# The intersection graph of ideals of $\mathbb{Z}_{n}$ is
weakly perfect ††thanks: Key Words: Intersection graph of ideals of
$\mathbb{Z}_{n}$, Clique number, Chromatic number. ††thanks: 2010 Mathematics
Subject Classification: 05C15, 05C69.

R. Nikandish and M.J. Nikmehr

Department of Mathematics, K. N. Toosi University of Technology, Tehran, Iran

<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A graph is called weakly perfect if its vertex chromatic number equals its
clique number. Let $R$ be a ring and $I(R)^{*}$ be the set of all left proper
non-trivial ideals of $R$. The intersection graph of ideals of $R$, denoted by
$G(R)$, is a graph with the vertex set $I(R)^{*}$ and two distinct vertices
$I$ and $J$ are adjacent if and only if $I\cap J\neq 0$. In this paper, it is
shown that $G(\mathbb{Z}_{n})$, for every positive integer $n$, is a weakly
perfect graph. Also, for some values of $n$, we give an explicit formula for
the vertex chromatic number of $G(\mathbb{Z}_{n})$. Furthermore, it is proved
that the edge chromatic number of $G(\mathbb{Z}_{n})$ is equal to the maximum
degree of $G(\mathbb{Z}_{n})$ unless either $G(\mathbb{Z}_{n})$ is a null
graph with two vertices or a complete graph of odd order.

1\. Introduction

The study of algebraic structures, using the properties of graphs, becomes an
exciting research topic in the last twenty years, leading to many fascinating
results and questions. There are many papers on assigning a graph to a ring,
for instance see [1] and [4]. Let $R$ be a ring with unity. By $I(R)$ and
$I(R)^{*}$, we mean the set of all left ideals of $R$ and the set of all left
proper non-trivial ideals of $R$, respectively.
Let $G$ be a graph with the vertex set $V(G)$. For any $x\in V(G)$, $d(x)$
represents the number of edges incident to $x$, called the degree of the
vertex $x$ in $G$. The maximum degree of vertices of $G$ is denoted by
$\Delta(G)$. A graph $G$ is connected if there is a path between every two
distinct vertices. The complete graph of order $n$, denoted by $K_{n}$, is a
graph in which any two distinct vertices are adjacent. A clique of $G$ is a
maximal complete subgraph of $G$ and the number of vertices in the largest
clique of $G$, denoted by $\omega(G)$, is called the clique number of $G$. For
a graph $G$, let $\chi(G)$ denote the vertex chromatic number of $G$, i.e.,
the minimal number of colors which can be assigned to the vertices of $G$ in
such a way that every two adjacent vertices have different colors. Clearly,
for every graph $G$, $\omega(G)\leq\chi(G)$. A graph $G$ is said to be weakly
perfect if $\omega(G)=\chi(G)$. Recall that a $k$-edge coloring of a graph $G$
is an assignment of $k$ colors $\\{1,\ldots,k\\}$ to the edges of $G$ such
that no two adjacent edges have the same color, and the edge chromatic number
$\chi^{\prime}(G)$ of a graph $G$ is the smallest integer $k$ such that $G$
has a $k$-edge coloring. To find some graph coloring methods, we refer the
reader to [5].
Let $F=\\{S_{i}|i\in I\\}$ be an arbitrary family of sets. The intersection
graph, $G(F)$, is a graph with $V(G(F))=F$ and two distinct vertices are
adjacent if and only if they have non-empty intersection. The following
theorem is an interesting fact about intersection graphs due to Marczewski
([6]).

###### Theorem 1

. Every simple graph is an intersection graph.

This result shows that intersection graphs are not weakly perfect in general
(for example, the clique number of a cycle with five vertices is 2, but its
vertex chromatic number is 3). The intersection graph of ideals of a ring $R$,
denoted by $G(R)$, is a graph with the vertex set $I(R)^{*}$ and two distinct
vertices $I$ and $J$ are adjacent if and only if $I\cap J\neq 0$. This graph
was first defined in [4]. While the authors were mainly interested in the
study of intersection graph of ideals of $\mathbb{Z}_{n}$, where
$\mathbb{Z}_{n}$ is the ring of integers modulo $n$. For instance, they
determined the values of $n$ for which the graph of $\mathbb{Z}_{n}$ is
complete, Eulerian or Hamiltonian. The present article is a natural
continuation of the study in this direction. The main aim of this paper is to
show that $G(\mathbb{Z}_{n})$ is a weakly perfect graph, for every positive
integer $n$. Moreover, we determine all integers $n$ when
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))$.

2\. Vertex Chromatic Number and Clique Number of $G(\mathbb{Z}_{n})$

In this section, we prove that
$\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))$, for every positive
integer $n$. Also, for some values of $n$, we give an explicit formula for
$\chi(G(\mathbb{Z}_{n}))$.

Let $n$ be a natural number. Throughout the paper, without loss of generality,
we assume that $n=p_{1}^{n_{1}}p_{2}^{n_{2}}\ldots p_{m}^{n_{m}}$, where
$p_{i}$’s are distinct primes and $n_{i}$’s are natural numbers and $n_{1}\leq
n_{2}\leq\cdots\leq n_{m}$.
We begin with the following remarks.

###### Remark 2

. Consider the ring $\mathbb{Z}_{n}$. It follows from Chinese Remainder
Theorem that
$\mathbb{Z}_{n}\cong\mathbb{Z}_{p_{1}^{n_{1}}}\times\cdots\times\mathbb{Z}_{p_{m}^{n_{m}}}$.

###### Remark 3

. $I\in I(\mathbb{Z}_{n})$ if and only if $I=I_{1}\times\cdots\times I_{m}$,
where $I_{i}\in I(\mathbb{Z}_{p_{i}^{n_{i}}})$.

By Remarks 2 and 3, one can easily see that
$|I(\mathbb{Z}_{n})^{*}|=\prod_{i=1}^{m}(n_{i}+1)-2$.

Let $I=I_{1}\times\cdots\times I_{m}\in I(\mathbb{Z}_{n})$. Each $I_{i}$,
$1\leq i\leq m$, is called a component of $I$. Suppose that $k$, $1\leq k\leq
m$, is an integer and $F$ is the family of all ideals of $\mathbb{Z}_{n}$ with
exactly $k$ non-zero components in which $I_{i_{j}}\neq 0$, $1\leq j\leq k$.
Obviously, one can consider $F$ as the set of ideals with no zero component of
the subring
$\mathbb{Z}_{p_{i_{1}}^{n_{i_{1}}}}\times\cdots\times\mathbb{Z}_{p_{i_{k}}^{n_{i_{k}}}}$.
$\mathbf{Definition.}$ (i) The number of elements of $F$ is denoted by
$\mathcal{W}(F)$.
(ii) Let $G$ be the family of ideals with no zero component of subring
$\mathbb{Z}_{p_{j_{1}}^{n_{j_{1}}}}\times\cdots\times\mathbb{Z}_{p_{j_{l}}^{n_{j_{l}}}}$.
We write $G\subseteq F$, if
$\\{p_{j_{1}}^{n_{j_{1}}},\ldots,p_{j_{l}}^{n_{j_{l}}}\\}\subseteq\\{p_{i_{1}}^{n_{i_{1}}},\ldots,p_{i_{k}}^{n_{i_{k}}}\\}$.
(iii) We write $F\cap G=\varnothing$, if
$\\{p_{j_{1}}^{n_{j_{1}}},\ldots,p_{j_{l}}^{n_{j_{l}}}\\}\cap\\{p_{i_{1}}^{n_{i_{1}}},\ldots,p_{i_{k}}^{n_{i_{k}}}\\}=\varnothing$.
(iv) The family $G$ is said to be complement to $F$, if $F\cap G=\varnothing$
and
$\\{p_{j_{1}}^{n_{j_{1}}},\ldots,p_{j_{l}}^{n_{j_{l}}}\\}\cup\\{p_{i_{1}}^{n_{i_{1}}},\ldots,p_{i_{k}}^{n_{i_{k}}}\\}=\\{p_{1}^{n_{1}},\ldots,p_{m}^{n_{m}}\\}$
and we write $F^{c}=G$. Clearly, $(F^{c})^{c}=F$.

The following example investigate families of ideals of $\mathbb{Z}_{n}$, when
$n=p_{1}^{n_{1}}p_{2}^{n_{2}}p_{3}^{n_{3}}$.

###### Example 4

. Let $n=p_{1}^{n_{1}}p_{2}^{n_{2}}p_{3}^{n_{3}}$. Then there exists one
family of ideals with no zero component. Suppose that $F_{0}$ is the family of
ideals of the form $I_{1}\times I_{2}\times I_{3}$, where each $I_{i}$ is a
non-zero ideal of $\mathbb{Z}_{p_{i}^{n_{i}}}$. Let $F_{1},F_{2},F_{3}$ be
families of ideals of the form $0\times I_{2}\times I_{3},I_{1}\times 0\times
I_{3},I_{1}\times I_{2}\times 0$, respectively, where each $I_{i}$ is a non-
zero ideal of $\mathbb{Z}_{p_{i}^{n_{i}}}$. Also, assume that
$F_{4},F_{5},F_{6}$ are families of ideals of the form $0\times 0\times
I_{3},0\times I_{2}\times 0,I_{1}\times 0\times 0$, respectively, where each
$I_{i}$ is a non-zero ideal of $\mathbb{Z}_{p_{i}^{n_{i}}}$. Then
$\mathcal{W}(F_{0})=n_{1}n_{2}n_{3},\mathcal{W}(F_{1})=n_{2}n_{3},\mathcal{W}(F_{2})=n_{1}n_{3}$
and $\mathcal{W}(F_{3})=n_{2}n_{3}$. Also, $F_{i}\subset F_{0}$, for
$i=1,\ldots,6$. Moreover, $F_{1}^{c}=F_{6},F_{2}^{c}=F_{5}$ and
$F_{4}^{c}=F_{3}$. If we let $F_{7}=0\times 0\times 0$, then
$\\{F_{0},\ldots,F_{7}\\}$ is a partition of ideals of
$\mathbb{Z}_{p_{1}^{n_{1}}}\times\mathbb{Z}_{p_{2}^{n_{2}}}\times\mathbb{Z}_{p_{3}^{n_{3}}}$.

###### Theorem 5

. Let $n=p_{i}^{n_{i}}$. Then
$\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=n_{i}-1$.

###### Proof.

It is clear. $\Box$

###### Theorem 6

. The graph $G(\mathbb{Z}_{n})$ is weakly perfect, for every $n>0$.

###### Proof.

Let $C=\\{G\,|\,\mathcal{W}(G)>\mathcal{W}(G^{c})\\}\cup A$, where $G\in A$ if
and only if $\mathcal{W}(G)=\mathcal{W}(G^{c})$ and $|A\cap\\{G,G^{c}\\}|=1$.
We show that $C_{1}=\bigcup_{G\in C}G$ is a clique of $G(\mathbb{Z}_{n})$.
Assume to the contrary that there exist ideals $I$ and $J$ in $C_{1}$ and
$I\cap J=0$. Then there exist two families $G_{1}$ and $G_{2}$ such that
$G_{1}\cap G_{2}=\varnothing$. This implies that $G_{1}\subseteq G_{2}^{c}$
and $G_{2}\subseteq G_{1}^{c}$. Consequently,
$\mathcal{W}(G_{2})\geq\mathcal{W}(G_{2}^{c})\geq\mathcal{W}(G_{1})\geq\mathcal{W}(G_{1}^{c})\geq\mathcal{W}(G_{2})$
and so
$\mathcal{W}(G_{2})=\mathcal{W}(G_{2}^{c})=\mathcal{W}(G_{1})=\mathcal{W}(G_{1}^{c})$.
Hence, $G_{1}=G_{2}^{c}$ and $G_{1},G_{2}\in C$, a contradiction. Therefore,
$|C_{1}|\leq\omega(G(\mathbb{Z}_{n}))$. To complete the proof, we show that
$\chi(G(\mathbb{Z}_{n}))\leq|C_{1}|$. We color all ideals in $C_{1}$ with
different colors and color each family $H$ of ideals out of $C$ with colors of
ideals of $H^{c}$. Now, we show that this is a proper vertex coloring of
$G(\mathbb{Z}_{n})$. Suppose that $I,J$ are adjacent vertices in
$G(\mathbb{Z}_{n})$. Without loss of generality, one can assume that there are
different families $F_{i}$ and $F_{j}$ such that $I\in F_{i}$ and $J\in F_{j}$
and at least one of $I,J$ is not contained in $C_{1}$. Since $F_{i}\neq
F_{j}$, we deduce that $F_{i}\neq F_{j}^{c},F_{i}^{c}\neq F_{j},F_{i}^{c}\neq
F_{j}^{c}$. Therefore, $I$ and $J$ have different colors. Thus we obtain a
proper vertex coloring for $G(\mathbb{Z}_{n})$, as desired.  $\Box$

###### Theorem 7

. Let $n_{m}\geq\prod_{i=1}^{n_{m-1}}n_{i}$. Then

$\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=n_{m}\prod_{i=1}^{n_{m-1}}(n_{i}+1)-1.$

###### Proof.

It is enough to see that $F\in C$ if for every $J\in F$ the ideal $J$ contains
$I=0\times\cdots\times 0\times(p_{m}^{n_{m}-1})$. $\Box$

From the previous theorem we have the following immediate corollaries.

###### Corollary 8

. Let $n=p_{1}^{n_{1}}p_{2}^{n_{2}}$. Then
$\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=n_{2}(n_{1}+1)-1$.

###### Proof.

By the assumption, $n_{2}\geq n_{1}$. So the result follows from Theorem 7.
$\Box$

###### Corollary 9

. Let $n_{1}=n_{2}=\cdots=n_{m}=1$. Then
$\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=2^{m-1}-1$.

Let $R$ be a ring and $R\cong F_{1}\times\cdots\times F_{m}$, where each
$F_{i}$ is a field. By a similar argument in the proof of Theorem 7 and
Corollary 9, one can show that $\omega(G(R))=\chi(G(R))=2^{m-1}-1$.

###### Theorem 10

. Let $m>1$ be an odd number such that
$\prod_{i=0}^{\lfloor\frac{m}{2}\rfloor-1}n_{m-i}\leq\prod_{i=1}^{m-\lfloor\frac{m}{2}\rfloor}n_{i}$.
Then $\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=$

$|\\{I\in
I(\mathbb{Z}_{n})^{*}\,|\,I\rm{\,has\,at\,most\lfloor\frac{m}{2}\rfloor\,zero\,components}\\}|.$

###### Proof.

It is enough to see that $F\in C$ if for every $J\in F$ the ideal $J$ has at
most $\lfloor\frac{m}{2}\rfloor$ zero components.  $\Box$

###### Corollary 11

. Let $m$ be an odd number and $n_{1}=n_{2}=\cdots=n_{m}=\alpha$. Then

$\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=\sum_{i=0}^{\lfloor\frac{m}{2}\rfloor}\big{(}\,^{m}_{i}\,\big{)}\alpha^{m-i}-1.$

###### Proof.

By Theorem 10, $\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=$

$|\\{I\in
I(\mathbb{Z}_{n})^{*}|I\rm{\,has\,at\,most\lfloor\frac{m}{2}\rfloor\,zero\,components}\\}|.$

It is clear that there exist $\big{(}\,^{m}_{i}\,\big{)}\alpha^{m-i}$ ideals
with $i$ non-zero components. Note that the ideal
$\mathbb{Z}_{p_{1}^{\alpha}}\times\cdots\times\mathbb{Z}_{p_{m}^{\alpha}}$ is
not a vertex of $G(\mathbb{Z}_{n})$. $\Box$

In the sequel, similar results in case $m$ is an even number are given.

###### Theorem 12

. Let $m>2$ be an even number such that
$\prod_{i=0}^{\frac{m}{2}-2}{n_{m-i}}\leq\prod_{i=1}^{\frac{m}{2}+1}{n_{i}}$.
Then $\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=$

$|\\{I\in
I(\mathbb{Z}_{n})^{*}\,|\,I\rm{\,has\,at\,most\frac{m}{2}-1\,zero\,components}\\}|+\sum_{F\in
A}\mathcal{W}(F).$

###### Proof.

It is enough to see that $F\in C$ if for every $J\in F$ either $J$ has at most
$\frac{m}{2}-1$ zero components or $J$ has exactly $\frac{m}{2}$ zero
components and $\mathcal{W}(F)=\mathcal{W}(F^{c})$.  $\Box$

We close this section with the following corollary.

###### Corollary 13

. Let $m$ be an even number and $n_{1}=n_{2}=\cdots=n_{m}=\alpha$. Then

$\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=\sum_{i=0}^{\frac{m}{2}-1}\big{(}\,^{m}_{i}\,\big{)}\alpha^{m-i}+\frac{\big{(}\,^{m}_{\frac{m}{2}}\,\big{)}\alpha^{\frac{m}{2}}}{2}-1.$

###### Proof.

Since $n_{1}=n_{2}=\cdots=n_{m}=\alpha$, it is easily seen that
$\prod_{i=0}^{\frac{m}{2}-2}{n_{m-i}}\leq\prod_{i=1}^{\frac{m}{2}+1}{n_{i}}$.
Moreover, there are $\big{(}\,^{m}_{i}\,\big{)}\alpha^{m-i}$ ideals with $i$
non-zero components. Note that, if $i=\frac{m}{2}$, exactly
$\frac{\big{(}\,^{m}_{\frac{m}{2}}\,\big{)}\alpha^{\frac{m}{2}}}{2}$ of proper
ideals are adjacent together and
$\mathbb{Z}_{p_{1}^{\alpha}}\times\cdots\times\mathbb{Z}_{p_{m}^{\alpha}}$ is
not a vertex of $G(\mathbb{Z}_{n})$. The result, now, follows from Theorem 12.
$\Box$

It follows from [2], for every ring $R$, if $\omega(G(R))<\infty$, then
$\chi(G(R))<\infty$. Also, we have not found any example of a ring $R$ such
that $G(R)$ is not weakly perfect. These positive results motivate the
following conjecture.

$\mathbf{Conjecture.}$ For every ring $R$, $G(R)$ is a weakly perfect graph.

3\. Edge Chromatic Number of $G(\mathbb{Z}_{n})$

In this section we study the edge chromatic number of $G(\mathbb{Z}_{n})$ and
prove that, for every positive integer $n$,
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))$, unless $n=pq$,
where $p,q$ are distinct primes, or $n=p^{m}$, where $m$ is an even number. If
$n=pq$, then $G(\mathbb{Z}_{n})$ is a null graph with two vertices and if
$n=p^{m}$, where $m$ is an even number, then $G(\mathbb{Z}_{n})$ is a complete
graph of order $m-1$. First, we need the following lemmas:

###### Lemma 14

. [8, p. 16] If $G$ is a simple graph, then either
$\chi^{\prime}(G)=\Delta(G)$ or $\chi^{\prime}(G)=\Delta(G)+1$.

###### Lemma 15

. [3, Corollary 5.4] Let $G$ be a simple graph. Suppose that for every vertex
$u$ of maximum degree, there exists an edge $u--v$ such that
$\Delta(G)-d(v)+2$ is more than the number of vertices with maximum degree in
$G$. Then $\chi^{\prime}(G)=\Delta(G)$.

###### Lemma 16

. [7, Theorem D] If $G$ has order $2s$ and maximum degree $2s-1$, then
$\chi^{\prime}(G)=\Delta(G)$. If $G$ has order $2s+1$ and maximum degree $2s$,
then $\chi^{\prime}(G)=\Delta(G)+1$ if and only if the size of $G$ is at least
$2s^{2}+1$.

Now, we are ready to state our main result in this section.

###### Theorem 17

. Let $n$ be a positive integer. Then
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))$, unless
$n=p_{1}p_{2}$, where $p_{1},p_{2}$ are distinct primes, or $n=p^{m}$, where
$m$ is an even number.

###### Proof.

Suppose that $n=p_{1}^{n_{1}}p_{2}^{n_{2}}\ldots p_{m}^{n_{m}}$, where
$p_{i}$’s are all distinct primes and $n_{i}$’s are all natural numbers. If
$m=1$, then $G(\mathbb{Z}_{n})$ is a complete graph of order $n_{1}-1$. If
$n_{1}$ is an even number, then
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))+1$, otherwise
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))$. Thus assume that
$m\geq 2$. We continue the proof in the following cases:

Case 1. $n_{1}=n_{2}=\cdots=n_{m}=1$. In this case, $\mathbb{Z}_{n}\cong
F_{1}\times\cdots\times F_{m}$, where each $F_{i}$ is a field. If $m=2$, then
$\mathbb{Z}_{n}\cong F_{1}\times F_{2}$ and so $G(\mathbb{Z}_{n})$ is a null
graph with two vertices. Therefore,
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))+1$. Then suppose
that $m\geq 3$. For each $j=1,\ldots,m$, define the ideal
$I_{j}=F_{1}\times\cdots\times F_{j-1}\times 0\times F_{j+1}\times\cdots\times
F_{m}$. Then $d(I_{j})=\Delta(G(\mathbb{Z}_{n}))=2^{m}-4$, for each $1\leq
j\leq m$. Also, $I_{1},\ldots,I_{m}$ are all vertices with maximum degree in
$G(\mathbb{Z}_{n})$. Let $u=I_{i}$ ($1\leq i\leq m$) be a vertex of maximum
degree, say $I_{1}$. Then $v=0\times F_{2}\times 0\times\cdots\times 0$ is an
adjacent vertex to $u$. Thus
$\Delta(G(\mathbb{Z}_{n}))-d(v)+2=2^{m}-4-(2^{m-1}-2)+2=2^{m}-2^{m-1}$. Since
$m\geq 3$, we conclude that $2^{m}-2^{m-1}>m$, for each $m$. By Lemma 15,
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))$.

Case 2. Every $n_{i}$ is an even number. Then $|V(G(\mathbb{Z}_{n}))|$ is an
odd number of the form $2s+1$. Note that every vertex of the form
$I_{1}\times\cdots\times I_{m}$, where $I_{i}$ is a non-zero ideal of
$\mathbb{Z}_{p_{i}^{n_{i}}}$ is adjacent to all vertices of
$G(\mathbb{Z}_{n})$. Since the size of a complete graph of order $2s+1$ is
$2s^{2}+s$, if we prove that the intersection graph $G(\mathbb{Z}_{n})$, in
this case, losses at least $s$ edges, then by Lemma 16,
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))$. One can consider
the ring $\mathbb{Z}_{n}$ of the form $R_{1}\times R_{2}$, where $R_{1},R_{2}$
are two rings with $|I(R_{1})|=\alpha$ and $|I(R_{2})|=\beta$. Therefore,
$\alpha\beta=2s+3$. Clearly, every vertex of the form $(I,0)$ is not adjacent
to every vertex of the form $(0,J)$. Then $G(R_{1}\times R_{2})$ losses at
least $\alpha\beta-(\alpha+\beta)+1$ edges. It is easily checked that
$\alpha\beta-(\alpha+\beta)+1>s=\frac{\alpha\beta-3}{2}$, as desired.

Case 3. There exists at least one $i$ such that $n_{i}$ is an odd number and
$n_{j}>1$ for some $j$. Therefore $\mathbb{Z}_{p_{j}^{n_{j}}}$ has at least
one non-trivial proper ideal and then $\mathbb{Z}_{n}$ has an ideal with no
zero complement which is adjacent to every other vertex. Since
$|V(G(\mathbb{Z}_{n}))|$ is an even number, by Lemma 16,
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))$.  $\Box$

Acknowledgements. The authors would like to express their deep gratitude to
the professor Saieed Akbari for his fruitful comments. Also, we thank to the
referee for his/her careful reading and his/her valuable suggestions.

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Y|\widetilde{Z}}}\leavevmode\nobreak\ =\leavevmode\nobreak\ $
$\displaystyle\frac{\big{\\{}\mathbb{E}_{Z_{m}}[(p_{X|Z_{m}}-q_{X\cdot|\widetilde{Z}})(p_{Y|Z_{m}}-q_{\cdot
Y|\widetilde{Z}})]\\}^{2}}{\mathbb{E}_{Z_{m}}[(p_{X|Z_{m}}+q_{X\cdot|\widetilde{Z}})(p_{Y|Z_{m}}+q_{\cdot
Y|\widetilde{Z}})]},$

where we denote $p_{X|Z_{m}}=p_{X|Z}(x|Z_{m})$ and
$p_{Y|Z_{m}}=p_{Y|Z}(y|Z_{m})$. Next we apply Jensen’s inequality several
times and obtain that

$\displaystyle\frac{\big{\\{}\mathbb{E}_{Z_{m}}[(p_{X|Z_{m}}-q_{X\cdot|\widetilde{Z}})(p_{Y|Z_{m}}-q_{\cdot
Y|\widetilde{Z}})]\\}^{2}}{\mathbb{E}_{Z_{m}}[(p_{X|Z_{m}}+q_{X\cdot|\widetilde{Z}})(p_{Y|Z_{m}}+q_{\cdot
Y|\widetilde{Z}})]}\leavevmode\nobreak\
\overset{(\text{i})}{\leq}\leavevmode\nobreak\ $
$\displaystyle\mathbb{E}_{Z_{m}}\Bigg{[}\frac{(p_{X|Z_{m}}-q_{X\cdot|\widetilde{Z}})^{2}(p_{Y|Z_{m}}-q_{\cdot
Y|\widetilde{Z}})^{2}}{(p_{X|Z_{m}}+q_{X\cdot|\widetilde{Z}})(p_{Y|Z_{m}}+q_{\cdot
Y|\widetilde{Z}})}\Bigg{]}$ $\displaystyle\leavevmode\nobreak\
\overset{(\text{ii})}{\leq}\leavevmode\nobreak\ $
$\displaystyle\mathbb{E}_{Z_{m},Z_{m}^{\prime},Z_{m}^{\prime\prime}}\Bigg{[}\frac{(p_{X|Z_{m}}-p_{X|Z_{m}^{\prime}})^{2}(p_{Y|Z_{m}}-p_{Y|Z_{m}^{\prime\prime}})]^{2}}{(p_{X|Z_{m}}+p_{X|Z_{m}^{\prime}})(p_{Y|Z_{m}}+p_{Y|Z_{m}^{\prime\prime}})}\Bigg{]}$

where we detail step (i) and step (ii) at the end of this section. Applying
Tonelli’s theorem yields

$\displaystyle
3\int\int\frac{\big{(}{q_{XY|\widetilde{Z}}}-{q_{X\cdot|\widetilde{Z}}q_{\cdot
Y|\widetilde{Z}}}\big{)}^{2}}{{q_{XY|\widetilde{Z}}}+3{q_{X\cdot|\widetilde{Z}}q_{\cdot
Y|\widetilde{Z}}}}d\mu_{X}d\mu_{Y}$ $\displaystyle\leavevmode\nobreak\
\leq\leavevmode\nobreak\ $ $\displaystyle
3\mathbb{E}_{Z_{m},Z_{m}^{\prime},Z_{m}^{\prime\prime}}\Bigg{[}\int\frac{(p_{X|Z_{m}}-p_{X|Z_{m}^{\prime}})^{2}}{p_{X|Z_{m}}+p_{X|Z_{m}^{\prime}}}d\mu_{X}\times\int\frac{(p_{Y|Z_{m}}-p_{Y|Z_{m}^{\prime\prime}})^{2}}{p_{Y|Z_{m}}+p_{Y|Z_{m}^{\prime\prime}}}d\mu_{Y}\Bigg{]}$
$\displaystyle\leq\leavevmode\nobreak\ $ $\displaystyle
12\mathbb{E}_{Z_{m},Z_{m}^{{}^{\prime}},Z_{m}^{{}^{\prime\prime}}}\big{[}\mathcal{D}_{\mathrm{H}}^{2}\big{(}P_{X|Z=Z_{m}},P_{X|Z=Z_{m}^{{}^{\prime}}}\big{)}\times\mathcal{D}_{\mathrm{H}}^{2}\big{(}P_{Y|Z=Z_{m}},P_{Y|Z=Z_{m}^{{}^{\prime\prime}}}\big{)}\big{]},$

where the last inequality holds since $(x+y)\leq(\sqrt{x}+\sqrt{y})^{2}\leq
2(x+y)$ for $x,y\geq 0$. Hence the result follows.

Details of step (i) and step (ii)

Throughout this part, we assume that $q_{X\cdot|\widetilde{Z}}>0$ and
$q_{\cdot Y|\widetilde{Z}}>0$. Otherwise we have
$q_{XY|\widetilde{Z}}-q_{X\cdot|\widetilde{Z}}q_{\cdot Y|\widetilde{Z}}=0$ and
the bound becomes trivial. Based on the identity

$\displaystyle\frac{\big{\\{}\mathbb{E}_{Z_{m}}[(p_{X|Z}-q_{X\cdot|\widetilde{Z}})(p_{Y|Z}-q_{\cdot
Y|\widetilde{Z}})]\\}^{2}}{\mathbb{E}_{Z_{m}}[(p_{X|Z}+q_{X\cdot|\widetilde{Z}})(p_{Y|Z}+q_{\cdot
Y|\widetilde{Z}})]}$ $\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\frac{\big{\\{}\mathbb{E}_{Z_{m}}[p_{X|Z_{m}}p_{Y|Z_{m}}]-q_{\cdot
Y|\widetilde{Z}}\mathbb{E}_{Z_{m}}[p_{X|Z_{m}}]-q_{X\cdot|\widetilde{Z}}\mathbb{E}_{Z_{m}}[p_{Y|Z_{m}}]+q_{X\cdot|\widetilde{Z}}q_{\cdot
Y|\widetilde{Z}}\\}^{2}}{\mathbb{E}_{Z_{m}}[p_{X|Z_{m}}p_{Y|Z_{m}}]+q_{\cdot
Y|\widetilde{Z}}\mathbb{E}_{Z_{m}}[p_{X|Z_{m}}]+q_{X\cdot|\widetilde{Z}}\mathbb{E}_{Z_{m}}[p_{Y|Z_{m}}]+q_{X\cdot|\widetilde{Z}}q_{\cdot
Y|\widetilde{Z}}},$

it is enough to show that the function

$\displaystyle f(x,y,z)=\frac{(x-ay-bz+ab)^{2}}{x+ay+bz+ab}$

is convex for $x,y,z\geq 0$ and $a,b>0$. Then the result follows by Jensen’s
inequality for a vector valued function. To this end, we prove that the
Hessian matrix of $f(x,y,z)$ is positive semidefinite. Some calculations show
that

$\displaystyle\frac{\partial^{2}f}{\partial x^{2}}\leavevmode\nobreak\
=\leavevmode\nobreak\ $ $\displaystyle\frac{8(ay+bz)^{2}}{(x+bz+ay+ab)^{3}},$
$\displaystyle\frac{\partial^{2}f}{\partial y^{2}}\leavevmode\nobreak\
=\leavevmode\nobreak\ $
$\displaystyle\frac{8a^{2}(x+ab)^{2}}{(x+bz+ay+ab)^{3}},$
$\displaystyle\frac{\partial^{2}f}{\partial z^{2}}\leavevmode\nobreak\
=\leavevmode\nobreak\ $
$\displaystyle\frac{8b^{2}(x+ab)^{2}}{(x+bz+ay+ab)^{3}},$
$\displaystyle\frac{\partial^{2}f}{\partial x\partial y}\leavevmode\nobreak\
=\leavevmode\nobreak\ $
$\displaystyle-\frac{8a(x+ab)(ay+bz)}{(x+bz+ay+ab)^{3}},$
$\displaystyle\frac{\partial^{2}f}{\partial x\partial z}\leavevmode\nobreak\
=\leavevmode\nobreak\ $
$\displaystyle-\frac{8b(x+ab)(ay+bz)}{(x+bz+ay+ab)^{3}},$
$\displaystyle\frac{\partial^{2}f}{\partial y\partial z}\leavevmode\nobreak\
=\leavevmode\nobreak\ $
$\displaystyle-\frac{8ab(x+ab)^{2}}{(x+bz+ay+ab)^{3}},$

and thus the Hessian matrix becomes

$\displaystyle\textbf{H}_{f}\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{8}{(x+bz+ay+ab)^{3}}\begin{bmatrix}ay+bz\\\ -a(x+ab)\\\
-b(x+ab)\end{bmatrix}\begin{bmatrix}ay+bz&-a(x+ab)&-b(x+ab)\end{bmatrix}.$

This completes the proof of step (i). Step (ii) follows similarly by
conditional Jensen’s inequality together with the observation that
$\frac{(a-x)^{2}}{b+x}$ is convex for $x\geq 0$ and $a,b\geq 0$.

###  D.2 Proof of Lemma 4

The first inequality (71) is stated in Corollary 2 of Kamps, (1989). Hence we
focus on the second inequality (72). We first claim that, for any
$1\leq\alpha\leq\beta$ such that $1/\alpha+1/\beta=1$,

$\displaystyle|x-y|\leq|x^{1/\alpha}-y^{1/\alpha}|\times|x^{1/\beta}+y^{1/\beta}|.$
(75)

Without loss of generality, assume that $x>y\geq 0$ (when $x=y$, there is
nothing to prove). Then, by expanding the terms, the claim becomes equivalent
to

$\displaystyle x^{1/\alpha}y^{1/\beta}\geq x^{1/\beta}y^{1/\alpha}.$

By taking the log on both sides,

$\displaystyle\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)\log
x\geq\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)\log y,$

which is true since $1\leq\alpha\leq\beta$ and $x>y\geq 0$. This verifies the
inequality (75). Leveraging this inequality, observe that

$\displaystyle\int\big{|}p^{1/\gamma}-q^{1/\gamma}\big{|}^{\gamma}d\mu\leq\int\big{|}p^{1/(\gamma\alpha)}-q^{1/(\gamma\alpha)}\big{|}^{\gamma}\big{|}p^{1/(\gamma\beta)}+q^{1/(\gamma\beta)}\big{|}^{\gamma}d\mu$

for any $1\leq\alpha\leq\beta$ and $1/\alpha+1/\beta=1$. By Hölder’s
inequality, we further have

$\displaystyle\bigg{[}\int\big{|}p^{1/\gamma}-q^{1/\gamma}\big{|}^{\gamma}d\mu\bigg{]}^{1/\gamma}\leq\bigg{[}\int\big{|}p^{1/(\gamma\alpha)}-q^{1/(\gamma\alpha)}\big{|}^{\gamma\alpha}d\mu\bigg{]}^{1/(\gamma\alpha)}\bigg{[}\int\big{|}p^{1/(\gamma\beta)}+q^{1/(\gamma\beta)}\big{|}^{\gamma\beta}d\mu\bigg{]}^{1/(\gamma\beta)}.$

Since $p,q$ are density functions, the second term in the product is bounded
above by

$\displaystyle\bigg{[}\int\big{|}p^{1/(\gamma\beta)}+q^{1/(\gamma\beta)}\big{|}^{\gamma\beta}d\mu\bigg{]}^{1/(\gamma\beta)}=\leavevmode\nobreak\
\bigg{[}2^{\gamma\beta}\int\bigg{|}\frac{p^{1/(\gamma\beta)}+q^{1/(\gamma\beta)}}{2}\bigg{|}^{\gamma\beta}d\mu\bigg{]}^{1/(\gamma\beta)}\leq\leavevmode\nobreak\
2,$

where the last step follows by Jensen’s inequality (note that $\gamma\beta\geq
1$). Therefore

$\displaystyle\mathcal{D}_{\gamma,\mathrm{H}}(P,Q)=\bigg{[}\frac{1}{2}\int\big{|}p^{1/\gamma}-q^{1/\gamma}\big{|}^{\gamma}d\mu\bigg{]}^{1/\gamma}\leq\leavevmode\nobreak\
$ $\displaystyle
2^{1-1/\gamma}\bigg{[}\int\big{|}p^{1/(\gamma\alpha)}-q^{1/(\gamma\alpha)}\big{|}^{\gamma\alpha}d\mu\bigg{]}^{1/(\gamma\alpha)}$
$\displaystyle=\leavevmode\nobreak\ $ $\displaystyle
2^{1-1/\gamma+1/(\gamma\alpha)}\mathcal{D}_{\gamma\alpha,\mathrm{H}}(P,Q).$

Note that by the restriction on $\alpha$ and $\beta$, the parameter $\alpha$
should be in the range of $1\leq\alpha\leq 2$. This completes the proof of
Lemma 4.

###  D.3 Proof of Lemma 6

By the definition of the quantile function, $q_{1-\alpha}$ can be written as

$\displaystyle q_{1-\alpha}\leavevmode\nobreak\ =\leavevmode\nobreak\ $
$\displaystyle\inf\bigg{\\{}x\in\mathbb{R}:1-\alpha\leq\frac{1}{K}\sum_{\bm{\pi}_{i}\in\bm{\Pi}}\mathds{1}\big{(}T_{\mathrm{CI}}^{\bm{\pi}_{i}}\leq
x\big{)}\bigg{\\}}$ $\displaystyle=\leavevmode\nobreak\ $
$\displaystyle\inf\bigg{\\{}x\in\mathbb{R}:1-\alpha\leq\frac{1}{K}\sum_{\bm{\pi}_{i}\in\bm{\Pi}}\mathds{1}\big{(}T_{\mathrm{CI}}^{\bm{\pi}_{i}}<x\big{)}\bigg{\\}}$
$\displaystyle=\leavevmode\nobreak\ $ $\displaystyle
T_{\mathrm{CI}}^{\bm{\pi}_{(k)}},$

where we recall $k=\lceil(1-\alpha)K\rceil$. Given this representation and by
noting that the permutation distribution is discrete, we see that
$T_{\mathrm{CI}}\leq q_{1-\alpha}$ holds if and only if

$\displaystyle\frac{1}{K}\sum_{\bm{\pi}_{i}\in\bm{\Pi}}\mathds{1}\big{(}T_{\mathrm{CI}}^{\bm{\pi}_{i}}<T_{\mathrm{CI}}\big{)}<1-\alpha,$

which is equivalent to $p_{\mathrm{perm}}>\alpha$. This completes the proof.

###  D.4 Proof of Proposition 2

Since we assume that $\phi_{\mathrm{perm},k}=0$ when $k=0$, it holds that

$\displaystyle\sup_{P_{X,Y,Z}\in\mathcal{P}_{0}^{\prime}}\bigg{\\{}\sum_{k=0}^{\infty}\mathbb{P}(N=k)\mathbb{E}_{P_{X,Y,Z}^{k}}[\phi_{\text{{perm}},k}]\bigg{\\}}\leavevmode\nobreak\
=\leavevmode\nobreak\ $
$\displaystyle\sup_{P_{X,Y,Z}\in\mathcal{P}_{0}^{\prime}}\bigg{\\{}\sum_{k=1}^{\infty}\mathbb{P}(N=k)\mathbb{E}_{P_{X,Y,Z}^{k}}[\phi_{\text{{perm}},k}]\bigg{\\}}$
$\displaystyle\leq\leavevmode\nobreak\ $
$\displaystyle\alpha+C_{1}\sum_{k=1}^{\infty}\mathbb{P}(N=k)k^{-\epsilon}.$

Furthermore, by transforming $k$ to $k=k^{\prime}+1$, we can simplify the
infinite sum as

$\displaystyle\sum_{k=1}^{\infty}\mathbb{P}(N=k)k^{-\epsilon}\leavevmode\nobreak\
=\leavevmode\nobreak\ $
$\displaystyle\sum_{k=1}^{\infty}\frac{n^{k}e^{-n}}{k!}k^{-\epsilon}=n\sum_{k^{\prime}=0}^{\infty}\mathbb{P}(N=k^{\prime})(k^{\prime}+1)^{-\epsilon-1}$
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\ $ $\displaystyle
n\mathbb{E}[(N+1)^{-\epsilon-1}],$

where we recall that $N\sim\text{Pois}(n)$. Now decompose and upper bound the
expectation by

$\displaystyle n\mathbb{E}[(N+1)^{-\epsilon-1}]\leavevmode\nobreak\
=\leavevmode\nobreak\ $ $\displaystyle
n\mathbb{E}[(N+1)^{-\epsilon-1}\mathds{1}\\{N\leq
n/2\\}]+n\mathbb{E}[(N+1)^{-\epsilon-1}\mathds{1}\\{N>n/2\\}]$
$\displaystyle\leq\leavevmode\nobreak\ $ $\displaystyle n\mathbb{P}(N\leq
n/2)+n\left[\frac{1}{n/2+1}\right]^{1+\epsilon}$
$\displaystyle\leq\leavevmode\nobreak\ $ $\displaystyle
ne^{-n/12}+2^{1+\epsilon}n^{-\epsilon},$

where the last inequality uses a Chernoff bound for a Poisson random variable
(e.g. Canonne,, 2017). This completes the proof of Proposition 2.

###  D.5 Proof of Lemma 7

Let $X$ have a Poisson distribution with parameter $\lambda>0$. Then by using
a Poisson tail bound (e.g. Canonne,, 2017), we have for $x>0$,

$\displaystyle\mathbb{P}(X\leq\lambda-x)\leq e^{-\frac{x^{2}}{\lambda+x}}.$

Now, by setting $x=\lambda/2$, the above inequality guarantees that

$\displaystyle\mathbb{P}(X\leq\lambda/2)\leq e^{-\frac{\lambda}{6}}.$ (76)

By the union bound along with (76), we have

$\displaystyle\mathbb{P}\Big{(}\min_{m\in[M]}X_{m}\leq
c/2\Big{)}\leq\sum_{m=1}^{M}\mathbb{P}(X_{m}\leq\lambda_{m}/2)\leq\sum_{m=1}^{M}e^{-\frac{\lambda_{m}}{6}},$

which proves the result.


# Microstructure-Resolved Degradation Simulation of Lithium-Ion Batteries in
Space Applications

Linda J. Bolay  Tobias Schmitt  Simon Hein  Omar S. Mendoza-Hernandez  Eiji
Hosono  Daisuke Asakura  Koichi Kinoshita  Hirofumi Matsuda  Minoru Umeda
Yoshitsugu Sone  Arnulf Latz  Birger Horstmann<EMAIL_ADDRESS>Institute of Engineering Thermodynamics, German Aerospace Center (DLR),
Pfaffenwaldring 38-40, 70569 Stuttgart, Germany  Helmholtz Institute Ulm
(HIU), Helmholtzstraße 11, 89081 Ulm, Germany  Institute of Electrochemistry,
University of Ulm, Albert-Einstein-Allee 47, 89081 Ulm, Germany  Institute of
Space and Astronautical Science, Japan Aerospace Exploration Agency (JAXA),
3-1-1 Yoshinodai, Chou-ku, Sagamihara, Kanagawa 252-5210, Japan  The Graduate
University of Advanced Studies (SOKENDAI), 3-1-1 Yoshinodai, Chou-ku,
Sagamihara, Kanagawa 252-5210, Japan  Global Zero Emission Research Center,
National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1
Umezono, Tsukuba, Ibaraki 305-8568, Japan  Research Institute of Energy
Conservation, National Institute of Advanced Industrial Science and Technology
(AIST), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan  Department of
Materials Science and Technology, Nagaoka University of Technology, 1603-1
Kamitomioka, Nagaoka, Niigata 940-2188, Japan

###### Abstract

In-orbit satellite REIMEI, developed by the Japan Aerospace Exploration
Agency, has been relying on off-the-shelf Li-ion batteries since its launch in
2005. The performance and durability of Li-ion batteries is impacted by
various degradation mechanisms, one of which is the growth of the solid-
electrolyte interphase (SEI). In this article, we analyse the REIMEI battery
and parameterize a full-cell model with electrochemical cycling data, computer
tomography images, and capacity fading experiments using image processing and
surrogate optimization. We integrate a recent model for SEI growth into a
full-cell model and simulate the degradation of batteries during cycling. To
validate our model, we use experimental and in-flight data of the satellite
batteries. Our combination of SEI growth model and microstructure-resolved 3D
simulation shows, for the first time, experimentally observed inhomogeneities
in the SEI thickness throughout the negative electrode for the degraded cells.

###### keywords:

Li-ion battery, solid-electrolyte interphase, REIMEI satellite, degradation
model, microstructure-resolved simulation, parameter identification

##  1 Introduction

Lithium-ion batteries are the technology of choice for various applications
like laptops, smartphones and electric vehicles. Since the early 2000s they
are considered applicable for space missions or more precisely for low earth
orbit (LEO) missions. The requirements for batteries in a LEO mission are to
be reliable for about 5 years with 30,000 cycles when charged for ca. 60
minutes and discharged for ca. 30 minutes. [1, 2, 3, 4]. Battery reliability
is hampered by the harsh conditions they have to withstand. Already during
lift-off they experience severe vibrations. In space they suffer from extreme
temperatures, radiation, vacuum, and microgravity [5, 6]. Thus, battery
satellites are quite costly (for a small scientific satellite: 4 million US
dollar [7]).

Failure of battery satellites may abort the satellite mission and can endanger
other missions through space debris. Therefore, determination and prediction
of battery state-of-health during operation is necessary. This is only
possible by remote control of the satellite system from a control station on
earth. The antenna receives the telemetry data of the battery and transmits
operation commands, while the satellite passes the ground station during a
small time window [7].

Typically, the cells are tested extensively prior to the mission start.
Cycling experiments are conducted over long periods of time, adjusted to the
planned mission. Additionally, the cells undergo various abuse tests. This
ensures safe operability during the whole mission.

If the mission changes for unforeseen reasons, the predictions for the aging
of the cells have to be adjusted. A long time might have passed, however,
since the start of the mission. Access to cells of the same kind than those
used in the mission might be difficult. For these reasons, it is essential to
have physics-based models available, which can predict the performance and
degradation of the cells for any given operation of the satellite. These
models are based on short term experiments. This yields the additional benefit
of shortening the time for tests prior to the mission start.

The major degradation process taking place in Li-ion batteries is the growth
of the solid-electrolyte interphase (SEI). This passivation layer builds on
the anode surface during the first cycles of the cell, when solvents from the
electrolyte react with Li-ions and electrons. This process prevents the
electrolyte from further reaction but does not passivate the reaction
completely. The SEI is growing continuously during storage as well as during
cycling. Thereby, Li-ions are consumed irreversibly, which causes capacity
loss of the cell.

The growth of the SEI is discussed extensively in literature, both in
experiment and in theory. Usually, either short-term growth, which considers
growth behavior during the first cycle, or long-term growth, which considers
continuous growth over months and years, is investigated. In this work, only
long-term degradation is discussed.

In experiments, a square-root-of-time ($\sqrt{t}$) behavior is observed for
long-term growth [8, 5]. Furthermore, experiments show that SEI growth is
accelerated during cycling compared to storage [8]. Attia et al. show that the
SEI is especially growing faster during charging of the cell [9]. Another
aspect is the inhomogeneous growth of the SEI, which is investigated on
nanoscale [10, 11, 12] and on cell level [13].

Short-term SEI growth, i.e. the initial build-up of the SEI during the first
cycles, is addressed by atomistic theories like density functional theory and
molecular dynamic simulations [14, 15]. The long-term growth behavior of the
SEI can be described with empirical models [16, 17]. But a better
understanding of the internal processes is given by physics-based models. For
these, several transport mechanisms are considered as the rate-limiting
process. The most important are electron conduction [18, 19], electron
tunneling [20, 21], reaction kinetics [22, 23], radical diffusion [24, 25],
and solvent diffusion [26, 27, 28, 29]. Single et al. compared all of these
transport processes to storage experiments and show that the most probable
mechanism for SEI growth during storage is radical diffusion [24]. Das et al.
model the asymmetric growth behavior which was shown in Ref. [9] by coupling
SEI growth to the Li-ion concentration in the SEI [19]. Von Kolzenberg et al.
extended the model of radical diffusion in Ref. [24] by an electron migration
effect during cycling, which also reproduces the asymmetric growth. They
describe the transitions between transport- and reaction-limited growth and
with that give a consistent explanation for the short-term and long-term
growth behavior [30]. A comprehensive review of SEI models is given by
Horstmann et al. in Ref. [31].

Until recently, electrode level continuum models had not reproduced the
experimentally observed SEI heterogeneity [26, 32]. In his PhD thesis, Schmitt
implemented a preliminary version of the model of Ref. [30] in the software
BEST [33] and simulated the growth behavior in a 3D resolved microstructure.
Thereby, it was shown that the SEI is growing inhomogeneously [34]. This was
the first time that experimentally observed inhomogeneities in the SEI
thickness could be shown with a physics-based model. Considering reaction-
limited growth, such heterogeneous SEI growth has also been recently analyzed
by Chouchane et al. [23].

In this article, we discuss how to incorporate the model of von Kolzenberg et
al. into the physics-based model for Li-ion batteries of Latz et al. [35] and
simulate the SEI growth in a pseudo-two-dimensional (P2D) framework and in 3D.
We are the first to show inhomogeneous long-term growth of the SEI in
microstructural-resolved simulations. To validate our model, we make use of
experiments and in-flight data of satellite REIMEI’s battery cells.

REIMEI is a small scientific satellite developed by the Japanese Aerospace
Exploration Agency (JAXA). It was launched in 2005 with the main scientific
mission to observe the aurora and demonstrate advanced satellite technology,
one of which is Li-ion batteries that use off-the-shelf pouch cells. In 2013
the aurora observation was terminated. The satellite operation was extended
with the main focus shifted to the prediction of performance and lifetime of
the on-board Li-ion batteries [6, 36].

For a profound analysis of the battery state, a physics-based model that
describes the cell degradation is required. However, only little is known
about the cells parameters, as a long time has passed since the start of the
mission. This makes the modeling a challenging task.

One part of this challenge is the parameterization of the P2D model. This is
addressed by different optimization approaches, where the error between
experiment and model simulation, depending on the parameters, is to be
minimized. Different approaches can be found in the literature. Forman et al.
use a genetic algorithm to identify the parameters [37]. Rahman et al. use a
particle swarm optimization algorithm [38]. Here, we extract some parameters
from CT images by image processing. The remaining parameters are determined
with a surrogate optimization. To the best of our knowledge, this optimization
method has not yet been used for battery models.

In sec. 2, we give a brief overview of REIMEI and its batteries. After that,
we describe the SEI model in detail in sec. 3, which we then use for
simulating the cycling and the degradation of the cells. For this purpose, in
sec. 4, we first determine the parameters of the fresh cell with an
optimization approach. Subsequently, in sec. 5 we determine the parameters of
the degradation model and discuss the results of the degradation and SEI
growth simulation.

##  2 Experimental Data of REIMEI Batteries

Satellite REIMEI, which flies on a polar low Earth orbit (LEO), has an orbital
period of 96 minutes. During the sunlight period, the batteries get charged
via the solar panels, while they discharge during the eclipse period to power
the loads [5]. This results in 15 cycles a day. To ensure a reliable
operability of 3 - 5 years, the temperature of the batteries is maintained at
$19-22^{\circ}$C [7, 6] and the depth of discharge around 20 % [5].

Two identical batteries are connected in parallel on board of the satellite
REIMEI, each consisting of seven pouch cells connected in series. The lithium-
ion cells are produced by NEC-Tokin Corporation. The positive active material
is spinel lithium manganese oxide (LMO) and the negative electrode consists of
graphite. The liquid electrolyte is 1 M LiPF6 in EC/DEC (3:7 by wt%) with
additives [6, 5]. The cells have a voltage range of 3.0 V - 4.1 V (4.2 V).
Their nominal capacity is 3 Ah. [8]

The in-flight operation of the cells depend on the loads in use. Given by the
on-board devices, the cells reach their end of life when the end-of-discharge
voltage (EoDV) falls below 3.75 V [5].

For the parameterization of the models, two kinds of terrestrial measurements
are used. The electric battery response was measured after the mission began
(see sec. 2.1). We identify parameters of the cell microstructure from CT
images (see sec. 2.2).

###  2.1 Terrestrial electrochemical measurements

Accompanying the REIMEI mission, Uno et al. performed terrestrial cycling
experiments [5] with the same kind of cells as those of the satellite. The
experiments span about 27,000 cycles, which is equivalent to five years. They
charged and discharged several cells with a cycling protocol similar to the
one of the satellite. To track the degradation of the cell, they measured the
remaining capacity and the EoDV of the cells.

In the experiments, the cells get discharged with a constant current (CC) of 1
A for 35 minutes, which corresponds to approximately 20 % DOD. Then they get
charged for 65 minutes with a constant current - constant voltage (CC-CV)
profile. The charging is performed at 1.5 A to the upper voltage limit of 4.1
V. After each discharging, the EoDV is measured. As the cells are degrading,
the EoDV falls below 3.75 V after several thousand cycles. At that point, the
upper voltage limit for the charging gets changed to 4.2 V to prolong the
remaining useful life of the cells. Before and during the cycling, they
measure the remaining capacity of the cells several times by discharging it
with a CC of 1 A to 3.0 V. We use these electrochemical measurements of Uno et
al. in sec. 4.1 to parameterize the P2D model. For the model we assume
constant temperature and constant pressure throughout the cell. This
assumption is based on the experiments in Ref. [5] and on the in-flight data
as can be seen in Ref. [39]. In sec. 5.1, the measurements are used to
parameterize the degradation.

In 2008, Brown et al. conducted comprehensive terrestrial experiments at JAXA.
They investigated the degradation of cycled and calendaric aged cells under
various conditions. For the cycling, they use the same cycling protocol than
Uno et al. and vary it regarding temperatures and discharge currents
respectively the depth of discharge (DOD). They examined full and half cells
before, during and after the aging process. Furthermore, they measured open-
circuit voltage (OCV) curves for both electrodes [8, 40]. The OCV curve
measurements of Brown et al. are used in sec. 4.1 for the P2D model.
Furthermore, our cycling protocols in sec. 5.2 are based on those of Browns et
al.

###  2.2 Computer tomography

One of the pouch cells of the kind used in the satellite was stored by JAXA.
Several years after the start of the mission, CT images of the cell, which was
stored at room temperature, were taken in various resolutions. Examples of the
CT images can be found in Fig. B.1 in the supporting information. From the CT
images we extract several parameters of the cell structure. These are the
thicknesses and porosities of the cell layers and the specific surface area of
the electrodes. These values are necessary to parameterize the model described
in sec. 3.

The thickness $L$ of the layers is determined using the CT images with 1
$\mu$m and 3.5 $\mu$m resolution. To separate the layers, we consider
grayvalues of voxels in the three dimensional CT image. Every type of layer in
the cell has a specific grayvalue distribution. We assign the median of
grayvalues to every voxel layer parallel to the cell layers. Hereby, we get a
one dimensional signal which is perpendicular to the cell layers. The changes
of grayvalue medians are determined by signal processing methods. These
represent the boundaries between cell layers. The thickness is then averaged
over several layers.

The porosities $\epsilon$ of the different electrodes are determined from a
representative section of the 1 $\mu$m resolution CT image. The histogram of
grayvalues within this volume is determined and approximated by three normal
distributions. For the cathode, the histogram and the corresponding
distribution approximation can be seen in Fig. B.2 in the supporting
information. The integral of these distributions yield the volume fractions of
the phases in the electrode, the pore space, the active solid phase, and the
passive solid phase, which consists of binder and conductive material. From
the distributions, we get a binarization of the electrode volume using k-means
clustering. We use a simplified version of the method described in Ref. [41].
Here, we only want to classify the voxels as pore space or solid phase. The
classification is assumed sufficient, when the grayvalue distributions classes
have the same integral than the approximated distributions. The binarization
is then used to determine the specific surface area $A_{\text{spec}}$ of the
electrodes. This is done with the software GeoDict [42].

In the case of the separator, we proceed in the same way. However, here the
histogram of the grayvalues is approximated by only two normal distributions.
The porosity is then again obtained from the integral of the distributions.

For the full cell area $A_{\text{cell}}$, the width, length, and number of
cell layers are needed. The length is taken from the CT image with 140 $\mu$m
resolution, which shows the whole cell. The width and number of layers are
taken from the CT image with 24 $\mu$m resolution, which shows the upper part
of the cell. We get $A_{\text{cell}}=0.139\ \mathrm{m^{2}}$. The other
identified parameters can be found in Table B.2 in the supporting information.

##  3 Computational Model

In order to understand and simulate the processes taking place in the cell, a
physics-based model for the short-term cycling as well as the long-term
degradation behavior of the cells is necessary. The long-term degradation of
the cell is mainly determined by capacity fading, which is caused by the loss
of Li-ion inventory. To quantify the long-term capacity fading under cycling
as well as under storage conditions, a new physics-based model has been
developed. Von Kolzenberg et al. describe the model in detail in Ref. [30].
Here, we summarize it and embed it into the thermodynamic consistent transport
theory for Li-ion batteries of Latz et al. [35].

a)b)c)

Figure 1: Lithium transport on different scales. a) Li-ion transport in
electrolyte between electrode particles. b) Li-ion current at electrode
interface. c) SEI growth during calendar aging (A) and during cycling (A + B).

This theory models transport and reactions of Li-ions in electrolyte and
electrode particles on the microstructural scale of batteries. Several
equations describe how the inner states, like Li-ion concentration or
electrical potential, are distributed in the cell and how they evolve over
time. The corresponding equations can be found in the supporting information
[35]. For the microstructure-resolved simulations, first, the morphology of
the electrodes needs to be reconstructed from tomographic image data or
simulated with given morphology characteristics. For example, this is done
with stochastic geometry methods by Hein et al. [43]. In our research group,
this 3D representation of the Li-ion cell is then used with the software BEST
to perform spatially resolved electrochemical simulations [33, 44]. In Ref.
[45] Hein et al. use these simulations to study and analyze the heterogeneous
surface effects (see also sec. 5).

Using volume averaging techniques the transport theory by Latz et al. can be
uspcaled into a pseudo-multidimensional model [46] like the one dimensional
P2D model, which is illustrated in Fig 1. This model has first been described
by Doyle et al. [47]. It has since been used a lot to describe Li-ion battery
dynamics [48, 37, 38]. In Fig. 1a the volume-averaged transport of Li-ions in
the electrolyte between electrodes is shown. The interfacial reaction kinetics
between electrolyte and electrode is depicted in Fig. 1b. It is modeled by a
Butler-Volmer rate, with the overpotential $\eta_{\text{s}}$ dependent current
density $j_{\text{Li}^{+}}(\eta_{\text{s}})$ (see eq. A.5 and A.6 in the
supporting information).

When a Li-ion cell gets charged for the first time, the electrolyte gets
reduced at the anode surface due to the low graphite potential. The products
build a passivation layer, the solid-electrolyte interphase (SEI), which
prevents the electrolyte from further reduction. However, the SEI continues
growing because of a leakage of electrons from the anode through the SEI to
the electrolyte, where it reacts with Li-ions and solvents. This reaction
consumes Li-ions irreversibly, which is one of the main reasons for capacity
fading.

The relation between irreversible capacity loss $Q_{\text{SEI,irr}}$, SEI
thickness $L_{\text{SEI}}$, and the growth causing flux $N_{\text{SEI}}$ is
given as

$\partial_{t}Q_{\text{SEI,irr}}=\frac{AFs_{\text{SEI}}}{V_{\text{SEI}}}\cdot\partial_{t}L_{\text{SEI}}=AF\cdot
N_{\text{SEI}},$ (1)

where $N_{\text{SEI}}\geq 0$, $A$ is the surface area of the anode,
$s_{\text{SEI}}$ is the mean stoichiometric coefficient of Li in the SEI
formation reaction, $V_{\text{SEI}}$ is the mean partial molar volume of the
SEI, and $F$ is the Faraday constant.

Single et al. provided evidence that electron diffusion through the SEI is
causing calendaric aging [24]. These electrons diffuse through localized
state, e.g., neutral Li-interstitials diffusion [49]. The corresponding flux
is deduced from Fick’s law and is given as

$N_{\text{SEI,diff}}=D^{e^{\text{-}}}\cdot\frac{c^{e^{\text{-}}}}{L_{\text{SEI}}},$
(2)

where $D^{e^{\text{-}}}$ is the diffusion coefficient and $c^{e^{\text{-}}}$
is the interstitial concentration at the anode-SEI interface, which is
dependent on the electrochemical potential of the anode relative to Li. This
process is depicted as process A in Fig. 1c.

The capacity loss increases when the cell is cycled. Attia et al. show that
the SEI grows almost exclusively during lithiation of the anode [9]. This
phenomenon is also modeled by an electron flux, which is influenced by the Li-
ion current through the SEI. This can be seen in Fig. 1c process B. To take
the influence of the Li-ion current into account, an electron migration flux
is incorporated and the SEI flux is derived from the Nernst-Planck equation

$N_{\text{SEI}}=D^{e^{\text{-}}}\cdot\frac{c^{e^{\text{-}}}}{L_{\text{SEI}}}\
-\ zF\kappa_{\text{SEI}}^{\text{e}^{\text{-}}}\nabla\Phi,$ (3)

with the valence $z=-1$, the conductivity of the localized electrons
$\kappa_{\text{SEI}}^{\text{e}^{\text{-}}}$, and the potential gradient
$\nabla\Phi$ in the SEI caused by the Li-ion flux. According to [24], the
electron concentration in the SEI is approximated by

$c^{e^{\text{-}}}=c^{{e^{\text{-}}}}_{0}\cdot\exp\left(-\frac{F}{RT}\eta_{\text{SEI}}\right),$
(4)

with the interstitial concentration $c^{{e^{\text{-}}}}_{0}$ at $0V$ anode
potential, the universal gas constant $R$, and the temperature $T$.
Furthermore, the potential gradient is approximated by

$\nabla\Phi=\frac{-U_{\text{SEI}}}{L_{\text{SEI}}},$ (5)

where the potential drop is defined as

$U_{\text{SEI}}=\frac{L_{\text{SEI}}}{\kappa_{\text{Li}^{+}}^{\text{SEI}}}\cdot
j_{\text{Li}^{+}},$ (6)

with the conductivity $\kappa_{\text{Li}^{+}}^{\text{SEI}}$ of Li-ions in the
SEI.

With the help of the Nernst-Einstein equation, the conductivity
$\kappa_{\text{SEI}}^{\text{e}^{\text{-}}}$ is approximated by the diffusion
coefficient

$D^{e^{\text{-}}}=RT\kappa_{\text{SEI}}^{\text{e}^{\text{-}}}/c^{e^{\text{-}}}\approx
2RT\kappa_{\text{SEI}}^{\text{e}^{\text{-}}}/c^{{e^{\text{-}}}}_{0}$ (7)

and eq. 3 is rearranged to

$\begin{split}N_{\text{SEI}}\ =\ &\frac{D^{e^{\text{-}}}\cdot
c^{{e^{\text{-}}}}_{0}}{L_{\text{SEI}}}\cdot\exp\left(-\frac{F}{RT}\eta_{\text{SEI}}\right)\\\
&\qquad\qquad\cdot\left(1-\frac{F}{2RT}U_{\text{SEI}}\right).\end{split}$ (8)

The overpotential for the reaction is given as

$\eta_{\text{SEI}}=\Phi_{\text{s}}-\varphi_{\text{e}}-U_{\text{SEI}},$ (9)

where $\Phi_{\text{s}}$ is the anode potential and $\varphi_{\text{e}}$ is the
electrochemical potential of the electrolyte.

The growth of the SEI not only affects the cells capacity but also its inner
resistance. This is taken into account in the Butler-Volmer rate for the
intercalation into graphite, where the overpotential is supplemented by the
potential drop in the SEI and becomes

$\tilde{\eta}_{\text{s}}=\eta_{\text{s}}-U_{\text{SEI}}.$ (10)

The overpotential of the Butler-Volmer term is given in Ref. [35] (see also
eq. A.6 in the supporting information).

##  4 Battery Cycling

We simulate battery cycling with the volume-averaged P2D model. Several
battery parameters are required for this. The parameters describing the
microstructure morphologies are extracted from CT data (see sec. 2.2). The
concentration-dependent electrolyte parameters are taken from Ref. [50]. The
OCV curves are fitted to the measured curves in Ref. [40] (see eq. A.8 and A.9
in the supporting information). The remaining cell parameters are determined
with an optimization algorithm, where experimental data of a fresh cell from
Ref. [5] are used. The parameters can be found in Table B.1 and B.2 in the
supporting information.

###  4.1 Parameterization

The parameters of the cell model, that are not known from literature or
measurements, are determined by an optimization, where the deviation between
experiment and simulation, given as an objective function, is minimized.

Those parameters are the initial SOC on the anode side and for both electrodes
the maximal concentration of Li-ions in the active material
$c_{\text{s,max}}$, the diffusion coefficient of Li-ions in the active
material $D_{\text{s}}$, and the rate constant $k$.

The optimization algorithm should find the global minimum and preferably it is
parallelizable as the evaluation of the time-evolution of our physics-based
model is time-consuming. To this aim, Forman et al. [37] used a genetic
algorithm to identify the parameters of the P2D model and Rahman et al. [38]
used particle swarm optimization.

Here, we will use the surrogate optimization, which is a stochastic
optimization algorithm. The benefits of this algorithm are that its
convergence to the global optimum is proven and that it can run in parallel
[51]. To the best of our knowledge, this is the first time that the surrogate
optimization has been used to identify the parameters of the P2D model.

####  4.1.1 Surrogate optimization

We briefly outline the algorithm for surrogate optimization. For further
details, we refer to Refs. [51, 52]. In a first phase, the algorithm evaluates
the objective function for a given number of points, which are randomly chosen
in the given parameter space. These points get interpolated by radial basis
functions to build a surrogate function. In the next phase, the point with the
smallest value is found. Around this point, multiple points are sampled and an
additional merit function is evaluated at these points. The merit function
consists of the surrogate and a function that describes the distance of
sampled points to already evaluated points. This ensures that the surrogate
function is minimized while searching in new regions. At the point, where the
merit function has its minimal value, the objective function is evaluated and
the surrogate function is updated. This procedure is repeated until a stopping
criterion is reached. Subsequently, the surrogate gets rejected and a new
surrogate is created with new random points, while the best fit of the
previous surrogate is kept for comparison. This overall procedure of
generating new surrogate functions can be repeated until the global optimum is
found. Usually the search for the optimum is stopped after a predefined time.

####  4.1.2 Fitting

The objective function for the optimization is a weighted sum of the discharge
curve error err${}_{\text{V}}$ and the EoDV error err${}_{\text{EoDV}}$. Here,
we chose the L1-Norm to measure the discharge curve error:

$\text{err}_{\text{V}}(\theta)=\int_{T}|V_{\text{exp}}(t)-V_{\text{sim}}(t;\theta)|dt,$
(11)

where $V_{\text{exp}}$ is the experimental voltage and $V_{\text{sim}}$ is the
voltage output of the simulation over the discharge time T for a given
parameter vector $\theta$. The EoDV error gets measured after a few cycles,
since the cycling range has to level off after full discharge in the
experiment as well as in the simulation.

The parameters are optimized in a reasonable parameter space. The results of
the fitting can be found in Table B.2 in the supporting information. As the
algorithm stopped due to a time stopping criterion, it cannot be assumed that
the resulting parameters are the optimal parameters. Also, it has not been
shown yet that the parameters of the P2D model are identifiable at all.
Bizeray et al. showed the identifiability for the single particle model, which
is a reduced version of the P2D model, in Ref. [53]. In Ref. [54], Li et al.
analysed the sensitivity of the P2D model parameters. They showed that only
part of the parameters is identifiable.

The parameters of the battery cycling and those of the degradation are
correlated. Thus, the parameterization of both models needs to be linked to
each other. For that purpose, the degradation parameters are taken into
account in this section.

Protocol | Charge (CC-CV) | Discharge (CC) | Temperature
---|---|---|---
P1 | 1.5 A / 4.1 V | 0.88 A 0.74 A | 20°C
(in-flight) | 63 min | 15 min 19 min |
P2 | 1.5 A / 4.1 V (4.2 V) | 1.0 A | 25°C
(P2D) | 65 min | 35 min |
P3 | 1.5 A / 4.1 V | 1.0 A | 25°C
(3D) | 65 min | 35 min |
P4 | 1.5 A / 4.1 V | 2.0 A | 25°C
(3D) | 65 min | 35 min |
P5 | 3.0 A / 4.1 V | 2.0 A | 25°C
(3D) | 65 min | 35 min |

Table 1: Cycling protocols used in experiments and simulations.

###  4.2 Simulation and validation with in-flight data

Figure 2: Three cycles of satellite REIMEI’s battery. In-flight data and
simulation corresponding to the satellite cycling protocol.

To validate the determined parameters, the satellite in-flight data are used.
On orbit, the cells get charged via the solar panels with a CC-CV profile.
During the eclipse period, they get discharged to power the cameras for the
aurora observation. There are two different discharge profiles, depending on
whether the aurora of the north or the south pole is observed. The cycling
protocols are described in detail in Ref. [8].

In Fig. 2 the simulation of the in-flight data is depicted. The data were
taken from the early life of the battery in 2005. The cycling protocol, that
has been used during this period of satellite operation, was for north pole
observation. It can also be found in Table 1 (P1). In the figure we show the
agreement between simulation and experiment over the cause of three cycles. It
can be seen that the discharge voltage curve is met by the simulation.

##  5 Battery Degradation

To determine the parameters of the degradation model, long-term cycling is
simulated with the volume-averaged P2D model (see sec. 5.1). Subsequently, 3D
simulations are performed with the determined parameters. Different cycling
protocols are used to examine the influence of charge and discharge currents
on the SEI growth in the microstructure (see sec. 5.2).

The degradation with the P2D model is implemented in MATLAB [55]. The 3D model
is implemented with the software BEST [33]. Both kinds of simulations are run
on the high performance computer cluster JUSTUS 2 which belongs to the bwHPC.

###  5.1 Parameterization with P2D model

We simulate the long-term cycling protocol according to the experiments of Uno
et al. [5]. They cycled the cells about 27,000 times. After around 24,000
cycles, when the EoDV reaches 3.75 V, the charge voltage is changed from 4.1 V
to 4.2 V. At 12 different points in time, the cells are fully discharged to
3.0 V to measure the remaining capacity. This can also be seen in Fig. 3. The
charge-discharge protocol used here can be found in Table 1 (P2).

The parameters, that are examined in this subsection, are the initial SEI
thickness $L_{\text{SEI,0}}$, the diffusion coefficient $D^{e^{\text{-}}}$ of
localized electrons in the SEI, and the conductivity
$\kappa_{\text{Li}^{+}}^{\text{SEI}}$ of Li-ions in the SEI. The other
parameters, like the mean partial molar volume of the SEI, contribute to the
model as multiplicative factors and hence are not identifiable. Those
parameters are taken from literature. All parameters of the degradation model
are given in Table B.4 in the supporting information.

The determination of the optimal parameter set is a multi-objective
optimization problem, where the capacity fading as well as the EoDV error have
to be minimized. As the simulation of around 27,000 cycles is quite expensive
in terms of computation time, a sophisticated optimization algorithm is
inappropriate. For that reason, a fixed parameter space is used, where the
parameters are uniformly distributed in a reasonable range. The cycling is
then simulated for all combinations of parameters simultaneously on the
computer cluster. The optimal parameter set of the fixed parameter space is
chosen.

In Fig. 3a the capacity fading over around 27,000 cycles is depicted. Fig. 3b
shows the corresponding EoDV. At around 24,000 cycles, when the charge voltage
is raised from 4.1 V to 4.2 V, the degradation accelerates. For the
simulation, we first consider the red lines. In both graphs, we see that the
degradation gets overrated at the beginning of the cycling. This can be
explained by a not optimal choice of cell parameters. As we already described
in sec. 4.1.2, the optimization of the parameters is a difficult task due to
computation time and the identifiability of the parameters. After around 2,000
cycles, over a wide range of the cycling, the trend of the simulation
corresponds quite well to that of the experiment, especially in the EoDV. At
the end, after about 20,000 cycles, the simulation underrates the degradation
of the experiment. At this late period of cycling, there are probably
additional degradation processes taking place in the cell, that are not
covered by our SEI growth model. This could be for example the cracking of
anode particles as described in Ref. [56] or Li plating. These processes lead,
among other things, to a sudden increase of anode surface, which accelerates
the SEI growth. Mendoza-Hernandez et al. [36] observe indicators for plating
in the satellite in-flight data. These occur more frequently at a later stage
of the cycling. Plating is responsible for or accelerates a number of
degradation processes. This may be another reason why accelerated aging is
observed that cannot be explained by SEI growth.

The difference between the new model of von Kolzenberg et al. [30] and the
model of Single et al. [24] is the additional consideration of the Li-ion flux
through the SEI during charging. This causes the migration of electrons due to
an electric potential drop in addition to diffusion. To discuss the influence
of the electron migration flux for SEI growth, we perform simulations of the
degradation with and without the term that describes the electron migration
($zF\kappa_{\text{SEI}}^{\text{e}^{\text{-}}}\nabla\Phi=0$) in eq. 3. The
simulation result is depicted in Fig. 3 as the yellow lines. It can be seen
that the degradation decreases significantly in that case already after a few
thousand cycles. Also the trend of the simulation does not correspond to the
experiments in both graphs.

a)

b)

Figure 3: Two simulations of degradation experiment with satellite cycling
protocol. Both simulations with equal parameters, one with and the other
without electron migration flux. a) Capacity and b) end of discharge voltage
(EoDV) of simulation and experiment.

###  5.2 3D microstructure-resolved simulations

a)b)c)d)e)f)

Figure 4: SEI thickness of anode particles connected to current collector for
two cycling protocols at three time points. Left column (a, b, and c) shows
cycling protocol P3, right column (d, e and f) shows P4 (cf. Table 1). a) and
d) 8 cycles. b) and e) 136 cycles. c) and f) 672 cycles.

The microstructure-resolved SEI growth is simulated in 3D with the same
degradation parameters as determined before with the P2D model. The cell and
degradation parameters for the 3D simulations can be found in Table B.3 and
B.4 in the supporting information. There are three different cycling protocols
used for the 3D simulations. They differ in charge and discharge currents to
investigate how the current influences the SEI growth behavior. The protocols
(P3 - P5) can be found in Table 1. P3 is equivalent to the cycling protocol
used with the P2D model (P2). In P4 the discharge current is doubled. This is
based on the experiments in Ref. [8]. In P5, both the charge current and the
discharge current are twice as large as in P3. The intervals for charging and
discharging are the same in all protocols and correspond to the typical
satellite battery cycling protocols.

The 3D simulations reveal an inhomogeneous growth of the SEI. In Fig. 4 the
results of the simulations with two different cycling protocols at three
points in time are shown. We see anode particles connected to a current
collector. The color represents the thickness of the SEI after a certain
amount of cycles. In the left column the cell is cycled with protocol P3, the
right one is cycled with P4. Fig. 4a and d show the SEI thickness after 8
cycles, Fig. 4b and e after 136 cycles, and Fig. 4c and f after 672 cycles.

For all states shown in Fig. 4, there are coexisting regions with a thicker
SEI and regions with a thinner SEI. In Fig. 4a and d we see that, at the
beginning of the cycling, the particles near the separator have a thicker SEI
than those near the current collector. Fig. 4b, c, e, and f show that after
several cycles, the inhomogeneity spreads over the whole anode. When comparing
Fig. 4b and e or Fig. 4c and f, we see that this happens sooner for the
protocol with the higher discharge current.

After many cycles, there are particular domains of the anode, where the SEI is
noticeably thicker. In Fig. 4f, we observe that these are the thinner parts of
the anode. Especially the peak-like structures have the thickest SEI.

The inhomogeneous growth is caused by the varying SOCs through the anode and
consequently varying overpotentials at the anode SEI interface. The thinner
parts of the particles reach higher SOCs faster, when charging the cell.

a)

b)

Figure 5: Variation of SEI thickness in 3D simulation for three different
cycling protocols (cf. Table 1). a) Stepwise growth behavior in each cycle. b)
SEI thickness and relative maximum deviation (dashed lines) over a long
cycling period.

For a quantitative analysis of the growth and variation of SEI thickness, its
mean, minimum, and maximum, throughout the anode, at every computed time step,
is depicted in Fig. 5. We investigate here the three cycling protocols P3 - P5
at a short period of 25 cycles, which corresponds to about two days, and a
long period of 400 - 800 cycles, which corresponds to approximately 1 - 2
months.

When regarding the SEI thickness at the short period, as shown in Fig. 5a, we
observe a stepwise growth behavior. During charging of the cell or more
precisely during lithiation of the anode, the SEI is growing faster than
during delithiation. These simulations reflect that the growth rate depends on
the current density, which has been observed experimentally in Ref. [9].

Furthermore, when comparing the cycling protocols in Fig. 5a, we note
differences in SEI thickness and its inhomogeneity. This becomes even clearer
in the longer period in Fig. 5b. The differences in the total length of the
cycling simulations are due to numerical instabilities, when using higher
currents. When comparing the mean SEI thickness of the different protocols, we
observe that for P3 and P4, which differ in the discharge current, there is no
significant difference in the thickness. In contrast, the SEI is growing
faster for P5, which has a higher charge current than the other two protocols.
This is also in accordance with the growth behavior observed in Ref. [9] and
rationalized in Ref. [30].

The relative maximum deviation, i.e. the difference between minimum and
maximum divided by the mean of the thickness, is shown as dashed lines in Fig.
5b. The simulations with doubled currents (P4 and P5) show a significantly
larger deviation, i.e. a more inhomogeneous SEI growth. We also see that the
deviation is already remarkable for the first few cycles and increases only
slightly thereafter.

##  6 Conclusion

In this work, we present models for the short-term cycling and the long-term
degradation behavior of a Li-ion cell. The modeling is based on experimental
work and in-flight data of the batteries of small scientific satellite REIMEI,
which was developed by JAXA. To describe the cycling behavior of the cell, we
determine the parameters of the thermodynamic consistent transport theory of
Latz et al. [35], which we incorporate in a P2D model. The parameters are
determined with a surrogate optimization, in order to fit cycling experiments
of the batteries. Subsequently, the model is validated by the satellite in-
flight data. We could show that the surrogate optimization is appropriate to
find parameters to fit simulation and experiments of a few cycles. But this
method has the same limitation as other optimization methods, which lies in
the unidentifiability of a few parameters of the P2D model. One could use the
surrogate optimization to fit a wider range of the experiment, e.g., the first
1000 cycles. This should improve the accuracy of the estimated parameters, and
all degradation parameters could also be included in the optimization process.
However, this would need significantly longer computation times.

Furthermore, we give an overview of a model for the cell degradation, caused
by SEI growth during storage as well as during cycling. This model describes
the increased growth rate during lithiation of the anode. We determine the
unknown parameters of the model with the help of long-term satellite battery
experiments, where we focus on capacity fading and end-of-discharge voltage of
the cells.

With the same model and parameters, we simulate cell degradation in 3D. The
simulations show an inhomogeneous growth of the SEI, which depends on the
cycling currents. This provides a deeper understanding of the processes, which
take place in a Li-ion battery.

Apart from the capacity fading due to SEI growth, there are more degradation
mechanisms, which have not been considered in this work. A degradation
mechanism, which can cause severe damages in the cell and also contributes to
capacity fading, is the deposition of metallic lithium on the anode surface.
It is referred to as Li plating. In Ref. [36], Mendoza et al. analyse the
battery in-flight data of satellite REIMEI over an interval of 14 years. They
observe a knee in the discharge voltage curves, which they suggest to be due
to plating. Hein et al. modeled Li plating and simulated it in a
microstructure-resolved framework [45].

In this study we chose physics-based continuum models for our simulations as
these are able to reproduce the processes taking place in Li-ion batteries and
make them more comprehensible. Apart from that, there are a lot more
approaches to model the battery behavior. In Ref. [57] Shi et al. give a
comprehensive overview on multi-scale computation methods. Similarly, Wang et
al. review multi-scale models of SEI formation [14]. To parameterize our
continuum models, we employ experiments, since atomistic models do not yet
provide the necessary accuracy.

In future work, we will incorporate plating into our model. This work can be
the foundation for performing state estimation and reliably predicting the
state of health of satellite batteries in orbit.

## Acknowledgments

This work was supported by the German Aerospace Center (DLR). The authors
acknowledge support by the state of Baden-Württemberg through bwHPC and the
German Research Foundation (DFG) through grant no INST 40/575-1 FUGG (JUSTUS 2
cluster). This work contributes to the research performed at CELEST (Center
for Electrochemical Energy Storage Ulm-Karlsruhe).

## Declaration of competing interest

The authors declare to have no competing interests.

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