacity loss of the
battery [36, 37]. Under storage conditions, the literature reports two
important observations. The first one is the dependence of capacity loss on
the state-of-charge (SOC) [36, 37, 38, 39], and the second one is the square-
root behavior in time [40, 41, 42, 43].

Although SEI growth under storage conditions still lacks a complete
understanding, some efforts were made to evaluate the SEI growth during
cycling of batteries [26, 44, 45]. The experimentally observed heterogeneity
of the SEI during cycling was rationalized by electron diffusion [18] in
contrast to solvent diffusion [46]. In fact, even self-discharge during
battery storage is a sort of battery operation and influences the growth of
the SEI [20].

To reexamine the usual square-root behavior, Attia et al. investigated the
exact behavior of SEI growth in time during storage and cycling [47]. They
provide measurements of capacity loss on carbon black over time during open-
circuit storage. The experimental data show a particularly large capacity loss
and a clear deviation from the usual square-root behavior in the time domain.
This evidences that it is crucial to consider additional effects like self-
discharge, at least for large capacity losses.

In this paper, we discuss the theory of SEI growth mediated by electron
diffusion and solvent diffusion. We compare the observed dependence of
capacity loss on SOC and time to the simulation results. We check first for
electron diffusion and second for solvent diffusion whether the two mechanisms
can describe both dependencies with the same parameters. For solvent
diffusion, we look additionally for an intermediate regime revealing a trade-
off between SOC and time dependence. In the last section, we highlight the
important role of self-discharge on SEI growth for large capacity losses.

##  2 Theory

We use continuum modeling to describe homogeneous long-term growth of the SEI.
From this perspective, the SEI formation consists of a diffusion and a
reaction step. There is a variety of possible SEI formation reactions, but we
restrict to a single reaction for simplicity,

$2\mathrm{Li}^{+}+2\mathrm{e}^{-}+2\mathrm{EC}\rightarrow\mathrm{Li}_{2}\mathrm{EDC}+\mathrm{R}_{\mathrm{g}},$
(1)

where lithium ions $\mathrm{Li}^{+}$, electrons $\mathrm{e}^{-}$, and solvent
molecules ethylene carbonate ($\mathrm{EC}$) react to lithium ethylene
dicarbonate ($\mathrm{Li}_{2}\mathrm{EDC}$) which is often considered as the
main SEI component. Here, $\mathrm{R_{g}}$ is a gaseous byproduct.

To enable the reaction (1), all the reagents need to be close together. We
consider two possible transport mechanisms to achieve this condition. The
first one is the diffusion of electrons through the SEI towards the SEI-
electrolyte interface, where the reaction takes place in this case. The second
one is the diffusion of solvent molecules through the SEI towards the
electrode-SEI interface, where the reaction takes place then. In the
following, we derive the SEI growth rate for both diffusion mechanisms.

###  2.1 Electron diffusion through localized states

For the derivation of the theory of electron diffusion in the transport
limited regime, we refer to Ref. [20]. This article illustrates the electron
diffusion mechanism with the diffusion of neutral lithium interstitials
supported by Ref. [24]. However, as the SEI is not a perfect layer, electrons
can diffuse as well via different localized states, for example created by
inhomogeneities [48], grain boundaries between different SEI components [49],
or radicals inside the SEI [25].

The capacity loss $Q_{\mathrm{SEI}}$ observed in experiments is related to the
thickness of the SEI $L_{\mathrm{SEI}}$ via

$L_{\mathrm{SEI}}=\frac{v}{s}\frac{Q_{\mathrm{SEI}}}{AF}+L_{\mathrm{SEI},0},$
(2)

where $v$ is the mean molar volume of the SEI, $s$ is the mean stoichiometric
coefficient of lithium in the SEI formation reaction, $A$ is the surface of
the anode, and $F$ is the Faraday constant. The initial capacity loss
$Q_{\mathrm{SEI,0}}$ is related to the initial thickness of the SEI by
$L_{\mathrm{SEI},0}=vQ_{\mathrm{SEI,0}}/(sAF)$.

According to Ref. [20], the capacity loss over time due to SEI growth mediated
by electron diffusion is given by

$\partial_{t}Q_{\mathrm{SEI}}=\frac{A^{2}sF^{2}D_{\mathrm{e^{-}}}}{v}\cdot\frac{c_{\mathrm{e^{-},0}}\cdot
e^{-\tilde{U}}}{Q_{\mathrm{SEI}}+Q_{\mathrm{SEI,0}}}$ (3)

with $\tilde{U}=FU/R_{\mathrm{gas}}T$, where $U$ is the anode open circuit
potential, $R_{\mathrm{gas}}$ is the universal gas constant, and $T$ is the
temperature. $D_{\mathrm{e^{-}}}$ is the diffusion coefficient of electrons
through the SEI and $c_{\mathrm{e^{-},0}}$ is the electron concentration at
$U=0\mathrm{V}$.

For constant voltage, Eq. 3 leads to the well-known square-root behavior in
time,

$Q_{\mathrm{SEI}}=AF\sqrt{\frac{2s}{v}D_{\mathrm{e^{-}}}c_{\mathrm{e^{-},0}}}e^{-\tilde{U}/2}\sqrt{t+t_{0}}-Q_{\mathrm{SEI,0}},$
(4)

where $t_{0}$ depends on the initial capacity loss such that
$Q_{\mathrm{SEI}}$ vanishes at $t=0$.

###  2.2 Solvent diffusion

Next, we need to derive the theory of SEI growth generated by solvent
diffusion in detail. In the transport limited regime, solvent diffusion shows
no SOC dependence contradicting experimental observations [20]. In contrast,
in the reaction limited regime, the time behavior is linear, which does not
match experiments as well [26]. Therefore, we consider an interplay between
the diffusion of solvent molecules and the SEI formation reaction.

The rate of charge consumption due to the SEI formation reaction is described
by a Butler-Volmer equation as

$R=\frac{j_{0}}{F}\left[\tilde{c}_{\mathrm{EC}}(0)e^{-(1-\alpha)\tilde{U}}-e^{\alpha\tilde{U}-\tilde{U}_{\mathrm{SEI}}}\right].$
(5)

Here, $j_{0}$ is the maximum reaction current, $\tilde{c}_{\mathrm{EC}}(0)$ is
the normalized solvent concentration at the electrode-SEI interface, $\alpha$
is the symmetry factor and
$\tilde{U}_{\mathrm{SEI}}=FU_{\mathrm{SEI}}/R_{\mathrm{gas}}T$ is the SEI
formation potential with $U_{\mathrm{SEI}}=0.8\mathrm{V}$. Typically, the
first exponential term in Eq. 5 describing SEI formation dominates as the SEI
product is assumed to be insoluble.

The transport of solvent molecules through the SEI is described by Fickian
diffusion,

$\frac{\partial\tilde{c}_{\mathrm{EC}}}{\partial
t}=\frac{D_{\mathrm{EC}}}{L_{\mathrm{SEI}}^{2}}\cdot\frac{\partial^{2}\tilde{c}_{\mathrm{EC}}}{\partial\tilde{x}^{2}},$
(6)

where $\tilde{c}_{\mathrm{EC}}=c_{\mathrm{EC}}/c_{\mathrm{EC,bulk}}$ is the
normalized concentration, $\tilde{x}=x/L_{\mathrm{SEI}}$ is the distance from
the electrode, and $D_{\mathrm{EC}}$ is the diffusion coefficient of solvent
through the SEI.

At the boundaries, the SEI reaction rate $R$ and the bulk concentration,
respectively, determine the concentration,

$\displaystyle\quad\frac{D_{\mathrm{EC}}c_{\mathrm{EC,bulk}}}{L_{\mathrm{SEI}}}\cdot\left.\frac{\partial\tilde{c}}{\partial\tilde{x}}\right|_{x=0}=R,$
(7) $\displaystyle\quad\tilde{c}_{\mathrm{EC}}(\tilde{x}\rightarrow 1)=1.$

From the equations (5), (6) and (7) we derive the concentration at the
electrode-SEI interface as

$\tilde{c}_{\mathrm{EC}}(0)=\frac{1+\frac{j_{0}L_{\mathrm{SEI}}}{FD_{\mathrm{EC}}c_{\mathrm{EC,bulk}}}e^{\alpha\tilde{U}-\tilde{U}_{\mathrm{SEI}}}}{1+\frac{j_{0}L_{\mathrm{SEI}}}{FD_{\mathrm{EC}}c_{\mathrm{EC,bulk}}}e^{-(1-\alpha)\tilde{U}}}.$
(8)

The resulting equation describing capacity loss due to SEI growth mediated by
solvent diffusion derived from equations (5) and (8) then reads

$\partial_{t}Q_{\mathrm{SEI}}=Aj_{0}\frac{e^{-(1-\alpha)\tilde{U}}-e^{\alpha\tilde{U}-\tilde{U}_{\mathrm{SEI}}}}{1+\frac{vj_{0}}{sAF^{2}D_{\mathrm{EC}}c_{\mathrm{EC,bulk}}}e^{-(1-\alpha)\tilde{U}}\left(Q_{\mathrm{SEI}}+Q_{\mathrm{SEI,0}}\right)}.$
(9)

In the transport limited regime, i.e.
$FD_{\mathrm{EC}}c_{\mathrm{EC,bulk}}/L_{\mathrm{SEI}}\ll j_{0}$, Eq. 9
reveals the square-root behavior in time for constant voltage neglecting the
term of the backward reaction,

$Q_{\mathrm{SEI}}=AF\sqrt{\frac{2s}{v}D_{\mathrm{EC}}c_{\mathrm{EC,bulk}}\left(t+t_{0}\right)}-Q_{\mathrm{SEI,0}}.$
(10)

In the reaction limited regime, i.e. $j_{0}\ll
FD_{\mathrm{EC}}c_{\mathrm{EC,bulk}}/L_{\mathrm{SEI}}$, Eq. 9 shows a linear
time dependence, but non-trivial SOC dependence,

$Q_{\mathrm{SEI}}=Aj_{0}\left(e^{-(1-\alpha)\tilde{U}}-e^{\alpha\tilde{U}-\tilde{U}_{\mathrm{SEI}}}\right)t.$
(11)

##  3 Results and Discussion

Now, we evaluate the dependence of capacity fade on SOC and time for open-
circuit storage. We analyze the data by Keil et al. [36], who measured the
capacity loss depending on the SOC for lithium-ion batteries with a nickel
cobalt aluminum oxide (NCA) cathode. By measuring the SOC dependence at
different times up to 9.5 months, they also provide the time dependence of
capacity fade at different states-of-charge. Importantly, they also measured
the anode open circuit voltage $U(\mathrm{SOC})$ for the same cells.

We support our analysis with new data for lithium iron phosphate (LFP) cells
provided by Naumann et al. [38] in the supplementary information. They focus
on measuring the time dependence at different states-of-charge but do not
provide the anode open circuit voltage curve for the investigated cells. Thus,
for this data set, we use the anode open circuit voltage measured for LFP
cells in Ref. [36].

The authors have already analyzed the SOC dependence in Ref. [20]. Here, we
evaluate not only the SOC dependence but also the time dependence. A theory
that aims at explaining the observed capacity fade has to describe both
dependencies simultaneously.

###  3.1 Electron diffusion: SOC and time dependence

Figure 2: Capacity loss simulated with electron diffusion fitted to the SOC
depending data measured in Ref. [36]. (a) Capacity loss depending on the SOC.
The plot shows a good agreement between simulation and data points. (b)
Capacity loss depending on time. The agreement is excellent at low and medium
SOC, whereas a minor deviation is visible at high SOC.

First of all, we want to review the electron diffusion mechanism. The
experimentally observed capacity loss $Q_{\mathrm{loss}}$ consists of a
contribution due to SEI growth depending on the SOC and an additional
contribution that does not depend on the SOC, i.e.
$Q_{\mathrm{loss}}=Q_{\mathrm{SEI}}+Q_{\mathrm{add}}$ [20]. At high SOC, both
terms contribute significantly to the total capacity loss. At low SOC, the
constant contribution, which serves as an offset for the capacity loss, is the
dominant part. We can investigate the time dependence of the SOC depending
capacity fade by subtracting the capacity fade observed at $\mathrm{SOC}=0$.

To determine the behavior of capacity loss that is independent of the SOC, we
analyze the time dependence of the capacity fade measured at $\mathrm{SOC}=0$.
As only a few data points exist for the time dependence in Ref. [36], we
support our findings with additional data from Ref. [38] in the supplementary
information. For both data sets, we find that a square-root behavior in time
agrees well with the data points (see LABEL:fig:CL-time-SOC0 and LABEL:fig:CL-
time-SOC0-naumann). Due to the initial SEI thickness, we observe a time offset
in the square root that can be related to the initial capacity loss
$Q_{\mathrm{SEI,0}}$. Furthermore, the anode overhang area leads to the
observation of an increase in the capacity at the beginning of the experiment
at low SOC [38]. We account for this with a negative capacity loss at $t=0$.
For further use, we shift the data such that the curve describing the capacity
loss vanishes initially, i.e. it starts at the origin.

The cause of the capacity loss at $\mathrm{SOC}=0$ is not known. Possible
reasons are SEI growth during the check-up cycles or mediated by a different
transport mechanism independent of the SOC. In addition, other degradation
than SEI growth, like particle fracture and contact loss, can cause the
observed capacity fade.

After discussing the SOC independent capacity loss, we want to analyze the SOC
dependence and the time dependence of the capacity fade now. For this, we
subtract the curve obtained at $\mathrm{SOC}=0$ from the data points to
investigate the time dependence of the SOC depending contribution to the
capacity fade. We use our model in Eq. 3 with the determined
$Q_{\mathrm{SEI,0}}$. The anode voltage $U$ depends on the SOC as shown in
Ref. [36], where the SOC changes during storage due to capacity loss and
capacity retention after check-ups. The constant factor in Eq. 3, namely
$A^{2}sF^{2}D_{\mathrm{e^{-}}}c_{\mathrm{e^{-},0}}/v$, serves as a single
fitting variable. In the following, we examine the data provided by Keil et
al. [36] and analyze the data by Naumann et al. [38] in the supplementary
information (see LABEL:fig:ED-SOC-time-naumann).

The fit of the electron diffusion model equation (3) to the capacity loss
depending on the SOC is shown in Fig. 2(a). Overall, the simulation agrees
well with the experimental data, i.e. electron diffusion can explain the SOC
dependence of capacity fade. Only at very low and very high SOC, we observe
major deviations that may account for additional effects.

With the same parameters, in Fig. 2(b) we compare the time dependence to the
experiment. We observe an excellent agreement between the simulated time
dependence and the data at low and medium SOC. The data as well as the
simulation approximately show a square-root behavior in time. At high SOC, the
deviation between simulation and data is more pronounced, indicating an
additional contribution to capacity loss at very high SOC. This is in
accordance with the deviation at high SOC observed in the SOC dependence.

Altogether, we have shown that electron diffusion is able to describe the SOC
dependence and the time dependence with the same parameters. Thus, electron
diffusion can explain the observed behaviors of SEI growth quite accurately.

###  3.2 Solvent diffusion: SOC and time dependence

Figure 3: Capacity loss simulated with solvent diffusion in comparison to the
data measured in Ref. [36]. (a) and (b) Fit of the solvent diffusion model to
the SOC dependence of capacity loss (a) Capacity loss depending on the SOC.
The plot shows a good agreement between simulation and data points. (b)
Capacity loss depending on time. Simulation and experiment do not match at
medium and high SOC. (c) and (d) Fit of the solvent diffusion model to the
time dependence of capacity loss at $\mathrm{SOC}=50\%$ (c) Capacity loss
depending on the SOC. The simulation does not show the experimentally observed
SOC dependence. (d) Capacity loss depending on time. Excellent agreement is
achieved at $\mathrm{SOC}=50\%$ but the simulation does not reproduce the
behavior at $\mathrm{SOC}=100\%$.

Figure 4: Parameter study for the capacity loss simulated with solvent
diffusion with a variation of the reaction current $j_{0}$ and the diffusion
coefficient $D_{\mathrm{EC}}$. (a) Logarithm of the root mean square deviation
(RMSD) of the simulated capacity loss depending on the SOC from the experiment
[36]. The smallest deviation is depicted in dark blue and reached for a small
reaction current $j_{0}$ and large diffusivity $D_{\mathrm{EC}}$. The best fit
of the SOC dependence with the parameters used in Fig. 3(a) is highlighted.
(b) Fitted time exponent $\beta$ to the simulation. The experiment can be
approximately described with $t^{0.5}$ depicted in dark blue. Either large
values of $j_{0}$ or small values of $D_{\mathrm{EC}}$ are needed to achieve a
good agreement with $t^{0.5}$. The best fit of the time dependence with the
parameters used in Fig. 3(d) is highlighted.

Next, we investigate the solvent diffusion mechanism, which competes with
electron diffusion. To explain the various experimental observations, our
model for solvent diffusion considers the interplay between reaction and
transport limitation. In the following, we set the parameter $\alpha=0.5$, as
we observe no qualitative influence of this parameter on our results. With the
same procedure as before, we subtract the data at $\mathrm{SOC}=0$ from the
experimental data to examine the SOC dependent capacity fade. As above, we
analyze the data by Keil et al. [36] here and provide the analysis for the
data by Naumann et al. [38] in the supplementary information (see
LABEL:fig:SD-SOC-time-naumann and LABEL:fig:SD-parameter-study-naumann).

First, we fit the solvent diffusion model equation (9) to the SOC depending
capacity fade data. The corresponding result is shown in Fig. 3(a). The
simulation shows a good agreement with the SOC dependence of the observed
capacity loss. The quality of the fit is similar to the one calculated for
electron diffusion. This result is contrary to Ref. [20], where solvent
diffusion is only described in the transport limited regime. However, Fig.
3(b) shows that with the same parameters, the simulation reproduces the time
dependence only at small SOC, where the offset from $\mathrm{SOC}=0$
contributes primarily to the capacity loss. At higher SOC, the time dependence
does not fit the data points. In this case, the SOC depending capacity fade,
which is approximately linear in time, contributes significantly. The
linearity of the capacity fade reveals that the simulation is in the reaction
limited regime to capture the SOC dependence. In conclusion, matching the SOC
dependence to the experiment does not reproduce the time dependence.

In comparison to the fit to the SOC dependence, which revealed the reaction
limited regime, we fit the time dependence of the capacity fade next. As we
assume to achieve the best accordance at medium SOC, we fit the solvent
diffusion model to the capacity loss over time at $\mathrm{SOC}=50\%$. As
shown in Fig. 3(d), we find a perfect agreement of simulation and experiment
at the fitted $\mathrm{SOC}=50\%$. In contrast, at high SOC, the simulation
gives the same result as at $\mathrm{SOC}=50\%$ and does not capture the
experimental data. The deviation at high SOC is explained when regarding Fig.
3(c). With the parameters obtained from the fit to the time dependence, the
simulation shows a constant contribution to capacity loss over a wide range of
SOC values. The simulation does not capture the dependence of the capacity
loss on the SOC at all. As we observe perfect agreement of the time dependence
at $\mathrm{SOC}=50\%$ but no agreement of the SOC dependence, the simulation
is in the transport limited regime. Thus, matching the time dependence to the
experiment, the simulation does not reproduce the SOC dependence.

As both fits reveal only one observed behavior, we try to find a possible
intermediate regime where the solvent diffusion mechanism captures the SOC
dependence and the time dependence with the same parameters. To find this
regime, we vary the reaction current and the diffusion coefficient in Eq. 9.
In this parameter study, we compare the root mean square deviation between the
simulation and the experimental data points describing the SOC dependence.
Additionally, we fit a simple power law $t^{\beta}$ to the simulated time
dependence and evaluate the agreement of the time exponent $\beta$ with the
square-root profile, i.e. $\beta=0.5$.

The outcome of the parameter study is depicted in Fig. 4. On the one hand,
Fig. 4(a) shows the logarithm of the root mean square deviation of the
simulation to the experimental data describing the SOC dependence. The
smallest values depicted in dark blue refer to the best accordance of
simulation and experiment. Therefore, in the given range of parameters, small
values of the reaction current $j_{0}$ and large values of the diffusion
coefficient $D_{\mathrm{EC}}$ are necessary to describe the experimentally
observed SOC dependence.

On the other hand, Fig. 4(b) depicts the fitted time exponent $\beta$. Here,
values around $\beta=0.5$ are required to describe the experimentally observed
time dependence, which shows approximately a square-root behavior. The
corresponding values are illustrated again in dark blue. From the same range
of parameters as before, now large values of the reaction current $j_{0}$ and
small values of the diffusion coefficient $D_{\mathrm{EC}}$ are needed.

To combine both results, all sets of parameters that describe the SOC
dependence do not reproduce the time dependence, and all sets of parameters
that describe the time dependence do not reproduce the SOC dependence. The
parameter study shows either good accordance with the SOC dependence or good
accordance with the $t^{0.5}$ dependence. There exists no intermediate regime,
where solvent diffusion reproduces the SOC dependence and the time dependence.
As the same experiment exhibits both dependencies, the solvent diffusion
mechanism fails to describe the observed SEI growth characteristics. However,
solvent diffusion could be one of the reasons for the contribution to the
capacity fade that is independent of the SOC.

In this section, we have shown that the diffusion of solvent molecules can not
capture the experimentally observed dependencies of capacity fade on SOC and
time. An alternative approach is considering additional species like the SEI
product or lithium ions inside the SEI, which diffuse through the SEI from the
electrode towards the electrolyte [50, 51]. However, this model requires an
explanation of why the SEI product does not precipitate directly at the
electrode but diffuses towards the SEI-electrolyte interface instead.
Alternatively, it has to be clarified why lithium ions should limit the
transport process as they can move fast through the SEI to enable battery
operation. The mathematical structure of this particular description would be
similar to the model of electron diffusion, as the diffusion of a species from
the electrode towards the electrolyte determines the SEI growth rate.

###  3.3 Self-discharge: Deviation from square-root behavior

So far, we depict the SOC dependence of the capacity loss as a separate
feature compared to the time dependence. Now, we want to highlight the
interaction between the SOC dependence and the $t^{0.5}$ time dependence due
to self-discharge. For a good demonstration of this effect, we refer to an
experiment by Attia et al. measuring a particularly large capacity loss on
carbon black over time [47]. The experimental data are shown in Fig. 5.

For this experiment, Attia et al. reviewed the square-root behavior in time
and fitted power laws $t^{\beta}$ to the experimentally observed capacity fade
[47]. For storage, they observed apparent time exponents lower than $0.5$,
e.g. $\beta\sim 0.3$ as depicted in Fig. 5, in contrast to the usual $t^{0.5}$
behavior expected for transport limitation. However, this can not be explained
by leaving the transport limited regime, as values larger than $0.5$ indicate
reaction limitations [26].

Figure 5: Capacity loss on carbon black during storage. The experimental data
from Ref. [47] show a clear deviation from $t^{0.5}$ behavior. The simulation
for electron diffusion can describe the observed time behavior when self-
discharge is taken into account.

As discussed before, the growth rate of the SEI depends on the anode voltage
via the SOC calculated with respect to the initial capacity. However, the SOC
can not be assumed constant in time but gets effectively reduced by the amount
of charge consumed due to SEI growth. This effect of self-discharge leads to a
deviation of the simulated SEI growth from a simple square-root behavior in
time, particularly for large capacity losses.

In contrast to the former sections, we can not determine the contribution to
the capacity loss independent of the SOC separately, as Ref. [47] provides no
data at $\mathrm{SOC}=0$. However, we assume that both SOC dependent and
independent parts can contribute significantly to the total capacity loss. Our
assumption is supported by results from cryogenic electron microscopy of the
SEI on carbon black [52]. This study reports a thin and a thick SEI layer on
different particles. In addition, Ref. [51] describes two SEI growth modes,
namely homogeneous and heterogeneous SEI growth. Therefore, it is reasonable
to assume that two different mechanisms may lead to the observed SEI
thickness. Due to our former results, we attribute the SOC dependent term to
our electron diffusion model (3) and the SOC independent term to an additional
mechanism. Solvent diffusion would be a candidate for this mechanism.

As motivated, we divide the total capacity loss into a contribution with
nontrivial SOC dependence and a contribution with trivial SOC dependence.
Here, we set $Q_{\mathrm{SEI,0}}=0$ to reduce the number of parameters. Fig. 5
illustrates the simulation of the capacity loss. It matches the experiment
reasonably with similar quality as the simple power law fit. Therefore, we
highlight that electron diffusion can well describe SEI growth as a transport
limited process, i.e. $t^{0.5}$ behavior for constant voltage, when carefully
considering any change in the SOC and the voltage $U(\mathrm{SOC})$,
respectively. Considering this self-discharge, our model for electron
diffusion is able to describe deviations from the $t^{0.5}$ dependence. Our
results reveal that smaller apparent exponents of the power law $t^{\beta}$,
as observed in Ref. [47], are still consistent with transport limited SEI
growth models. Instead, exponents larger than $0.5$ imply the transition to
the reaction limited regime [26].

##  4 Conclusions

We have investigated the SOC and time dependence of capacity fade during
calendar aging and compared the electron diffusion with the solvent diffusion
mechanism. The relevant transport mechanism leading to SEI growth has to
capture the SOC and the time dependence with the same parameters.

We have found that the electron diffusion mechanism can reproduce the SOC
dependence and the time dependence of the capacity loss simultaneously. We
conclude that electron diffusion is able to describe the characteristics of
the SOC dependent capacity loss. Only at very low and high SOC do additional
degradation effects not captured in our model lead to deviations. Their
interplay deserves further investigation.

In contrast, we have shown that solvent diffusion can reproduce either the SOC
dependence or the time dependence of capacity fade. Our parameter study
approves that there is no intermediate regime, which captures both
dependencies with the same parameters. Thus, solvent diffusion fails to
describe the observed features of the SOC dependent capacity loss.

Finally, we have emphasized the importance of the interaction between SOC
dependence and time dependence of capacity fade. We have demonstrated that our
model for electron diffusion captures experimentally observed deviations from
the usual $t^{0.5}$ law by considering the effect of self-discharge.
Consequently, smaller apparent time exponents, as reported in Ref. [47], do
not contradict transport limited SEI growth. In contrast, larger exponents may
indicate reaction limitations [26].

To conclude our study, we have shown that only the electron diffusion
mechanism can explain the SOC and time dependence simultaneously. This
mechanism can even reproduce capacity fade experiments that deviate from the
square-root behavior in time. In agreement with Refs. [51, 52], we have
divided the total capacity loss into two contributions. However, the reason
for the SOC independent capacity loss needs further investigation.

To take future calendar aging studies into account, we propose to
appropriately state experimental data of capacity loss as a function of SOC
and time. The check-up cycles have to be done carefully to reduce their
influence on the self-discharge effect, i.e. it is important to state to which
SOC the cells are charged after the check-ups. Additionally, it is crucial to
provide the anode open circuit voltage curve for the investigated cells to
analyze the SOC dependence of SEI growth.

## Acknowledgments

This work was funded by the German Research Foundation (DFG) in the framework
of the research training group SiMET – Simulation of Mechano-Electro-Thermal
Processes in Lithium-Ion Batteries (281041241/GRK 2218).

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