# Voltage Hysteresis of Silicon Nanoparticles: Chemo-Mechanical Particle-SEI
Model

Lukas Köbbing  Institute of Engineering Thermodynamics, German Aerospace
Center (DLR), Wilhelm-Runge-Straße 10, 89081 Ulm, Germany  Helmholtz Institute
Ulm (HIU), Helmholtzstraße 11, 89081 Ulm, Germany  Arnulf Latz  Birger
Horstmann<EMAIL_ADDRESS>Institute of Engineering Thermodynamics,
German Aerospace Center (DLR), Wilhelm-Runge-Straße 10, 89081 Ulm, Germany
Helmholtz Institute Ulm (HIU), Helmholtzstraße 11, 89081 Ulm, Germany
Institute of Electrochemistry, Ulm University, Albert-Einstein-Allee 47, 89081
Ulm, Germany

###### Abstract

Silicon is a promising anode material for next-generation lithium-ion
batteries. However, the volume change and the voltage hysteresis during
lithiation and delithiation are two substantial drawbacks to their lifetime
and performance. We investigate the reason for the voltage hysteresis in
amorphous silicon nanoparticles covered by a solid-electrolyte interphase
(SEI). Concentration gradients inside the nanoscale silicon can not produce
the massive stresses necessary to cause the reported voltage hysteresis. Our
chemo-mechanical model shows that plastic deformation of the stiff, inorganic
SEI during lithiation and delithiation reproduces the observed silicon open-
circuit voltage hysteresis. Additionally, the viscous behavior of the SEI
explains the difference between the voltage hysteresis observed at low
currents and after relaxation. We conclude that the visco-elastoplastic
behavior of the SEI is the origin of the voltage hysteresis in silicon
nanoparticle anodes. Thus, consideration of the SEI mechanics is crucial for
further improvements.

Keywords: lithium-ion batteries, solid-electrolyte interphase, battery aging,
silicon anode, voltage hysteresis, plastic flow, SEI mechanics

##  I Introduction

The improvement of lithium-ion batteries is essential for facing the pressing
global challenges. Due to their high theoretical capacity, pure silicon anodes
are a promising candidate for the next generation of lithium-ion batteries [1,
2, 3, 4, 5]. Therefore, research makes an effort to investigate this
advantageous and abundant material.

Silicon confronts research with two challenging features on the way to
commercial silicon anodes. The first is the significant expansion and
constriction with a volume change of up to $300\%$ during lithiation and
delithiation, respectively. The deformation leads to the cracking of large
silicon particles and the formation of networks with nanometer-sized silicon
particles [6]. However, no cracking occurs below a critical particle diameter
of about $150\,\mathrm{nm}$ [7]. Due to the superior mechanical stability,
recent research concentrates on investigating nano-structured silicon anodes
[8, 9, 10, 11, 12]. Additionally, experiments observe a volume hysteresis,
i.e. a difference in the silicon particle or anode volume between lithiation
and delithiation [13, 14, 15, 16].

The second major issue is the voltage hysteresis of silicon anodes observed in
experiments for various silicon structures, namely thin films [17, 18, 19, 20,
21, 13, 22, 23], nanowires [24, 25], and nanoparticles [26, 27, 28, 29]. The
voltage hysteresis considerably reduces energy efficiency and leads to
detrimental heat generation during cycling [30]. Thus, the hysteresis
phenomenon is a very important challenge for the commercialization of silicon-
anode lithium-ion batteries. Most literature reports the hysteresis of pseudo-
open-circuit voltages at currents smaller than $\mathrm{C}/10$. However, even
for the open-circuit voltage measured by the galvanostatic intermittent
titration technique (GITT) after relaxation periods, a reduced, but still
significant voltage hysteresis remains [20, 28, 29]. We emphasize that the
measurements reveal a clear difference in the size of the voltage hysteresis
observed for small currents and after relaxation periods.

Phase transformations can explain a voltage hysteresis for crystalline silicon
or the first cycle of amorphous silicon anodes [31, 32, 33, 34]. However, also
amorphous silicon anodes show a voltage hysteresis after the first cycle. In
this case, the literature commonly considers plastic flow of silicon as the
reason for the hysteresis phenomenon. In thin-film silicon electrodes, massive
stresses arise naturally due to the restricted expansion in the in-plane
direction [35, 36, 22]. For micron-sized silicon particles or high applied
currents, the slow diffusion in silicon can lead to concentration gradients
during lithiation and delithiation, inducing a voltage hysteresis [37, 38,
39]. Nevertheless, the reason for substantial stresses inside amorphous
silicon nanoparticles during slow lithiation and delithiation remains unclear.

Similarly to other anode materials, the solid-electrolyte interphase (SEI)
naturally covers silicon particles. The SEI protects the anode from the
electrolyte [40, 41] and grows via electron transport from the anode through
the SEI [42, 43]. Additionally, between the silicon particle and the SEI,
there is a native silicon oxide layer influencing the interface between
particle and SEI [44]. Research invests much effort into characterizing and
improving the SEI on silicon anodes [45, 46, 47, 48, 49, 50, 51]. Due to the
considerable volume changes of silicon anodes, the plasticity and cracking of
the SEI on silicon deserve particular interest [52, 53, 54, 55, 56].

As the SEI deforms due to the chemical expansion and shrinkage of the
particle, considerable strains and stresses occur inside the SEI. Certain
studies even claim that the compressive stress of the SEI acting on the
particle might be beneficial to avoid particle cracking [57, 58, 4]. Further,
we note that a carbon coating on top of the silicon particle reduces the
observed voltage hysteresis of the silicon anode [27]. The reason might be a
different SEI composition on carbon and silicon. In conclusion, we propose
that the composition and stress of the SEI have a crucial impact on the
lithiation behavior and, particularly, on the voltage hysteresis of silicon
particles.

In this paper, we consider a visco-elastoplastic SEI model based on non-
equilibrium thermodynamics [59]. We evaluate the influence of the stress
generated by the SEI on the lithiation and delithiation behavior of an
amorphous silicon nanoparticle. Our model can faithfully reproduce the broadly
recognized but empirical Plett model. Furthermore, it allows to investigate
the reason for the difference between the voltage hysteresis for small
currents and GITT data points. Finally, we discuss the influence of the
mechanical parameters of the SEI and the silicon nanoparticle on the silicon
voltage hysteresis.

##  II Theory

When the silicon anode particle expands and contracts, distinct strains emerge
inside the SEI as illustrated in Fig. 1. Elastic deformations lead to
significant stresses in tangential and radial directions. The radial stress
component impacts the stress inside the particle, which crucially influences
the silicon potential. We discuss a chemo-mechanical silicon particle model
and a visco-elastoplastic SEI model.

Figure 1: Scheme of the radial stress inside the SEI during lithiation and
delithiation of a silicon nanoparticle.

###  II.1 Silicon particle model

Our silicon particle model is based on the work of Kolzenberg et al.
complemented by plasticity [56, 60]. During lithiation and delithiation, the
silicon particle experiences significant expansion and shrinkage,
respectively. The deformation tensor
$\mathbf{F}=\partial\vec{x}/\partial\vec{X}_{0}$ describes the transformation
from the undeformed Lagrangian frame to the deformed Eulerian frame [61]. We
use the large deformation approach to describe the deformation inside the
silicon particle [62]. Therefore, the total deformation $\mathbf{F}$
consisting of a chemical deformation $\mathbf{F}_{\mathrm{ch}}$, an elastic
deformation $\mathbf{F}_{\mathrm{el}}$, and a plastic deformation
$\mathbf{F}_{\mathrm{pl}}$ reads

$\mathbf{F}=\mathbf{F}_{\mathrm{pl}}\mathbf{F}_{\mathrm{el}}\mathbf{F}_{\mathrm{ch}}.$
(1)

The chemical deformation $\mathbf{F}_{\mathrm{ch}}$ is described by the
concentration of lithium $c_{\mathrm{Li,0}}$ inside the silicon particle in
the Lagrangian frame according to

$\mathbf{F}_{\mathrm{ch}}=\lambda_{\mathrm{ch}}\mathbf{Id}=(1+v_{\mathrm{Li}}c_{\mathrm{Li,0}})^{1/3}\mathbf{Id},$
(2)

where $v_{\mathrm{Li}}$ is the molar volume of lithium inside silicon.

The lithiation behavior and the mechanical deformations are derived in our
model based on non-equilibrium thermodynamics. Our model builds up on the free
energy according to the first law of thermodynamics. In section SI in the
supporting information, we discuss non-negativity of the entropy production
$\mathcal{R}\geq 0$ in our model stated in Eq. (S12), which means accordance
with the second law.

The free energy density $\rho_{0}\varphi$ of the silicon anode consists of a
chemical contribution depending on the lithium concentration and a mechanical
contribution due to elastic deformation. The free energy of the silicon anode
reads

$\displaystyle\rho_{0}\varphi=$
$\displaystyle-\int_{0}^{c_{\mathrm{Li}}}FU_{0}\left(c_{\mathrm{Li}}^{\prime}\right)\mathrm{d}c_{\mathrm{Li}}^{\prime}$
(3)
$\displaystyle+\frac{1}{2}\left(\lambda_{\mathrm{Si}}\left(\operatorname{tr}\left(\mathbf{E}_{\mathrm{el}}\right)\right)^{2}+2G_{\mathrm{Si}}\operatorname{tr}\left(\mathbf{E}_{\mathrm{el}}^{2}\right)\right)$

with the Faraday constant $F$ and the ideal open-circuit voltage of silicon
$U_{0}$. The elastic strain tensor $\mathbf{E}_{\mathrm{el}}$ in Eq. 3 is
defined by

$\mathbf{E}_{\mathrm{el}}=\frac{1}{2}\left(\mathbf{F}_{\mathrm{el}}^{\mathrm{T}}\mathbf{F}_{\mathrm{el}}^{\phantom{\mathrm{T}}}-\mathbf{Id}\right).$
(4)

The first and second Lamé constants read
$\lambda_{\mathrm{Si}}=2G_{\mathrm{Si}}\nu_{\mathrm{Si}}/\left(1-2\nu_{\mathrm{Si}}\right)$
and $G_{\mathrm{Si}}=E_{\mathrm{Si}}/2(1+\nu_{\mathrm{Si}})$, where
$\nu_{\mathrm{Si}}$ is the Poisson ratio of silicon. The elastic deformation
$\mathbf{F}_{\mathrm{el}}$ depends on the Piola-Kirchhoff stress $\mathbf{P}$
inside the particle determined by the constitutive equation

$\displaystyle\mathbf{P}$
$\displaystyle=2\mathbf{F}\frac{\partial\rho_{0}\varphi}{\partial\mathbf{C}}$
(5)
$\displaystyle=\lambda_{\mathrm{ch}}^{-2}\mathbf{F}\mathbf{F}_{\mathrm{pl}}^{-\mathrm{T}}\mathbf{F}_{\mathrm{pl}}^{-1}\left(\lambda_{\mathrm{Si}}\operatorname{tr}(\mathbf{E}_{\mathrm{el}})\mathbf{Id}+2G_{\mathrm{Si}}\mathbf{E}_{\mathrm{el}}\right),$

where $\mathbf{C}$ is the right Cauchy-Green tensor
$\mathbf{C}=\mathbf{F}^{\mathrm{T}}\mathbf{F}$. The Piola-Kirchhoff stress is
related to the Cauchy stress $\sigma$ as
$\mathbf{P}=J\sigma\mathbf{F}^{-\mathrm{T}}$ with
$J=\operatorname{det}\mathbf{F}$. The Piola-Kirchhoff stress $\mathbf{P}$
defined in Eq. 5 has to fulfill the momentum balance in the Lagrangian frame

$0=\nabla_{0}\cdot\mathbf{P}.$ (6)

In addition to the elastic deformation, the particle deforms plastically when
reaching the von Mises yield criterion

$f=\frac{\frac{3}{2}|\mathbf{M}^{\mathrm{dev}}|^{2}}{\sigma_{\mathrm{Y,Si}}^{2}}-1\leq
0,$ (7)

where $\mathbf{M}^{\mathrm{dev}}=\mathbf{M}-1/3\operatorname{tr}\mathbf{M}$ is
the deviatoric part of the Mandel stress
$\mathbf{M}=\mathbf{F}_{\mathrm{rev}}^{\mathrm{T}}\sigma\mathbf{F}_{\mathrm{rev}}^{-\mathrm{T}}$
with the reversible part of the deformation
$\mathbf{F}_{\mathrm{rev}}=\mathbf{F}_{\mathrm{el}}\mathbf{F}_{\mathrm{ch}}$.
The yield stress $\sigma_{\mathrm{Y}}$ depends on the material and reveals how
much stress the particle can withstand with reversible deformations. The
plastic flow
$\mathbf{L}_{\mathrm{pl}}=\dot{\mathbf{F}}_{\mathrm{pl}}\mathbf{F}_{\mathrm{pl}}^{-1}$
is calculated by

$\mathbf{L}_{\mathrm{pl}}=\phi\frac{\partial f}{\partial\mathbf{M}},$ (8)

where the plastic multiplier $\phi$ is determined from the consistency
condition $\dot{f}=0$.

The defining equation for the lithiation and delithiation of a silicon
particle reads

$\dot{c}_{\mathrm{Li},0}=-\nabla_{0}\cdot\vec{N}_{\mathrm{Li},0}$ (9)

with the lithium flux $\vec{N}_{\mathrm{Li},0}=-L\nabla_{0}\mu_{\mathrm{Li}}$
and the electro-chemo-mechanical potential $\mu_{\mathrm{Li}}$. The mobility
is determined by $L=D_{\mathrm{Li}}\left(\partial\mu_{\mathrm{Li}}/\partial
c_{\mathrm{Li,0}}\right)^{-1}$ with $D_{\mathrm{Li}}$ the diffusion
coefficient of lithium in silicon. At the boundary of the particle $r=R$, the
lithium flux $\vec{N}_{\mathrm{Li},0}(R)$ is determined by the applied
(de)lithiation rate.

The lithium concentration and the stress inside the particle influence the
electro-chemo-mechanical potential of lithium according to the constitutive
equation

$\mu_{\mathrm{Li}}=\frac{\partial\rho_{0}\varphi}{\partial
c_{\mathrm{Li,0}}}=-FU_{0}-\frac{v_{\mathrm{Li}}}{3\lambda_{\mathrm{ch}}^{3}}\mathbf{P}:\mathbf{F}.$
(10)

The chemical potential at the outer border of the anode determines the
measurable silicon open-circuit voltage (OCV) by $U=-\mu_{\mathrm{Li}}/F$.
Thus, the OCV reads

$U=U_{0}+\frac{v_{\mathrm{Li}}}{3F\lambda_{\mathrm{ch}}^{3}}\mathbf{P}:\mathbf{F}.$
(11)

We determine the ideal open-circuit voltage $U_{0}$ as the mean value of the
open-circuit voltages measured via GITT during lithiation and delithiation in
Ref. [28]. However, stress inside the particle crucially affects the open-
circuit voltage during lithiation and delithiation.

###  II.2 SEI model

Inside the SEI, there is no chemical deformation, as no lithiation of the SEI
is possible. Therefore, the SEI only deforms elastically and plastically

$\mathbf{F}_{\mathrm{SEI}}=\mathbf{F}_{\mathrm{SEI,pl}}\mathbf{F}_{\mathrm{SEI,el}}.$
(12)

In our visco-elastoplastic SEI model, elastoplastic deformations and viscous
flow contribute to the total stress. The stress due to elastoplastic
deformations is defined similarly to Eq. 5 as

$\displaystyle\mathbf{P}_{\mathrm{SEI,el}}=2\mathbf{F}_{\mathrm{SEI}}\frac{\partial\rho_{\mathrm{0,SEI}}\varphi_{\mathrm{SEI}}}{\partial\mathbf{C}_{\mathrm{SEI}}}$
(13)
$\displaystyle=\mathbf{F}_{\mathrm{SEI}}\mathbf{F}_{\mathrm{SEI,pl}}^{-\mathrm{T}}\mathbf{F}_{\mathrm{SEI,pl}}^{-1}\left(\lambda_{\mathrm{SEI}}\operatorname{tr}(\mathbf{E}_{\mathrm{SEI,el}})\mathbf{Id}+2G_{\mathrm{SEI}}\mathbf{E}_{\mathrm{SEI,el}}\right)$

with the strain tensor $\mathbf{E}_{\mathrm{SEI,el}}$, the right Cauchy-Green
tensor $\mathbf{C}_{\mathrm{SEI}}$, and the Lamé constants
$\lambda_{\mathrm{SEI}}$ and $G_{\mathrm{SEI}}$ defined analog to the particle
model. The free energy density of the SEI is defined according to merely the
mechanical contribution in Eq. 3.

Plastic flow can occur inside the SEI similarly to the particle when reaching
the von Mises yield criterion

$f_{\mathrm{SEI}}=\frac{\frac{3}{2}|\mathbf{M}_{\mathrm{SEI,el}}^{\mathrm{dev}}|^{2}}{\sigma_{\mathrm{Y,SEI}}^{2}}-1\leq
0$ (14)

leading to the plastic flow

$\mathbf{L}_{\mathrm{SEI,pl}}=\phi_{\mathrm{SEI}}\frac{\partial
f_{\mathrm{SEI}}}{\partial\mathbf{M}_{\mathrm{SEI,el}}}.$ (15)

The consistency condition $\dot{f}_{\mathrm{SEI}}=0$ determines again the
value of the plastic multiplier $\phi_{\mathrm{SEI}}$. We emphasize that
plastic flow in the SEI is driven by the deviatoric part of the elastoplastic
stress contribution $\mathbf{M}_{\mathrm{SEI,el}}^{\mathrm{dev}}$.

In addition to the elastoplastic model discussed in Ref. [56], we consider
stress generated by the viscous flow of the SEI. The viscous Cauchy stress is
defined as

$\mathbf{\sigma}_{\mathrm{SEI,visc}}=\eta_{\mathrm{SEI}}\dot{\mathbf{E}}_{\mathrm{SEI}}$
(16)

with the viscosity of the SEI $\eta_{\mathrm{SEI}}$. Radial symmetry in our
system ensures
$\dot{\mathbf{E}}_{\mathrm{SEI}}=\mathbf{F}_{\mathrm{SEI}}^{\mathrm{T}}\dot{\mathbf{F}}_{\mathrm{SEI}}$.
Therefore, the viscous Piola-Kirchhoff stress corresponding to Eq. 16 reads

$\mathbf{P}_{\mathrm{SEI,visc}}=J_{\mathrm{SEI}}\eta_{\mathrm{SEI}}\dot{\mathbf{F}}_{\mathrm{SEI}}$
(17)

with $J_{\mathrm{SEI}}=\operatorname{det}\mathbf{F}_{\mathrm{SEI}}$.

We describe the SEI as a power-law shear thinning material. Therefore, the
viscosity decreases with increasing strain rate

$\eta_{\mathrm{SEI}}\left(\dot{\mathbf{E}}_{\mathrm{SEI}}\right)=\eta_{\mathrm{SEI,0}}\dot{\mathbf{E}}_{\mathrm{SEI}}^{n-1},$
(18)

where $\eta_{\mathrm{SEI,0}}$ is a constant value and $n<1$ is the shear-
thinning exponent.

In our model, we consider the viscous behavior in parallel to the
elastoplastic model. Thus, the sum of the elastic and viscous contributions
describes the total Piola-Kirchhoff stress as

$\mathbf{P}_{\mathrm{SEI}}=\mathbf{P}_{\mathrm{SEI,el}}+\mathbf{P}_{\mathrm{SEI,visc}}.$
(19)

Analog to the particle model, the momentum balance

$0=\nabla_{0}\cdot\mathbf{P}_{\mathrm{SEI}}$ (20)

determines the stress inside the SEI.

Finally, we emphasize that our model describes the SEI mechanics on a
continuum level. The visco-elastoplastic behavior is not a single-crystal
property but results from subsequent partial SEI fracture and healing [56].
This approach is valid for small currents, where the SEI fractures only
partially and healing is fast compared to the lithiation timescale.

###  II.3 Particle-SEI interface

At the interface between the anode particle and SEI, we assume a perfect
sticking of the SEI on top of the silicon particle. Therefore, the stresses
inside the particle and SEI interact at the interface. Precisely, the radial
component of the Cauchy stress has to be equal in particle and SEI at the
boundary. Due to the radial symmetry, this also implies the equivalence of the
radial part of the Piola-Kirchhoff stress at the interface

$\mathbf{P}_{\mathrm{rr}}\big{\rvert}_{r=R}=\mathbf{P}_{\mathrm{SEI,rr}}\big{\rvert}_{r=R}.$
(21)

Inside the particle, the chemical contribution dominates the total
deformation. In contrast, the SEI deforms only elastically and plastically.
Thus, mechanical deformations of the SEI have to afford the large deformation
implied by the particle. These deformations result in significant strains and
stresses inside the SEI. Consequently, the stress inside the SEI can
substantially affect the stress and lithiation behavior of the silicon
particle.

###  II.4 Material parameters

The literature states values of Young’s modulus of the SEI $E_{\mathrm{SEI}}$
between only a few hundreds of $\mathrm{MPa}$ and values greater than
$100\,\mathrm{GPa}$. For a better overview, we divide the reported values into
the categories: soft with $E_{\mathrm{SEI}}<1\,\mathrm{GPa}$ [63, 64, 65, 66,
67], medium with $1\,\mathrm{GPa}\leq E_{\mathrm{SEI}}<10\,\mathrm{GPa}$ [63,
68, 69, 70, 71, 72, 66], stiff with $10\,\mathrm{GPa}\leq
E_{\mathrm{SEI}}<100\,\mathrm{GPa}$ [73, 68, 69, 70, 53, 74, 75], and very
stiff SEI with $E_{\mathrm{SEI}}\geq 100\,\mathrm{GPa}$ [68, 75]. In general,
references report lower values of Young’s modulus for thicker SEI layers. In
contrast, the initial SEI or the inner, inorganic SEI layer reveals higher
values of Young’s modulus. Due to the small scale, the measurement values of
the inner SEI might underestimate the corresponding Young’s modulus, as the
measured point might be in the transition zone between the inner and the outer
SEI. Further, the literature reports a size effect of Young’s modulus on the
nanoscale for different materials. Structures on the nanometer level typically
reveal a higher Young’s modulus compared to bulk values [76, 77, 78, 79, 80].
Therefore, we use $E_{\mathrm{SEI}}=100\,\mathrm{GPa}$ for the SEI in our
studies.

Experiments attempt to determine the viscosity of the SEI but face severe
difficulties due to the small length scale of interest. Recently, research has
advanced in estimating the viscosity value of the outer, organic SEI layer
[75, 67]. However, the inner, inorganic SEI layer is known to be much stiffer
than the outer layer. We assume this coincides with a considerably higher
viscosity of the inner layer. Consequently, reasonable values of the viscosity
of the inner SEI are between $\eta=10^{7}\,\mathrm{Pa\,s}$ measured for pitch,
a highly viscous polymer [81], and $\eta=10^{15}\,\mathrm{Pa\,s}$ measured for
silicon oxide [82, 83, 84]. We estimate the shear-thinning exponent to be
$n=0.15$ from Ref. [85].

Measurements for Young’s modulus of silicon show a wide range of values [86,
87]. For amorphous macroscopic silicon, literature reports values around
$90\,\mathrm{GPa}$ [88, 89] and $125\,\mathrm{GPa}$ [90, 91]. Crystalline
silicon nanowires reveal values between $50\,\mathrm{GPa}$ and
$250\,\mathrm{GPa}$ [87, 92]. At the nanoscale, we expect that the values of
Young’s modulus of amorphous and crystalline silicon will approach each other
as crystalline phases inside nanoparticles gain more importance. In comparison
to nanowires, nanoparticles may even show more pronounced surface effects
leading to higher values of Young’s modulus. Therefore, we use
$E_{\mathrm{Si}}=200\,\mathrm{GPa}$ in our simulations.

If not mentioned explicitly, we use the parameters presented in Table S1 in
the supporting information.

###  II.5 Computational details

We implement our model in MATLAB using a finite difference approach. We solve
the partial differential equations (6), (8), (9), (15), and (20) with the
solver ode15i by discretizing the radial dimension. The principal variables
inside the particle are the lithium concentration $c_{\mathrm{Li,0}}$, the
deformed radius $r$, and the radial component of the plastic deformation
$\mathbf{F}_{\mathrm{pl,rr}}$ of each element. Inside the SEI domain, the
principal variables are the deformed radius $r_{\mathrm{SEI}}$ and the radial
component of the plastic deformation $\mathbf{F}_{\mathrm{SEI,pl,rr}}$ of each
SEI element.

##  III Results and Discussion

Based on our chemo-mechanical model, we evaluate the influence of the
mechanics of the silicon particle and its covering interphase on the voltage
hysteresis of silicon anodes in this section. Although we term the covering
layer SEI throughout this paper, our model also describes a particle covered
by a silicon oxide layer.

Without the impact of the SEI, we can create three different scenarios
presented in the supporting information in sections SII to SIV reasoning the
experimental voltage hysteresis. Firstly, literature often considers silicon
thin-film anodes, where the expansion and contraction are restricted to the
normal direction of the film as illustrated in Figure S1. This leads to
significant stresses and plastic flow, resulting in a voltage hysteresis shown
in Figure S2. Secondly, in a real multi-particle electrode, the expansion of a
single particle is constrained. We model this situation with a simplified
particle confined by a fixed wall (see Figure S3). As depicted in Figure S4,
considerable stresses and a voltage hysteresis occur only during the first
lithiation due to permanent plastic deformation. Thirdly, massive stresses
generated by concentration gradients can arise in silicon particles without
constraints due to the slow diffusion process. However, S5 shows that no
voltage hysteresis is visible for silicon nanoparticles cycled at
$\mathrm{C}/20$. Only currents as large as $1\mathrm{C}$ create substantial
concentration gradients and a voltage hysteresis in silicon nanoparticles
without constraints. We emphasize that these concentration gradients vanish
after relaxation and can not explain the open-circuit voltage hysteresis.

Consequently, plastic flow due to slow diffusion or constrained particles can
not reproduce the voltage hysteresis observed in GITT measurements with a
silicon nanoparticle anode. Thus, we investigate the influence of the SEI on
the stress and potential inside the silicon particle.

###  III.1 Open-circuit voltage hysteresis

In our model, we consider the SEI as a visco-elastoplastic material. Inside
the particle, a substantial amount of the total deformation is the chemical
contribution due to the change in lithium concentration. In contrast, the SEI
merely shows mechanical deformations. As the deformation of the SEI has to
adjust to the deformation of the particle, considerable strains occur
naturally inside the SEI. Assuming a high Young’s modulus of the SEI, strains
inside the SEI cause significant stresses. As discussed, the radial component
of the stress inside the SEI at the particle-SEI interface determines the
radial stress component inside the particle at the interface. For currents as
small as $\mathrm{C}/20$, the radial stress component of the SEI is the main
reason for the stress inside the particle. The particle stress distributes
uniformly, reasoning vanishing deviatoric stress. Therefore, plastic flow will
not occur inside the particle. Nevertheless, the substantial stresses inside
the particle can generate a voltage hysteresis.

In our simulation, we determine the open-circuit voltage after a relaxation
period to reproduce GITT measurements. Figure 2 depicts the open-circuit
voltage for lithiation and delithiation simulated with our visco-elastoplastic
SEI model depending on the state-of-charge (SOC). Notably, the simulation
reveals a significant voltage hysteresis. Furthermore, our results agree well
with the experimentally observed voltage hysteresis [28]. Only for very low
and very high SOC does the experiment show a steeper decrease and increase,
respectively, compared to the simulation. We attribute this to the use of the
mean value of the lithiation and delithiation GITT voltages as the ideal open-
circuit voltage curve. However, the situation is asymmetric at the endpoints
of lithiation and delithiation. Before swapping the current direction, the
stress stays constant but changes gradually after. Therefore, the ideal open-
circuit voltage will differ from the mean value of the lithiation and
delithiation open-circuit voltages at very low and very high SOC.

Figure 2: Open-circuit voltage hysteresis generated by a visco-elastoplastic
SEI in comparison to GITT measurement [28]. A complete cycle and a partial
cycle are depicted for the simulation and the Plett model.

Figure 3: Stress and size effects of a visco-elastoplastic SEI. Lithiation is
depicted with solid and delithiation with dashed lines. (a) Hysteresis of the
stress components inside the SEI at the SEI-particle interface. (b) Hysteresis
of the particle radius generated by the impact of the visco-elastoplastic SEI.

The standard fitting model to describe transitions between the lithiation and
delithiation voltages is the Plett model [93, 94, 95]. We briefly present the
equations describing the Plett model in the supporting information in section
V. In Fig. 2, we depict our simulation of a complete cycle and a partial cycle
between $25\%$ and $75\%$ in comparison to the behavior of the Plett model.
The illustration reveals that our physical model is in reasonable accordance
with the phenomenological Plett model.

In our model, the radial stress component of the SEI impacts the stress inside
the particle and implicitly generates the voltage hysteresis of the silicon
nanoparticle. In Fig. 3(a), we illustrate the stress components inside the SEI
at the particle-SEI interface. During lithiation, the particle expands, and
the SEI deforms purely elastically until reaching the yield criterion. As the
particle surface area increases, the SEI has to stretch in the tangential
direction leading to tensile tangential stress (yellow). This tangential
expansion leads to a constriction in the SEI thickness and compressive radial
stress (red). Upon meeting the yield criterion, the SEI starts to flow
plastically. The tangential stress shows a significant kink at the transition
between elastic and plastic deformation. In the purely plastic regime, the
stress components change only slightly. The radial component stays
compressive, and the tangential tensile. Switching to delithiation, the SEI
immediately leaves the plastic regime and deforms purely elastically again.
The tensile tangential stress reduces, and compressive stress occurs.
Simultaneously, compressive radial stress reduces, and tensile stress occurs.
Reaching the plastic limit, the compressive tangential stress and the tensile
radial stress stay almost constant until the end of the delithiation process.
Eventually, the radial stress and the tangential stress of the SEI show a
hysteresis behavior due to path-dependent plastic deformations.

The maximum stress in the system results from the yield condition, which
determines the maximum deviatoric stress independent of the particle size.
However, expressing Eq. 6 in spherical coordinates indicates that the radial
stress component increases with the ratio of SEI thickness and particle
radius. Nevertheless, the particle size of interest is around the optimum size
for silicon nanoparticles in terms of degradation found in Ref. [55]. That
means smaller particles with a higher specific surface area are covered by a
thinner SEI compared to larger particles. We conclude that a change in
particle size has only a minor influence on the size of the OCV hysteresis in
the realm of interest.

As the stresses inside particle and SEI influence the particle deformations,
the stress hysteresis induces a hysteresis phenomenon in the deformation
tensor. This leads to a hysteresis behavior in the radius and volume of the
particle. In Fig. 3(b), we depict the particle radius depending on the SOC,
revealing a clear hysteresis effect. The particle radius shows larger values
for delithiation than for lithiation at the same SOC. The simulation result
matches qualitatively the volume hysteresis reported for silicon anodes [13,
14, 15, 16].

###  III.2 Voltage hysteresis during cycling

The literature states a significant difference between the voltage hysteresis
of silicon anodes for small currents and the open-circuit voltage hysteresis
measured with GITT. As discussed at the beginning of section III, we can not
attribute this to concentration gradients inside the particle. Instead, we
assign the potential difference to the viscous behavior of the SEI. As the
literature provides no value for the viscosity of the inner SEI, we fit the
viscosity to the experimental data.

Figure 4: Impact of visco-elastoplastic SEI on the silicon potential. (a)
Simulation of the voltage hysteresis generated by a visco-elastoplastic SEI
during a GITT procedure in comparison to the measurement [28]. (b) Simulation
of lithiation pulse and rest period in comparison to a single GITT pulse [29].

To investigate the influence of viscous SEI behavior on the voltage during
slow cycling, we simulate the GITT measurement procedure with alternating
$\mathrm{C}/20$ current steps and relaxation periods. We show the simulated
voltage together with the voltage measured during the GITT experiment [28] in
Fig. 4(a). The simulated voltage profile shows a good agreement with the
experiment. The fitted viscosity value in Fig. 4(a) is
$\eta_{\mathrm{SEI,0}}=15\,\mathrm{GPa\,s}^{n}$. Only the potential difference
between the two observed hysteresis phenomena stays approximately constant in
the experiment but slightly decreases with increasing SOC in the simulation.
The declining strain rate of the SEI for higher SOC causes the observed
decrease of the viscous contribution in the simulation. More pronounced shear
thinning behavior could reduce the deviation between simulation and
experiment.

In Fig. 4(b), we depict the simulation of a single lithiation pulse with a
subsequent rest period in comparison to an experimental GITT pulse [29]. The
voltage curve demonstrates the transition between the lithiation potential and
the measured GITT data points and vice versa. During the lithiation step, the
simulation exhibits a slower decay from the rest potential to the lithiation
potential compared to the experiment. Therefore, the timescale of the
simulated lithiation step with the applied parameters is larger than the
experimental value. For the subsequent rest period, our simulation and the
experiment reveal similar timescales. However, the experiment reveals an
additional longer timescale as the voltage is not constant after three hours
of relaxation. A more sophisticated rheological model describing the visco-
elastoplastic behavior could improve the timescales by the cost of additional
parameters. Further, we will evaluate the influence of Young’s modulus on the
simulated timescales of GITT pulses in section III.3.

In the supporting information, we compare shear-thinning viscosity with
Newtonian behavior. As illustrated in Fig. S6(a), Newtonian behavior leads to
a more pronounced decrease of the viscous contribution to the voltage
hysteresis with increasing SOC, which disagrees with the experiment.
Therefore, we consider the power-law shear thinning model in our studies.
However, due to the constant value of the viscosity, we can compare the
Newtonian viscosity to the range of values presented in section II.4. The
value $\eta_{\mathrm{SEI}}=1.25\times 10^{14}\,\mathrm{Pa\,s}$ found for the
Newtonian viscosity agrees with the range of
$10^{7}\,\mathrm{Pa\,s}<\eta_{\mathrm{SEI}}<10^{15}\,\mathrm{Pa\,s}$ discussed
as reasonable in section II.4. The accordance indicates the suitability of
considering viscous behavior to explain the amplified hysteresis observed
during cycling. Furthermore, the high fitting value of the viscosity matches
the assumption of a stiff inner SEI layer.

Regarding the timescales, Newtonian viscosity and power-law shear-thinning
viscosity reveal similar transition times during the GITT pulse depicted in
S6(b). Thus, the choice of shear-thinning or Newtonian behavior does not
significantly influence the transition profile between the different potential
curves.

###  III.3 Variation of Young’s modulus

Figure 5: Effect of Young’s modulus of the SEI and the silicon nanoparticle on
the voltage hysteresis. (a) Impact of Young’s modulus of the SEI on the shape
of the silicon voltage hysteresis in comparison to experimental data [28]. (b)
Influence of Young’s modulus of the silicon nanoparticle on the transition
voltage profile in comparison to a single GITT pulse [29].

As the exact mechanical parameters of the SEI are ambiguous in experiments, we
provide a parameter study to estimate the influence of Young’s modulus
$E_{\mathrm{SEI}}$. In Fig. 5(a), we compare the open-circuit voltage
hysteresis simulated for three different values of Young’s modulus. For a
small Young’s modulus of the SEI, the simulated voltage hysteresis is small,
and its shape does not match the experiment. In contrast, there is a
reasonable agreement between simulation and experiment for medium and high
values. The value $E_{\mathrm{SEI}}=200\,\mathrm{GPa}$ shows the best
agreement. This indicates that Young’s modulus of the SEI is often
underestimated and the nanoscale effect discussed in section II.4 induces a
further increase.

In the supporting information, Fig. S7(a) illustrates the timescale of a
single lithiation pulse and relaxation for the medium and high parameter
values of $E_{\mathrm{SEI}}$. Overall, the simulated timescales are comparable
with the ones observed in the experiment. Increasing Young’s modulus of the
SEI leads to a slightly faster transition. However, the influence of Young’s
modulus of the SEI on the revealed timescale is essentially negligible.

Therefore, we analyze the mechanical properties of the silicon nanoparticle
and its impact on the transition times. Due to the increase of Young’s modulus
reported at the nanoscale [76, 77, 78, 79, 80], we vary Young’s modulus of the
silicon nanoparticle for constant Young’s modulus of the SEI. We compare the
literature values of Young’s modulus for bulk amorphous silicon of
$90-125\,\mathrm{GPa}$ [88, 89, 90, 91] with the modulus for silicon estimated
at the nanoscale of $200\,\mathrm{GPa}$ from Refs. [87, 92]. In Fig. 5(b), we
depict the voltage of a simulated GITT pulse for the variation of Young’s
modulus of silicon. The highest value $E_{\mathrm{Si}}=200\,\mathrm{GPa}$
shows the best agreement with the experiment for the transition profile. The
voltage relaxation for the lower values of $E_{\mathrm{Si}}$ is slightly too
slow. Nevertheless, the lower values are still able to qualitatively describe
the GITT data. The variation reveals that a value for Young’s modulus of the
silicon nanoparticle, which is approximately a factor two higher than the
literature bulk value, can best reproduce the timescale of the experiment.
This indicates the importance of considering nanoscale effects for the
mechanical description.

At the same time, we investigate the influence of Young’s modulus of the
silicon nanoparticle on the shape and size of the voltage hysteresis in the
supporting information. Fig. S7(b) reveals that Young’s modulus of silicon has
only a minor influence on the voltage hysteresis in our model.

Therefore, the parameter study shows that high values of Young’s modulus of
the silicon nanoparticle and the SEI achieve the best agreement between the
chemo-mechanical simulation and experimental data. This indicates the
importance of nanoscale effects on the silicon voltage hysteresis.

##  IV Conclusions

To conclude, the slow diffusion process or particle constriction inside
electrodes can not cause the voltage hysteresis observed for silicon
nanoparticle anodes. However, the expansion of silicon nanoparticles leads to
an impact of the SEI or the silicon oxide layer on the stress and potential
inside the anode. We propose a visco-elastoplastic silicon-SEI model, which
can for the first time explain the open-circuit voltage hysteresis of a
silicon nanoparticle anode measured with GITT. The plasticity of the SEI leads
to different stresses inside the particle during lithiation and delithiation,
inducing the observed voltage hysteresis. In addition, our SEI model
qualitatively reproduces the volume hysteresis of silicon anodes. Furthermore,
the viscous SEI behavior describes the difference between the voltage
hysteresis observed for cycling at low currents and the open-circuit voltage
from GITT after a relaxation period. A variation of Young’s modulus of the SEI
reveals its crucial influence on the shape of the voltage hysteresis. The
results indicate a stiff inner SEI layer. Consequently, our work suggests that
a soft SEI will mitigate the voltage hysteresis of silicon nano-anodes.

###### Acknowledgements.

Lukas Köbbing gratefully acknowledges funding and support by the German
Research Foundation (DFG) within the research training group SiMET under
project number 281041241/GRK2218.

## References

  * Zuo  _et al._ [2017] X. Zuo, J. Zhu, P. Müller-Buschbaum, and Y.-J. Cheng, Silicon based lithium-ion battery anodes: A chronicle perspective review, Nano Energy 31, 113 (2017).
  * Kim  _et al._ [2021a] M. Kim, Z. Yang, and I. Bloom, Review—The Lithiation/Delithiation Behavior of Si-Based Electrodes: A Connection between Electrochemistry and Mechanics, Journal of The Electrochemical Society 168, 010523 (2021a).
  * Zhang  _et al._ [2021a] C. Zhang, F. Wang, J. Han, S. Bai, J. Tan, J. Liu, and F. Li, Challenges and Recent Progress on Silicon‐Based Anode Materials for Next‐Generation Lithium‐Ion Batteries, Small Structures 2, 2100009 (2021a).
  * Sun  _et al._ [2022] L. Sun, Y. Liu, R. Shao, J. Wu, R. Jiang, and Z. Jin, Recent progress and future perspective on practical silicon anode-based lithium ion batteries, Energy Storage Materials 46, 482 (2022).
  * Lin  _et al._ [2023] J. Lin, X. Zhang, E. Fan, R. Chen, F. Wu, and L. Li, Carbon neutrality strategies for sustainable batteries: from structure, recycling, and properties to applications, Energy & Environmental Science 16, 745 (2023).
  * Wetjen  _et al._ [2018] M. Wetjen, S. Solchenbach, D. Pritzl, J. Hou, V. Tileli, and H. A. Gasteiger, Morphological Changes of Silicon Nanoparticles and the Influence of Cutoff Potentials in Silicon-Graphite Electrodes, Journal of The Electrochemical Society 165, A1503 (2018).
  * Liu  _et al._ [2012] X. H. Liu, L. Zhong, S. Huang, S. X. Mao, T. Zhu, and J. Y. Huang, Size-Dependent Fracture of Silicon Nanoparticles During Lithiation, ACS Nano 6, 1522 (2012).
  * Chakraborty  _et al._ [2015] J. Chakraborty, C. P. Please, A. Goriely, and S. J. Chapman, Combining mechanical and chemical effects in the deformation and failure of a cylindrical electrode particle in a Li-ion battery, International Journal of Solids and Structures 54, 66 (2015).
  * Stein  _et al._ [2016] P. Stein, Y. Zhao, and B.-X. Xu, Effects of surface tension and electrochemical reactions in Li-ion battery electrode nanoparticles, Journal of Power Sources 332, 154 (2016).
  * Yang  _et al._ [2020] Y. Yang, W. Yuan, W. Kang, Y. Ye, Q. Pan, X. Zhang, Y. Ke, C. Wang, Z. Qiu, and Y. Tang, A review on silicon nanowire-based anodes for next-generation high-performance lithium-ion batteries from a material-based perspective, Sustainable Energy & Fuels 4, 1577 (2020).
  * Xiao  _et al._ [2022] Z. Xiao, C. Wang, L. Song, Y. Zheng, and T. Long, Research progress of nano-silicon-based materials and silicon-carbon composite anode materials for lithium-ion batteries, Journal of Solid State Electrochemistry 26, 1125 (2022).
  * Zhang  _et al._ [2022] L. Zhang, M. Al-Mamun, L. Wang, Y. Dou, L. Qu, S. X. Dou, H. K. Liu, and H. Zhao, The typical structural evolution of silicon anode, Cell Reports Physical Science 3, 100811 (2022).
  * Yoon  _et al._ [2016] I. Yoon, D. P. Abraham, B. L. Lucht, A. F. Bower, and P. R. Guduru, In Situ Measurement of Solid Electrolyte Interphase Evolution on Silicon Anodes Using Atomic Force Microscopy, Advanced Energy Materials 6, 1600099 (2016).
  * Duay  _et al._ [2016] J. Duay, K. W. Schroder, S. Murugesan, and K. J. Stevenson, Monitoring Volumetric Changes in Silicon Thin-Film Anodes through In Situ Optical Diffraction Microscopy, ACS Applied Materials & Interfaces 8, 17642 (2016).
  * Jerliu  _et al._ [2018] B. Jerliu, E. Hüger, L. Dörrer, B. K. Seidlhofer, R. Steitz, M. Horisberger, and H. Schmidt, Lithium insertion into silicon electrodes studied by cyclic voltammetry and: Operando neutron reflectometry, Physical Chemistry Chemical Physics 20, 23480 (2018).
  * Dong  _et al._ [2019] K. Dong, H. Markötter, F. Sun, A. Hilger, N. Kardjilov, J. Banhart, and I. Manke, In situ and Operando Tracking of Microstructure and Volume Evolution of Silicon Electrodes by using Synchrotron X‐ray Imaging, ChemSusChem 12, 261 (2019).
  * Baggetto  _et al._ [2009] L. Baggetto, J. Oudenhoven, T. van Dongen, J. Klootwijk, M. Mulder, R. Niessen, M. de Croon, and P. Notten, On the electrochemistry of an anode stack for all-solid-state 3D-integrated batteries, Journal of Power Sources 189, 402 (2009).
  * Sethuraman  _et al._ [2010a] V. A. Sethuraman, M. J. Chon, M. Shimshak, V. Srinivasan, and P. R. Guduru, In situ measurements of stress evolution in silicon thin films during electrochemical lithiation and delithiation, Journal of Power Sources 195, 5062 (2010a).
  * Sethuraman  _et al._ [2010b] V. A. Sethuraman, V. Srinivasan, A. F. Bower, and P. R. Guduru, In Situ Measurements of Stress-Potential Coupling in Lithiated Silicon, Journal of The Electrochemical Society 157, A1253 (2010b).
  * Sethuraman  _et al._ [2013] V. A. Sethuraman, V. Srinivasan, and J. Newman, Analysis of Electrochemical Lithiation and Delithiation Kinetics in Silicon, Journal of The Electrochemical Society 160, A394 (2013).
  * Verbrugge  _et al._ [2015] M. W. Verbrugge, D. R. Baker, X. Xiao, Q. Zhang, and Y.-T. Cheng, Experimental and Theoretical Characterization of Electrode Materials that Undergo Large Volume Changes and Application to the Lithium–Silicon System, The Journal of Physical Chemistry C 119, 5341 (2015).
  * Lu  _et al._ [2016] B. Lu, Y. Song, Q. Zhang, J. Pan, Y.-T. Cheng, and J. Zhang, Voltage hysteresis of lithium ion batteries caused by mechanical stress, Physical Chemistry Chemical Physics 18, 4721 (2016).
  * Baker  _et al._ [2017] D. R. Baker, M. W. Verbrugge, and X. Xiao, An approach to characterize and clarify hysteresis phenomena of lithium-silicon electrodes, Journal of Applied Physics 122, 165102 (2017).
  * Chan  _et al._ [2008] C. K. Chan, H. Peng, G. Liu, K. McIlwrath, X. F. Zhang, R. A. Huggins, and Y. Cui, High-performance lithium battery anodes using silicon nanowires, Nature Nanotechnology 3, 31 (2008).
  * Wu  _et al._ [2012] H. Wu, G. Chan, J. W. Choi, I. Ryu, Y. Yao, M. T. McDowell, S. W. Lee, A. Jackson, Y. Yang, L. Hu, and Y. Cui, Stable cycling of double-walled silicon nanotube battery anodes through solid–electrolyte interphase control, Nature Nanotechnology 7, 310 (2012).
  * Chandrasekaran and Fuller [2011] R. Chandrasekaran and T. F. Fuller, Analysis of the Lithium-Ion Insertion Silicon Composite Electrode/Separator/Lithium Foil Cell, Journal of The Electrochemical Society 158, A859 (2011).
  * Bernard  _et al._ [2019] P. Bernard, J. P. Alper, C. Haon, N. Herlin-Boime, and M. Chandesris, Electrochemical analysis of silicon nanoparticle lithiation – Effect of crystallinity and carbon coating quantity, Journal of Power Sources 435, 226769 (2019).
  * Pan  _et al._ [2019] K. Pan, F. Zou, M. Canova, Y. Zhu, and J.-H. Kim, Systematic electrochemical characterizations of Si and SiO anodes for high-capacity Li-Ion batteries, Journal of Power Sources 413, 20 (2019).
  * Pan [2020] K. Pan,  _A Systematic Methodology for Characterization and Prediction of Performance of Si-based Materials for Li-ion Batteries_ , Ph.D. thesis, Ohio State University (2020).
  * McDowell  _et al._ [2013] M. T. McDowell, S. W. Lee, W. D. Nix, and Y. Cui, 25th Anniversary Article: Understanding the Lithiation of Silicon and Other Alloying Anodes for Lithium-Ion Batteries, Advanced Materials 25, 4966 (2013).
  * Yang  _et al._ [2012] H. Yang, S. Huang, X. Huang, F. Fan, W. Liang, X. H. Liu, L.-Q. Chen, J. Y. Huang, J. Li, T. Zhu, and S. Zhang, Orientation-Dependent Interfacial Mobility Governs the Anisotropic Swelling in Lithiated Silicon Nanowires, Nano Letters 12, 1953 (2012).
  * Cui  _et al._ [2013] Z. Cui, F. Gao, and J. Qu, Interface-reaction controlled diffusion in binary solids with applications to lithiation of silicon in lithium-ion batteries, Journal of the Mechanics and Physics of Solids 61, 293 (2013).
  * Zhao  _et al._ [2019] Y. Zhao, P. Stein, Y. Bai, M. Al-Siraj, Y. Yang, and B.-X. Xu, A review on modeling of electro-chemo-mechanics in lithium-ion batteries, Journal of Power Sources 413, 259 (2019).
  * Jiang  _et al._ [2020] Y. Jiang, G. Offer, J. Jiang, M. Marinescu, and H. Wang, Voltage Hysteresis Model for Silicon Electrodes for Lithium Ion Batteries, Including Multi-Step Phase Transformations, Crystallization and Amorphization, Journal of The Electrochemical Society 167, 130533 (2020).
  * Bower  _et al._ [2011] A. Bower, P. Guduru, and V. Sethuraman, A finite strain model of stress, diffusion, plastic flow, and electrochemical reactions in a lithium-ion half-cell, Journal of the Mechanics and Physics of Solids 59, 804 (2011).
  * Di Leo  _et al._ [2015] C. V. Di Leo, E. Rejovitzky, and L. Anand, Diffusion–deformation theory for amorphous silicon anodes: The role of plastic deformation on electrochemical performance, International Journal of Solids and Structures 67-68, 283 (2015).
  * Chandrasekaran  _et al._ [2010] R. Chandrasekaran, A. Magasinski, G. Yushin, and T. F. Fuller, Analysis of Lithium Insertion/Deinsertion in a Silicon Electrode Particle at Room Temperature, Journal of The Electrochemical Society 157, A1139 (2010).
  * Zhao  _et al._ [2011] K. Zhao, M. Pharr, S. Cai, J. J. Vlassak, and Z. Suo, Large Plastic Deformation in High-Capacity Lithium-Ion Batteries Caused by Charge and Discharge, Journal of the American Ceramic Society 94, s226 (2011).
  * Cui  _et al._ [2012] Z. Cui, F. Gao, and J. Qu, A finite deformation stress-dependent chemical potential and its applications to lithium ion batteries, Journal of the Mechanics and Physics of Solids 60, 1280 (2012).
  * Horstmann  _et al._ [2021] B. Horstmann, J. Shi, R. Amine, M. Werres, X. He, H. Jia, F. Hausen, I. Cekic-Laskovic, S. Wiemers-Meyer, J. Lopez, D. Galvez-Aranda, F. Baakes, D. Bresser, C. Su, Y. Xu, W. Xu, P. Jakes, R.-A. Eichel, E. Figgemeier, U. Krewer, J. M. Seminario, P. B. Balbuena, C. Wang, S. Passerini, Y. Shao-Horn, M. Winter, K. Amine, R. Kostecki, and A. Latz, Strategies towards enabling lithium metal in batteries: interphases and electrodes, Energy & Environmental Science 14, 5289 (2021).
  * Horstmann  _et al._ [2019] B. Horstmann, F. Single, and A. Latz, Review on multi-scale models of solid-electrolyte interphase formation, Current Opinion in Electrochemistry 13, 61 (2019).
  * Kolzenberg  _et al._ [2020] L. Kolzenberg, A. Latz, and B. Horstmann, Solid–Electrolyte Interphase During Battery Cycling: Theory of Growth Regimes, ChemSusChem 13, 3901 (2020).
  * Köbbing  _et al._ [2023] L. Köbbing, A. Latz, and B. Horstmann, Growth of the solid-electrolyte interphase: Electron diffusion versus solvent diffusion, Journal of Power Sources 561, 232651 (2023).
  * Cao  _et al._ [2019] C. Cao, I. I. Abate, E. Sivonxay, B. Shyam, C. Jia, B. Moritz, T. P. Devereaux, K. A. Persson, H.-G. Steinrück, and M. F. Toney, Solid Electrolyte Interphase on Native Oxide-Terminated Silicon Anodes for Li-Ion Batteries, Joule 3, 762 (2019).
  * Pinson and Bazant [2013] M. B. Pinson and M. Z. Bazant, Theory of SEI Formation in Rechargeable Batteries: Capacity Fade, Accelerated Aging and Lifetime Prediction, Journal of The Electrochemical Society 160, A243 (2013).
  * Nie  _et al._ [2013] M. Nie, D. P. Abraham, Y. Chen, A. Bose, and B. L. Lucht, Silicon Solid Electrolyte Interphase (SEI) of Lithium Ion Battery Characterized by Microscopy and Spectroscopy, The Journal of Physical Chemistry C 117, 13403 (2013).
  * Wetjen  _et al._ [2017] M. Wetjen, D. Pritzl, R. Jung, S. Solchenbach, R. Ghadimi, and H. A. Gasteiger, Differentiating the Degradation Phenomena in Silicon-Graphite Electrodes for Lithium-Ion Batteries, Journal of The Electrochemical Society 164, A2840 (2017).
  * Zhang  _et al._ [2019] Y. Zhang, N. Du, and D. Yang, Designing superior solid electrolyte interfaces on silicon anodes for high-performance lithium-ion batteries, Nanoscale 11, 19086 (2019).
  * Kim  _et al._ [2021b] J. Kim, O. B. Chae, and B. L. Lucht, Perspective—Structure and Stability of the Solid Electrolyte Interphase on Silicon Anodes of Lithium-ion Batteries, Journal of The Electrochemical Society 168, 030521 (2021b).
  * Zhang  _et al._ [2021b] X. Zhang, S. Weng, G. Yang, Y. Li, H. Li, D. Su, L. Gu, Z. Wang, X. Wang, and L. Chen, Interplay between solid-electrolyte interphase and (in)active LixSi in silicon anode, Cell Reports Physical Science 2, 100668 (2021b).
  * Dopilka  _et al._ [2023] A. Dopilka, Y. Gu, J. M. Larson, V. Zorba, and R. Kostecki, Nano-FTIR Spectroscopy of the Solid Electrolyte Interphase Layer on a Thin-Film Silicon Li-Ion Anode, ACS Applied Materials & Interfaces 15, 6755 (2023).
  * Kumar  _et al._ [2017] R. Kumar, P. Lu, X. Xiao, Z. Huang, and B. W. Sheldon, Strain-Induced Lithium Losses in the Solid Electrolyte Interphase on Silicon Electrodes, ACS Applied Materials & Interfaces 9, 28406 (2017).
  * Tanaka  _et al._ [2018] M. Tanaka, J. B. Hooper, and D. Bedrov, Role of Plasticity in Mechanical Failure of Solid Electrolyte Interphases on Nanostructured Silicon Electrode: Insight from Continuum Level Modeling, ACS Applied Energy Materials 1, 1858 (2018).
  * Guo  _et al._ [2020] K. Guo, R. Kumar, X. Xiao, B. W. Sheldon, and H. Gao, Failure progression in the solid electrolyte interphase (SEI) on silicon electrodes, Nano Energy 68, 104257 (2020).
  * Jin  _et al._ [2021] M. Y. Jin, K. Guo, X. Xiao, M. W. Verbrugge, H. Gao, and B. W. Sheldon, Optimum Particle Size in Silicon Electrodes Dictated by Chemomechanical Deformation of the SEI, Advanced Functional Materials 31, 2010640 (2021).
  * Kolzenberg  _et al._ [2022] L. Kolzenberg, A. Latz, and B. Horstmann, Chemo‐Mechanical Model of SEI Growth on Silicon Electrode Particles, Batteries & Supercaps 5, 1 (2022).
  * Li  _et al._ [2019] Q. Li, X. Liu, X. Han, Y. Xiang, G. Zhong, J. Wang, B. Zheng, J. Zhou, and Y. Yang, Identification of the Solid Electrolyte Interface on the Si/C Composite Anode with FEC as the Additive, ACS Applied Materials & Interfaces 11, 14066 (2019).
  * Chen  _et al._ [2020] J. Chen, X. Fan, Q. Li, H. Yang, M. R. Khoshi, Y. Xu, S. Hwang, L. Chen, X. Ji, C. Yang, H. He, C. Wang, E. Garfunkel, D. Su, O. Borodin, and C. Wang, Electrolyte design for LiF-rich solid–electrolyte interfaces to enable high-performance microsized alloy anodes for batteries, Nature Energy 5, 386 (2020).
  * Schammer  _et al._ [2021] M. Schammer, B. Horstmann, and A. Latz, Theory of Transport in Highly Concentrated Electrolytes, Journal of The Electrochemical Society 168, 026511 (2021).
  * Castelli  _et al._ [2021] G. F. Castelli, L. von Kolzenberg, B. Horstmann, A. Latz, and W. Dörfler, Efficient Simulation of Chemical–Mechanical Coupling in Battery Active Particles, Energy Technology 9, 2000835 (2021).
  * Holzapfel [2000] G. Holzapfel,  _Nonlinear Solid Mechanics: A Continuum Approach for Engineering_ (John Wiley & Sons, 2000).
  * Gurtin  _et al._ [2010] M. E. Gurtin, E. Fried, and L. Anand,  _The Mechanics and Thermodynamics of Continua_ (Cambridge University Press, 2010).
  * Zheng  _et al._ [2014] J. Zheng, H. Zheng, R. Wang, L. Ben, W. Lu, L. Chen, L. Chen, and H. Li, 3D visualization of inhomogeneous multi-layered structure and Young’s modulus of the solid electrolyte interphase (SEI) on silicon anodes for lithium ion batteries, Phys. Chem. Chem. Phys. 16, 13229 (2014).
  * Kuznetsov  _et al._ [2015] V. Kuznetsov, A.-H. Zinn, G. Zampardi, S. Borhani-Haghighi, F. La Mantia, A. Ludwig, W. Schuhmann, and E. Ventosa, Wet Nanoindentation of the Solid Electrolyte Interphase on Thin Film Si Electrodes, ACS Applied Materials & Interfaces 7, 23554 (2015).
  * Yoon  _et al._ [2020] I. Yoon, S. Jurng, D. P. Abraham, B. L. Lucht, and P. R. Guduru, Measurement of mechanical and fracture properties of solid electrolyte interphase on lithium metal anodes in lithium ion batteries, Energy Storage Materials 25, 296 (2020).
  * Gu  _et al._ [2023] M. Gu, A. M. Rao, J. Zhou, and B. Lu, In situ formed uniform and elastic SEI for high-performance batteries, Energy & Environmental Science 16, 1166 (2023).
  * Chen  _et al._ [2023] Y. Chen, W. Wu, S. Gonzalez-munoz, L. Forcieri, C. Wells, S. P. Jarvis, F. Wu, R. Young, A. Dey, M. Isaacs, M. Nagarathinam, R. G. Palgrave, N. Tapia-ruiz, and O. V. Kolosov, Nanoarchitecture factors of solid electrolyte interphase formation via 3D nano-rheology microscopy and surface force-distance spectroscopy, Nature Communications 14, 1321 (2023).
  * Shin  _et al._ [2015] H. Shin, J. Park, S. Han, A. M. Sastry, and W. Lu, Component-/structure-dependent elasticity of solid electrolyte interphase layer in Li-ion batteries: Experimental and computational studies, Journal of Power Sources 277, 169 (2015).
  * Shkrob  _et al._ [2015] I. A. Shkrob, J. F. Wishart, and D. P. Abraham, What Makes Fluoroethylene Carbonate Different?, The Journal of Physical Chemistry C 119, 14954 (2015).
  * He  _et al._ [2017] Y. He, H. Hu, K. Zhang, S. Li, and J. Chen, Mechanical insights into the stability of heterogeneous solid electrolyte interphase on an electrode particle, Journal of Materials Science 52, 2836 (2017).
  * Kamikawa  _et al._ [2020] Y. Kamikawa, K. Amezawa, and K. Terada, Elastic-Plastic Deformation of a Solid Electrolyte Interface Formed by Reduction of Fluoroethylene Carbonate: A Nanoindentation and Finite Element Analysis Study, Journal of Physical Chemistry C 124, 22488 (2020).
  * Liu  _et al._ [2022] Y. Liu, X. Xu, O. O. Kapitanova, P. V. Evdokimov, Z. Song, A. Matic, and S. Xiong, Electro‐Chemo‐Mechanical Modeling of Artificial Solid Electrolyte Interphase to Enable Uniform Electrodeposition of Lithium Metal Anodes, Advanced Energy Materials 12, 2103589 (2022).
  * Zhang  _et al._ [2015] Q. Zhang, X. Xiao, W. Zhou, Y.-T. Cheng, and M. W. Verbrugge, Toward High Cycle Efficiency of Silicon-Based Negative Electrodes by Designing the Solid Electrolyte Interphase, Advanced Energy Materials 5, 1401398 (2015).
  * Zhang  _et al._ [2020] H. Zhang, D. Wang, and C. Shen, In-situ EC-AFM and ex-situ XPS characterization to investigate the mechanism of SEI formation in highly concentrated aqueous electrolyte for Li-ion batteries, Applied Surface Science 507, 145059 (2020).
  * Chai  _et al._ [2021] Y. Chai, W. Jia, Z. Hu, S. Jin, H. Jin, H. Ju, X. Yan, H. Ji, and L.-J. Wan, Monitoring the mechanical properties of the solid electrolyte interphase (SEI) using electrochemical quartz crystal microbalance with dissipation, Chinese Chemical Letters 32, 1139 (2021).
  * Chen  _et al._ [2006] C. Q. Chen, Y. Shi, Y. S. Zhang, J. Zhu, and Y. J. Yan, Size Dependence of Young’s Modulus in ZnO Nanowires, Physical Review Letters 96, 075505 (2006).
  * Mathur and Erlebacher [2007] A. Mathur and J. Erlebacher, Size dependence of effective Young’s modulus of nanoporous gold, Applied Physics Letters 90, 061910 (2007).
  * Agrawal  _et al._ [2008] R. Agrawal, B. Peng, E. E. Gdoutos, and H. D. Espinosa, Elasticity Size Effects in ZnO Nanowires - A Combined Experimental-Computational Approach, Nano Letters 8, 3668 (2008).
  * He and Lilley [2008] J. He and C. M. Lilley, Surface Effect on the Elastic Behavior of Static Bending Nanowires, Nano Letters 8, 1798 (2008).
  * Chen  _et al._ [2015] Y. Chen, Q. Gao, Y. Wang, X. An, X. Liao, Y.-W. Mai, H. H. Tan, J. Zou, S. P. Ringer, and C. Jagadish, Determination of Young’s Modulus of Ultrathin Nanomaterials, Nano Letters 15, 5279 (2015).
  * Edgeworth  _et al._ [1984] R. Edgeworth, B. J. Dalton, and T. Parnell, The pitch drop experiment, European Journal of Physics 5, 198 (1984).
  * Sutardja and Oldham [1989] P. Sutardja and W. G. Oldham, Modeling of Stress Effects in Silicon Oxidation, IEEE Transactions on Electron Devices 36, 2415 (1989).
  * Senez  _et al._ [1994] V. Senez, D. Collard, B. Baccus, M. Brault, and J. Lebailly, Analysis and application of a viscoelastic model for silicon oxidation, Journal of Applied Physics 76, 3285 (1994).
  * Ojovan [2008] M. I. Ojovan, Viscosity and Glass Transition in Amorphous Oxides, Advances in Condensed Matter Physics 2008, 1 (2008).
  * LePage  _et al._ [2019] W. S. LePage, Y. Chen, E. Kazyak, K.-H. Chen, A. J. Sanchez, A. Poli, E. M. Arruda, M. D. Thouless, and N. P. Dasgupta, Lithium Mechanics: Roles of Strain Rate and Temperature and Implications for Lithium Metal Batteries, Journal of The Electrochemical Society 166, A89 (2019).
  * Allred  _et al._ [2004] C. L. Allred, X. Yuan, M. Z. Bazant, and L. W. Hobbs, Elastic constants of defected and amorphous silicon with the environment-dependent interatomic potential, Physical Review B 70, 134113 (2004).
  * Nasr Esfahani and Alaca [2019] M. Nasr Esfahani and B. E. Alaca, A Review on Size‐Dependent Mechanical Properties of Nanowires, Advanced Engineering Materials 21, 1900192 (2019).
  * Kluge and Ray [1988] M. D. Kluge and J. R. Ray, Elastic constants and density of states of a molecular-dynamics model of amorphous silicon, Physical Review B 37, 4132 (1988).
  * Shenoy  _et al._ [2010] V. Shenoy, P. Johari, and Y. Qi, Elastic softening of amorphous and crystalline Li–Si Phases with increasing Li concentration: A first-principles study, Journal of Power Sources 195, 6825 (2010).
  * Tan  _et al._ [1972] S. I. Tan, B. S. Berry, and B. L. Crowder, Elastic and Anelastic Behavior of Ion-Implanted Silicon, Applied Physics Letters 20, 88 (1972).
  * Szabadi  _et al._ [1998] M. Szabadi, P. Hess, A. J. Kellock, H. Coufal, and J. E. E. Baglin, Elastic and mechanical properties of ion-implanted silicon determined by surface-acoustic-wave spectrometry, Physical Review B 58, 8941 (1998).
  * Tang  _et al._ [2012] D.-M. Tang, C.-L. Ren, M.-S. Wang, X. Wei, N. Kawamoto, C. Liu, Y. Bando, M. Mitome, N. Fukata, and D. Golberg, Mechanical Properties of Si Nanowires as Revealed by in Situ Transmission Electron Microscopy and Molecular Dynamics Simulations, Nano Letters 12, 1898 (2012).
  * Plett [2004] G. L. Plett, Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs, Journal of Power Sources 134, 262 (2004).
  * Graells  _et al._ [2020] C. P. Graells, M. S. Trimboli, and G. L. Plett, Differential hysteresis models for a silicon-anode Li-ion battery cell, in  _2020 IEEE Transportation Electrification Conference & Expo (ITEC)_, Vol. 1 (IEEE, 2020) pp. 175–180.
  * Wycisk  _et al._ [2022] D. Wycisk, M. Oldenburger, M. G. Stoye, T. Mrkonjic, and A. Latz, Modified Plett-model for modeling voltage hysteresis in lithium-ion cells, Journal of Energy Storage 52, 105016 (2022).