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Dynamics and morphology of solid electrolyte

interphase (SEI)†
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Cite this: DOI: 10.1039/c6cp02816k

Fabian Single,‡ab Birger Horstmann‡*ab and Arnulf Latzabc
Received 27th April 2016,
Accepted 13th June 2016

DOI: 10.1039/c6cp02816k

www.rsc.org/pccp

We develop a novel theory for the continuous electrochemical for- capacity fade scales with the square root of time,18–20 a strong
mation of porous films to study the solid electrolyte interphase (SEI) on indication that SEI formation is a transport limited process.
lithium ion battery anodes. Existing SEI studies model a homogeneous This observation is explored in numerous theoretical studies
morphology and a single relevant transport mechanism. Our approach, which use a single rate determining transport mechanism to
in contrast, is based on two transport mechanisms and enables us to describe SEI growth. SEI formation controlled by solvent diffu-
track SEI porosity in a spatially resolved way. SEI thickness evolution sion is assumed by Pinson and Bazant21 and Ploehn,4 whereas
agrees with existing studies and is validated with experiments. This electron conduction mechanisms are considered by Peled,22
consistent approach is unprecedented in SEI modeling. We predict a Christensen,23 Li24 and Lin.25 Most studies describe the evolution
non-zero SEI porosity and the dependence of morphology on of a homogeneous SEI layer with a sharp interface and do not
transport properties. Additionally, we capture dual-layer chemistry attempt to account for spatial heterogeneity. Only a few models
and morphology. Analytic expressions which describe the parameter consider a spatially resolved interface with the electrolyte or an
dependence of all key properties are derived and discussed. inhomogeneous SEI.26,27
Despite substantial differences in the chosen rate-limiting trans-
The formation of a stable interfacial layer, the so-called solid port mechanism, all available models predict SEI thickness evolu-
electrolyte interphase (SEI), on graphite anodes has enabled the tion in agreement with experiments. Thus, they remain inconclusive
success of Li-ion batteries (LIBs).1 In these batteries, electrolyte with respect to the rate limiting process. Conclusions require
solvent is unstable at typical working potentials.2,3 Solvent further observable predictions with respect to SEI morphology,
reduction products form a thin layer separating anode and e.g., porosity and dual-layer structure. For this reason, we
electrolyte, the SEI. A well-formed SEI significantly slows down develop a theory for the growth of a porous and inhomogeneous
further electrolyte reduction, resulting in the excellent cycling layer. Our model studies the dynamics of film porosity in one
stability of LIBs. However, electrolyte reduction and SEI formation dimension, perpendicular to the substrate surface. This is
are never fully suppressed and remain the major cause for long- possible because the transport of all film precursors within
term capacity fade.4–6 the porous structure is taken into account.
This critical role has led to numerous experimental and In this work we formulate and parameterize our model
theoretical studies of the SEI. Experimental results are sum- specifically to describe SEI evolution, as depicted in Fig. 1.
marized in review articles and systematic studies.7–14 Recently, We apply the popular porous electrode theory to the nano-
isotope tracer experiments demonstrated the potential-dependent porous SEI. To this aim SEI composition and morphology are
dual-layer structure of the SEI.15–17 It is generally accepted that averaged in slabs parallel to the anode surface. Thus film
the SEI consists of a dense inner layer close to the electrode and growth is modeled along a single coordinate x, see Fig. 1(b).
a porous outer layer. Several long-term measurements find that Within the simulation domain we trace the transport of all
species involved in SEI formation. Here we assume electron
a
German Aerospace Center (DLR), Institute of Engineering Thermodynamics, conduction in the SEI material.23 In the electrolyte, solvent
Pfaffenwaldring 38-40, 70569 Stuttgart, Germany. molecules diffuse towards the electrode.21 The electrochemical
E-mail: birger.horstmann@dlr.de potential of lithium ions is assumed to be constant at all times
b
Helmholtz Institute Ulm (HIU), Helmholtzstraße 11, 89081 Ulm, Germany
c
and does not result in inhomogeneous reaction rates. This
Ulm University, Institute of Electrochemistry, Albert-Einstein-Allee 47, 89069 Ulm,
Germany
assumption is justified because lithium ion transport in the
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6cp02816k SEI28 is very fast compared to SEI growth, i.e., SEI growth
‡ These authors contributed equally to this work. consumes small amounts of lithium and transport quickly

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coupled to sink terms from SEI formation with mass balance

equations

@eci  
¼ div jD;i þ jC;i  n i s_i ; (3)
@t
P
where e ¼ 1  ei is the local porosity and nEC = 2/nDMC = 1 are
stoichiometric coefficients. According to Fick’s law, diffusion is
driven by concentration gradients jD,i = Digradci. Convection
is determined by the velocity v of the electrolyte jC,i = civ.
By treating the mixture as an incompressible fluid, we use the
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P E
volume constraint V ci ¼ 1 to eliminate the co-solvent
i
concentration.34 Because v is the center-of-mass velocity, we
require DDMC = DECMECVDMC/(MDMCVEC) with molar masses Mi.
Volume constraint and mass balance eqn (3) together deter-
mine the convective velocity35,36
X S E

E
divv ¼ V i  n i V i s_i þ V EC divðDEC  DDMC ÞgradcEC :
(4)
Fig. 1 (a) Cross section through graphite electrode, SEI and electrolyte
depicting all relevant species: solvent molecules EC, lithium ions Li+, and
Conservation of ‘‘electronic charges’’ is ensured via
electrons e. EC and e move in opposite directions (single headed
arrows). (b) Profile of the averaged SEI volume fraction along the axis : :
0 = div jE + F(2sLi2EDC + sLiMC), (5)
perpendicular to the electrode surface.
where the electron current depends on the electric potential F
through Ohm’s law jE = k gradF. We solve eqn (2)–(5) for the
restores local equilibrium. SEI is formed when reactions between
spatially-resolved dynamics of eLi2EDC, eLiMC, cEC, F, and v within
charge moving away from the electrode and solvent moving
the simulation domain [0,xmax].
towards the electrode occur. In summary, we model diffusion
Volume-averaged transport parameters Di and k contain the
of solvent and conduction of electrons. Therefore, electronic
effects of porosity and tortuosity. The Bruggeman ansatz relates
conductivity and solvent diffusivity are key parameters.
them to their bulk values using the local porosity and SEI
The bulk electrolyte phase is a binary mixture of ethylene
volume fraction eSEI = 1  e,
carbonate (EC) with co-solvent dimethyl carbonate (DMC), i.e.,
EC:DMC 3:7. As we focus on morphology, SEI chemistry is Di = ebD0i and k = e1.5 0
SEIk , (6)
further simplified by neglecting the salt anion. Because ionic
species are neglected, double layer effects29 cannot be included where 1.5 is the standard Bruggeman coefficient for conduction37
in our model. Only a single representative reduction reaction in porous media. For simplicity, we choose the same electronic
per solvent species is considered bulk conductivity k0 for all SEI compounds. We use large values
of b B 20 in our model, representing the difficulty of electrolyte
2EC + 2Li+ + 2e - Li2EDCk + RECm, (1a) transport in nano-pores.
: :
The compound production rates si = AiGri depend on specific
DMC + Li+ + e - LiMCk + RDMCm. (1b) :
surface areas Ai, surface site density G, and reaction rates ri. The
We choose lithium ethylene dicarbonate (Li2EDC) as a product latter are given by a symmetric Butler–Volmer expression,38,39
of EC reduction30,31 and lithium methyl carbonate (LiMC) from  n i  
DMC reduction.32 Gaseous reaction byproducts Ri are neglected. 1 kB T ci 2 EA FZ
r_ i ¼ exp sinh i ; (7)
Hereinafter, indices i refer to i = EC, DMC or i = Li2EDC, LiMC 2 h c0i kB T RT
depending on the phase (electrolyte/SEI) of the corresponding
variable/parameter. where solvent reduction is driven by the overpotentials
:  
The production rate si of each SEI compound drives the   RT ci
evolution of the volume fraction ei Zi ¼  F  F0i þ ln 0 : (8)
F ci
@ei S
¼ V i s_i ; (2) Reduction reactions are in equilibrium when potential and
@t
concentrations are F0i and c0i , respectively. The activation
where V% Si is the molar volume of SEI compound i. This means a barrier of the reaction EA is twice the desolvation energy of
solvation/precipitation mechanism33 is not considered. Solvent Li+ in EC.40,41 We represent the irreversibility of these reactions
:
molecules move through the electrolyte phase via diffusion by setting ri = 0 for negative Z, i.e., we disregard the oxidation
and convection. The evolution of solvent concentration ci is (SEI components are oxidized at F E 3.25 V42).

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A continuous expression is used for the specific surface area. ensure good passivation and are below the ones for artificial
As derived in the ESI,† SEIs.43 The microscopic mechanism underlying this conductivity
  is still under investigation. Besides conventional conduction,
6 a02 @ 2~ei
Ai ¼ e ~ei þ ; ~ei ¼ ei þ einit : (9) successive electron tunneling25 or neutral lithium interstitial
a0 6 @x2
diffusion28 are potential mechanisms.
This smoothes the porosity profile and distributes growth such In Fig. 2(c) we show that the potential increases linearly
that the SEI front has finite thickness. Additionally it enables from the value Ffinal at the electrode to F0EC at the SEI front.
propagation of SEI into the electrolyte as well as numerical The linearity demonstrates that crystallization inside the SEI
convergence. is negligible. A potential drop over the SEI interface is absent
Simulation details, such as initialization, boundary condi- because the formation reaction is fast. For a constant porosity
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tions and parameters are discussed in the ESI.† e* and a linear potential F(x,t) we can approximate the electric
current through the bulk SEI phase and calculate the thickness
Inert co-solvent evolution
We start our discussion with the special case of an inert S 1=2 S
: @L jE V Li2 EDC eSEI k0 DFEC V Li2 EDC 1
co-solvent, i.e., we disable co-solvent reduction (rDMC = 0) and ¼  ; (10)
@t 2F eSEI 2F L
study the growth of an SEI with homogeneous chemistry. A
typical evolution of SEI volume fraction is depicted in Fig. 2(a). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
We find that growth is concentrated at the SEI front whose 1=2 S
eSEI k0 DFEC V Li2 EDC pffiffi
spatial width lies in the order of a0. Thus, electron conduction ) LðtÞ ¼  t: (11)
F
through the SEI becomes the rate limiting process in our
model. The porosity inside the SEI attains a nearly constant We note that SEI growth is essentially governed by the potential
value eðxÞ  e ¼ 1  eSEI which we explain below. On closer difference DFEC = F0EC  Ffinal. Fig. 2(b) proofs the excellent
inspection we find that SEI volume fraction increases slightly agreement between experiment, simulation and eqn (11).
with distance from electrode. We derive an expression for the nearly constant SEI porosity
In our model the SEI thickness grows with the square root e* in this SEI layer. Our approach traces the SEI formation rate
of time in agreement with experiments (see Fig. 2(b)). It has in the frame co-moving with the SEI front
been shown previously that this can be rationalized based on a deðL; tÞ @e @e @L
single rate limiting transport process.4,21 Therefore, we obtain ¼ þ : (12)
dt @t @L @t
SEI conductivity by fitting the simulated thickness evolution to
experimental data. With capacity fade measurements of Liu With few assumptions, i.e., no convection and infinitely fast
et al.19 and an estimate for the electrode surface area by Pinson reactions, we find that e(L,t) in eqn (12) has a stationary
et al.21 we find k0 = 0.3 pS m1 at T = 15 1C and k0 = 4.5 pS m1 solution e*. This means that in time, the porosity in the co-
at T = 60 1C (with b = 25). These low electron conductivities moving frame tends towards this stable value. An implicit
expression for e* can be derived from eqn (12)
 
F 2 c0EC 1 1  e
kðe Þ ¼ Dðe Þ þb  : (13)
RT 2 e

Our simulations show that eqn (13) gives an excellent estimate

for the dependence of the porosity e* on the transport para-
meters. Very good quantitative agreement is observed for small
EC concentrations, see Fig. 2(d). It can be seen that b is the
parameter with the highest influence on film porosity. The film
porosity is a direct consequence of an interplay between solvent
species crossing the moving SEI front and SEI expansion. As the
film becomes denser, solvent transport into the film is slowed
down. Eventually further growth is distributed such that the
film expands and the density no longer increases. As shown in
Fig. 2(d), large values of b are needed to see this effect. At b = 10,
film density is nearly one, eSEI  0:98.
Fig. 2 Results with inert co-solvent. (a) Temporal evolution of the SEI
volume fraction eLi2EDC + einit (k0 = 0.3 pS m1, b = 25, T = 15 1C). (b) SEI Reactive co-solvent
thickness evolution from experiment19,21 (dots), simulation (dashed) and In the following, we discuss simulations with simultaneous
eqn (11) (lines). (c) SEI potential distribution at different stages of SEI
solvent and co-solvent reduction. Fig. 3(a) depicts the corres-
evolution, corresponding to (a). (d) Influence of b and k0 on eSEI , analytic
results from eqn (13) (dashed lines) compared to simulation results (dots).
ponding evolution of both SEI volume fractions. Next to the
The values were obtained by averaging eSEI(x) between 2 nm and 55 nm electrode, LiMC grows ‘‘on top’’ of the Li2EDC phase. This
after simulating the growth of a 60 nm thick layer. forms a dense inner layer with eSEI(x) E 1 while the porous

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It allows the validation of our model and/or an estimate of

unknown reaction properties from observable SEI properties.
In conclusion, we formulate a novel SEI growth model which
extends the common approach of transport limited models. Our
theory predicts long-term SEI thickness evolution in agreement
with previous models and experiments, i.e., we retain square-root
like growth. Additionally, we present the first continuum model
Fig. 3 (a) Temporal evolution of the SEI volume fraction with two
which predicts properties of SEI structure. The competition
reduction reactions (k0 = 4.5 pS m1, b = 25, T = 60 1C). (b) Same simulation, between two counter-moving transport mechanisms, i.e., electron
evolution of total and dense SEI layer thickness (lines) compared to conduction and solvent diffusion, leads to a non-zero SEI porosity.
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numerical solutions of the analytical approximation eqn (14a) (dashed). This is a novel insight into porous film growth beyond the standard
Additional numerical solutions with different Ljdense(t0) ( j = 1,. . .,10) indicate, case of SEI formation on graphite anodes. Two distinct formation
how SEI growth continues when R a Rstat (thin grey lines). The inset in the
bottom-right corner shows the corresponding evolution of R j =L j/Ljdense.
reactions result in a dual-layer structure with a dense inner layer
and a porous outer layer. Porosity is constant within each layer.
We can understand these properties and derive formulas for
outer layer with eSEI ðxÞ  eSEI remains. At F0EC = 0.8 V EC starts SEI thickness, SEI porosity, and thickness of the dense layer.
to create a SEI layer with pores containing DMC as shown in Long-term in situ experiments, similar to ref. 50 and 51, should
Fig. 2(a). When the potential drops below F0DMC = 0.3 V, DMC is allow to test and refine our predictions. We hope that this kind of
reduced to form LiMC. Consequently the dense layer appears modeling can be extended to lithium transport through the SEI
near the electrode where the potential is lower. This dual-layer and the effect of electrochemical double layers. This would
structure agrees with experimental observations.17 Similar to allow better understanding of SEI impedance spectra.
co-solvent reduction, it would emerge from a conversion of the This work was supported by the German Federal Ministry
primary SEI compound at low potentials.44 Because the reduction of Education and Research (BMBF) in the project Li-EcoSafe
potential of EC is higher than the one of the co-solvent (see (03X4636A). The authors would like to thank Erkmen Karaca for
Borodin et al.45,46), SEI mostly consists of EC reduction products, fruitful discussions. Further support was provided, by the bwHPC
as recently validated by Grey et al.47 initiative and the bwHPC-C5 project through associated compute
We compare the evolution of total SEI thickness L and the services of the JUSTUS HPC facility at the University of Ulm.
thickness of the dense layer Ldense in Fig. 3(b). Both quickly
attain their asymptotic values corresponding to square root like
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