A Thermodynamic Perspective for Formation of Solid Electrolyte Interphase in Lithium-Ion Batteries

Electrochimica Acta 173 (2015) 736–742

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A Thermodynamic Perspective for Formation of Solid Electrolyte Interphase in Lithium-Ion Batteries Xiaowen Zhan, Mona Shirpour, Fuqian Yang * Department of Chemical and Materials Engineering, University of Kentucky, Lexington, KY 40506, United States

A R T I C L E I N F O

A B S T R A C T

Article history: Received 31 March 2015 Received in revised form 19 May 2015 Accepted 24 May 2015 Available online 27 May 2015

This work studied the formation and growth of solid electrolyte interphase (SEI) under the assumption that the formation of an SEI layer is associated with the nucleation and growth of disk-like islands at the interface between an active material and nonaqueous electrolyte solution (NES). The supersaturation of NES favors the nucleation of the islands via lowering the Gibbs free energy with the contribution of interfacial energy and mismatch strain enegy to the nucleation process. Using a modified LifshitzSlyozov-Wagner model, the growth of the disk-like islands is analyzed. Explicit kinetic estimation for the change of the density and mean dimension of the islands as well as the degree of supersaturation of NES with time is obtained. The degree of supersaturation at time t is found to be proportional to the square root of time. ã2015 Elsevier Ltd. All rights reserved.

Keywords: Lithium-ion batteries SEI LSW theory Precipitation Nucleation Growth

1. Introduction Rechargeable lithium-ion batteries (LIBs) are the most popular power source for portable electronic devices due to their long cycle life and high energy density. Recent development in hybrid electric vehicles and electric grids has witnessed new demands for lithium-ion batteries of high energy density and stability. One of the critical issues in the development of LIBs is the formation of solid electrolyte interphase (SEI, named by Peled [1]), which forms at the electrolyte/electrode interface in the first stage of electrochemical discharge/charge [2]. Ideally, SEI will allow Li ions to pass and block electrons to the surface of electrodes to avoid electrolyte decomposition. Currently, the structure and composition of SEI, though widely studied, are still not fully understood. Major components in SEI, including inorganic substances, i.e., Li2CO3, Li2O, and LiF, and organic substances, i.e., LiCH3, LiOCO2CH3 and ROLi (R is an organic group dependent on the solvent), have been reported, as reviewed by Verma [3]. In general, an SEI layer, once formed, is not fixed in chemistry and properties. Lee et al. [4] observed that the SEI layer in cells with high degradation is thicker than that in cells with low degradation. Bryngelsson et al. [5] suggested that the SEI layer becomes thicker at low potentials. In addition, temperature has an even more pronounced effect on the evolution of SEI [6,7]. Matsuoka et al. [8] and Jeong et al. [9]

* Corresponding author. E-mail address: [email protected] (F. Yang). http://dx.doi.org/10.1016/j.electacta.2015.05.142 0013-4686/ ã 2015 Elsevier Ltd. All rights reserved.

suggested that adding additives, such as vinylene carbonate (VC), can effectively improve the performance of LIBs via facilitating the formation of a dense and solid SEI layer and thus greatly suppressing the decomposition of solvent. In general, the evolution of SEI layers, including growth, dissolution, and aging, plays an important role in determining the long-term performance of LIBs [10,11]. There are various models for the formation of SEI. Based on the X-ray photoelectron spectroscopy results, Kanamura et al. [12] suggested that SEI is a multi-layer structure. Peled et al. [11] suggested that an SEI layer is a mosaic microphase. Aurbach [13,14] presented a scenario for the formation of a multilayer film on the Li electrodes. Edström et al. [15,16] suggested a two-layer SEI structure with a dense inorganic layer close to the electrode surface and a porous organic or polymeric layer on top of it. All of the works reveal the layered structure of SEI in which the main components consist of the substances from the reductive decomposition of nonaqueous electrolyte solution (NES). However, they did not address the formation and growth mechanism of SEI, which likely play an important role in determining the performance of lithium-ion batteries. Recently, Zheng et al. [17] visualized the SEI formed on the silicon anode material three-dimensionally through a scanning force curve method, using a scanning probe microscope. A partial coverage (95%) of SEI on the anode was observed even after adding 2 wt% VC in electrolyte. Their results reveal the presence of single-, double-, and multi-layered structure of highly inhomogeneous SEI and suggest that the insoluble species in the NES form the first

X. Zhan et al. / Electrochimica Acta 173 (2015) 736–742

A list of symbols

C C1 CR Rc0 R Rc t S v R v x

DGT DGS DGV DGT (vc)max Dm

g e V r/z rmax D D0 l: u

l-

molar concentration of NES molar concentration of saturated NES equilibrium molar concentration at the edge of an SEI island initial radius of an SEI island mean radius of SEI islands critical radius of SEI islands time spreading parameter growth rate of SEI islands critical volume of total molecules or atoms in an SEI island critical dimension of an SEI island change of total Gibbs free energy change of surface energy change of volumetric Gibbs free energy maximum change of total Gibbs free energy change of the Gibbs free energy introduced by moving unit volume of molecules or atoms from the bulk NES phase to the interior of an SEI island interfacial free energy mismatch strain atomic volume of solute atoms dimensionless size of an SEI island maximum dimensionless size of an SEI island degree of supersaturation of NES initial degree of supersaturation of NES coefficient of convective mass transfer dimensionless time

ayer of the SEI film on the anode surface via absorption, nucleation and growth, following with the deposition of the reduced and formed lithium salts either on the formed-in-advance SEI islands or on bare areas which depends on thermodynamic and/or kinetic factors. Winter [10] pointed out that the protective properties of SEI were overemphasized. He suggested that the limited SEI performance may result from the fact that the protective SEI components are less (or even not) cation-conductive and conductive components are less protective. However, it doesn't preclude the consistent acceptance of the concept of passivation film due to simplicity and reasonability, i.e., a denser SEI layer usually brings in better performance of LIBs [8,9,17]. Due to the complex nature of SEI, the very first layer of SEI formed upon the contact between anode and an electrolyte solution may be of particular importance to the protective function of SEI [1,2], and thus should be understood theoretically. Well-known models like single particle model and porous electrode model proposed by Pinson et al. [18] have demonstrated the capability in the prediction of the capacity fade of LIBs, based on the formation mechanism of SEI. However, the studies were focused on the prediction of life time rather than on the mechanism of the SEI formation and growth. Yan et al. [19,20] applied the classical nucleation theory (CNT) model to explain the formation of SEI on graphite electrode substrates. They assumed a nucleus of spherical cap and used the change of total Gibbs free energy to determine the critical size of nucleus. It is believed that the SEI thickness ranges from
20 to several hundred angstroms [15–17,21]. Thus, the model of a spherical cap for the SEI nucleation [19] may not be able to capture the limit to the thickness of the SEI layer. In addition, the observation of the formation of flat flake

737

structures of
0.9 nm in thickness on the surface of HOPG (highly oriented pyrolytic graphite) electrode [22] suggests that a model different from the model of a spherical cap is needed to explain the formation of flat flake. Considering the observation of the formation of the flat flake structures on the surface of HOPG electrode [22], the model proposed by Yan et al. [19] is modified with the constraint of the SEI thickness. Instead of using a spherical cap for the nuclei of SEI, disklike islands of a fixed thickness qualitatively in accord with the flat flake structures are used in analyzing the formation and growth of SEI. In addition, the mismatch strain between the SEI and active materials is introduced in the model, and the LSW (LifshitzSlyozov-Wagner) theory is modified to provide a kinetic image for the growth of SEI. 2. Mechanistic model Consider the nucleation of an SEI island on the surface of an active material, as schematically shown in Fig. 1. The following assumptions are made to establish the mechanistic model: a) The surface of the active material is flat and smooth. b) There is little change of temperature and pressure during the formation and growth of SEI. c) The small nucleus is in a disk-like shape with a fixed height of h. d) The composition of the nucleus is homogeneous and uniform. For the heterogeneous nucleation of a disk-like island, the total change of Gibbs free energy DGT can be calculated as

DGT ¼ DGV þ DGs ;

(1)

where DGV is the change of the volumetric Gibbs free energy and DGs represents the contribution of interfacial free energies. For a spontaneous nucleation process, DGT must be negative. The term DGs, which is controlled by the absorption and de-absorption of atoms or molecules near the interfacial region and proportional to the interfacial area, is positive. Thus, DGV plays a critical role in the SEI nucleation which can be expressed as

DGV ðvÞ ¼ vDm þ vMe2 ;

(2)

where v is the volume of total molecules or atoms in the SEI island, and Dm is the change of the Gibbs free energy when moving unit volume of molecules or atoms from the bulk NES phase to the interior of the SEI island. Here, we assume that Dm is independent of the size of the SEI island. Note that Dm is positive for unsaturated NES and negative for supersaturated NES. The second term represents the mismatch strain energy due to the composition difference between the active electrode material and the NES components, where M is the biaxial modulus and M2 represents the mismatch strain energy density (strain energy per unit volume). The term DGs is calculated as
   DGs ¼ 2pRh þ pR2 g 1 þ pR2 g 2  g 3 ; (3) where g 1,g 2 and g 3 are the interfacial energy of SEI-NES, SEI-AMS (Active Material Substrate) and AMS-NES interfaces, respectively.

Fig. 1. Schematic of the nucleation of a disk-like island.

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For simplification, the interfacial energy of g1 is used for the lateral surface and top surface of the SEI island, which will not affect the results qualitatively. Note that more complicated model is needed to address the effect of multiple components and the contribution of individual interface energy between two layers. For a disk-like solid particle with height of h and volume v, the radius R is R2 ¼

v

:

(4)

ph

Substituting Eq. (4) into Eq. (3) yields
pffiffiffiffiffiffiffiffiffi v  v DGs ðvÞ ¼ 2 phv þ g 1 þ g 2  g 3 : h h

(5)

Substituting Eq. (5) and (2) into Eq. (1), one obtains the expression of the total change of the Gibbs free energy as

DGT ðvÞ ¼ DGV ðvÞ þ DGs ðvÞ pffiffiffiffiffiffiffiffiffi v  v ¼ vDm þ ð2 phv þ Þg 1 þ g 2  g 3 þ vMe2 ; h

h

(6)

which is simplified as pffiffiffiffiffiffiffiffiffi v  DGT ðvÞ ¼ 2g 1 phv þ g 1 þ g 2  g 3 þ hDm þ hMe2 : h

(7)

3. Results and discussion 3.1. Nucleation of an SEI island According to Eq. (7), the only parameter determining the characteristics of DGT (v) is v for the formation and growth of a disk-like island of a fixed height at the interface between an electrolyte and an active material. The following two cases are discussed. Case I: g 1 þ g 2  g 3 þ hDm þ hMe2  0: Fig. 2 shows the variation of DGT (v) with v. The value of DGT (v) increases with increasing the nucleus size v; nuclei are unstable and no islands or SEI will form. Case II: g 1 þ g 2  g 3 þ hDm þ hMe2 < 0:

Fig. 3. Variation of the change of total Gibbs free energy DGT ðvÞ with nucleus size v for case II.

It requires the change of the Gibbs free energy less than zero for the formation of disk-like islands. Fig. 3 shows the variation of the change of the Gibbs free energy DGT (v) with v. The value of DGT first increases and then decreases with increasing the value of v. The change of the Gibbs free energy DGT reaches a maximum value at vc as calculated by

@DGT ðvÞ ¼ 0; @v

(8)

which gives

ph3 g 1 2  ; g 1 þ g 2  g 3 þ hMe2 þ hDm 2

vc ¼ 

(9)

and

DGT ðvc Þmax ¼ 

ph2 g 1 2 : g 1 þ g 2  g 3 þ hMe2 þ hDm

(10)

Here DGT (vc) max can be considered as the energy barrier for the nucleation of a disk-like island. The critical size vc decreases with the increase of the driving force and the decrease of the energy barrier. The growth of disk-like islands starts for the size of nuclei larger than the critical size, and a continuous growth follows. For a fixed thickness, lateral growth will occur and the formation of a single layer structure prevails. Equation (7) can be rearranged as pffiffiffiffiffiffiffiffiffi

v h

DGT ðvÞ ¼ 2g 1 phv þ vDm þ ðg 1 þ g 2  g 3 þ hMe2 Þ:

Fig. 2. Variation of the change of total Gibbs free energy DGT ðvÞ with nucleus size v for case I.

(11)

For g 1 þ g 2 þ hMe2 < g 3, the decrease in the interface free energy will introduce the precipitation of originally soluble substance from unsaturated NES (Dm > 0). For g 1 þ g 2 þ hMe2  g 3 , the SEI formation will occur only for supersaturated NES (Dm < 0). No stable nuclei will form on the active material surface if g 1 þ g 2  g 3 þ hDm þ hMe2  0 i.e. the case I. Note that, in the above discussion, the spreading parameter S   ¼ g 3  g 1  g 2 is assumed to be less than zero, i.e. there is a decrease in the interface energy due to the SEI formation. The

X. Zhan et al. / Electrochimica Acta 173 (2015) 736–742

results indicate that unsaturated NES (Dm > 0) does not necessarily prohibit stable nucleation (e.g. wheng 1 þ g 2 þ hMe2 < g 3 ), and supersaturation assists the nucleation process. For a disk-like island of radius R, Eq. (7) can be written as   (12) GT ðhÞ ¼ pRh 2g 1 þ RDm þ RMe2 þ pR2 ðg 1 þ g 2  g 3 Þ: Let DGT ðhÞ ¼ 0, one can calculate the thickness h as S ; 2g 1 =R þ Dm þ Me2

h¼

  function f r2 ; u is

@f @ þ ðf V Þ ¼ 0: @u @r2 R

hupp

(19)

The mass conservation gives Q 0 ¼ D0 þ q0 ¼ D þ q;

(13)

(14)

(20)

where Q 0 is the total supersaturation and q0 is materials initially in the island with a fixed thickness of h. The parameter q is calculated as

which gives the upper bound of the SEI thickness, hupp , as S ¼ ; Dm þ Me2

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q ¼ pR2c0

Z1

f r2 d r2 :

(21)

0

under the condition of 2g 1 =R þ Dm þ Me2 < 0 for all R. It is evident that the maximum SEI thickness is dependent on the spreading parameter, the change of chemical potential, and the strain energy density due to the mismatch. For Dm þ Me2 > 0, no SEI film will form.

Dividing Eq. (20) by Q 0, one obtains 1¼

D0 1 Q0x

Z1 þ k f r2 d r2

3.2. Growth of SEI islands Generally, the growth of the SEI islands will prevail for the islands with radius larger than the critical radius. To examine this growth behavior, the classical LSW theory is modified to analyze the lateral growth of the SEI islands with a given thickness of h, in which the growth behavior is controlled by atomic/molecular diffusion. Note that the growth of the SEI islands is 2D in contrast to the 3D growth in the classical LSW theory [23] and related theories [24–27]. Vengrenovich et al. [28,29] proposed a mechanism in which 3D growth of quantum dot was treated as 2D growth of disklike islands and considered radial 2D high-speed diffusion channels controlled by dislocation diffusion. Their analyses are essentially different from the model presented in this work, in which surface diffusion is the mechanism for mass transport.

,(22)

0

with k ¼ pR2c0 ð1=Q 0 Þ. It can be seen that n ¼

Z1

f dr2 is the

0

number of islands in a volume v= (unit area) h. It can be considered as the number of islands per unit area since h is fixed. Let z ¼ r2 =x2 ðuÞ and t ¼ lnx2 ðuÞ and   fðz; t Þdz ¼ f r2 ; u dr2 : (23) Substituting Eq. (23) into Eq. (19) yields

@f @ þ fvðz; g Þ ¼ 0; @t @z

(24)

which gives 3.2.1. Formulation Define C 1 as the concentration of saturated NES and C R as the equilibrium concentration at the edge of an SEI island. There is

a

CR ¼ C1 þ ; R

(15)

dz=dt ¼ vðz; g Þ ¼


dR a ; ¼ lðC  C R Þ ¼ l D  dt R

(16)

where C is the concentration of NES, D ¼ C  C 1 is the degree of superesaturation, and l is the coefficient of convective mass transfer. Let Rc0 be the radius of the SEI island and D0 be the degree of supersaturation at t=0. Introduce the following two parameters T¼

Rc 0

2

la

and Rc0 ¼ a=D0 :

(17)

Using the dimensionless variables of r ¼ R=Rc0 ; xðtÞ ¼ D0 =D; and u ¼ t=T, Eq. (16) can be written as 
r dr2 1 : ¼2 x du

(18)

Let f ðr2 ; u Þ be the area distribution function of the islands over the substrate with a growth rate of V R ¼ dr2 =du : According to the LSW theory [23], the equation of continuity for the unknown

(25)

where g ðt Þ ¼ 2du=dx2 . Thus Eq. (22) can be written as 1¼

where a ¼ ðg 1 =kT ÞVC 1 and V is atomic volume of solute atoms. The diffusion current at the edge of the island is j¼

pffiffiffi  z  1 g ðt Þ  z;

D0 Q0

t

e2 þ ket

Z1

fðz; t Þzdz;

(26)

0

with the initial conditions as fju¼t ¼0 ¼ f 0 ðzÞ and zjt ¼0 ¼ r2 . If the solution of Eq. (25) under the initial condition zjt¼0 ¼ y is written as zðy; t Þ, Eq. (26) can be then expressed in terms of the initial distribution function f 0 ðzÞ as 1

D0 Q0

t

e2 ¼ ket

Z1

f 0 ðyÞzðy; t Þdy;

(27)

y0 ðt Þ

where y0 ðt Þ is the solution of z½y0 ðt Þ; t  ¼ 0. 3.2.2. Asymptotical analysis During the growth of SEI islands, the degree of superesaturation D will decrease with time. Note thatx ¼ D0 =D, so 1=x will also decrease with time, and g ðt Þ can approach 1, 0 or constant ast ! 1. Following similar analysis as LSW theory [23], one can conclude that g ðt Þ must approach asymptotically a constant value g 0 so that 0 < q=Q 0 < 1 and Eq. (26) holds. With g ðt Þ being constant in Eq. (25), the general solution has the following form

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X. Zhan et al. / Electrochimica Acta 173 (2015) 736–742

1 : gðz; g Þ

fðz; t Þ ¼ xðt þ cÞ

where gðz; g Þ ¼ vðz; g Þ, c ¼

(28) Zz

g1 ðz; g Þdz and x is an unknown

0

function. Inserting Eq. (28) into Eq. (26), it can be seen that, to satisfy Eq. (26) or to make xðt þ cÞ asymptotically timeindependent, x must be in the form of

xðt þ cÞ ¼ AeðtþcÞ :

(29)

This result suggests that fðz; t Þ can be expressed as fðz; t Þ ¼ et Fðz; g Þ. Thus, the asymptotic forms of Eq. (24)–(26) can be written respectively as F þ vðz; g Þ

dF dv þ F ¼ 0; dz dz

For g ¼ g 0, Fðz; g 0 Þ can be expressed as 8   < Aecðz;g 0 Þ  1 ; z  z0 ¼ 4 ; F z; g 0 ¼ g z; g 0 : 0; z  z0 ¼ 4 where     g z; g 0 ¼ v z; g 0  0; Zz

Z1 1 ¼ k Fzdz:

(37)

(30) 1

pffiffiffi  dz ¼ vðz; t Þ ¼ z  1 g  z with g ¼ constant; dt

(36)

  pffiffiffi  4 z j g1 ðz; g Þdz ¼ 2ln 2  z  pffiffiffi z2 0 0 pffiffiffi

pffiffiffi z 2 z  pffiffiffi ; ¼ 2 ln 2 z2

c¼

(35)

k

A¼Z

z0

(31) 0

ec g zdz ðz;g 0 Þ

¼

Q0

pRc0 2 ð1:11Þ

Q  0:29 02 : Rc 0

(38)

Note that (32)

0

Zz0

c

e

lim

z0 !4 0

One can plot vðz; g Þ as a function of z (see Fig. 4) to determine the value of g 0 that corresponds to the stable solution of Eq. (30). From the curves, the value of g 0 represents the condition for the   growth rate of the island size, v z; g 0 ; which satisfies the following equations pffiffiffi   v z; g 0 ¼ ð z  1Þg 0  z ¼ 0 (33)

pffi Zz0 p2ffi z zdz 4ze z2   ¼ lim pffiffiffi 4  1:11; z0 !1 g z; g 0 z2

(39)

0

which is calculated by Wolfram Mathematica 10. Thus, one has 8 c     < Aet e  ¼ nðt Þp z; g 0 ; z  z0 F z; g 0 ¼ ; (40) g z; g 0 : 0; z  z0 with

dv g j ¼ p0ffiffiffi  1 ¼ 0 dz z¼z0 2 z

(34)

nðt Þ ¼

Z1

fðz; t Þdz ¼ Aet

(41)

0

Solving Eq. (33) and (34) gives g ¼ g 0 ¼ 4 and z0 ¼ 4 ¼ g 0 .

pffiffiffi 2 z pffiffiffi   ec 4e z  2 p z; g 0 ¼   ¼ pffiffiffi 4 ; z  z0 : > g z; g > z2 0 > : 0; z  z0 8 > > > <

(42)

Here, nðt Þ is the number of islands per unit area. Note that the variable p, though not mathematically proper to represent probability, is used here due to its efficiency in characterizing the island density over certain size range. Letting

pffiffiffi  pffiffiffi     dz pffiffiffi ¼ 2 z p z; g 0 ; p z; g 0 ¼ p z; g 0 (43) d z we have 8 > > > > <

pffiffiffi 2 z pffiffiffi p ffiffi ffi pffiffiffi  ec 8 ze z  2 pffiffiffi pffiffiffiffiffi p z; g 0 ¼   ¼ pffiffiffi 4 ; z  z0 ; > g z; g 0 > z2 > > p ffiffi ffi pffiffiffiffiffi : 0; z > z0

Fig. 4. Rate of change vðz; t Þ as a function of z wheng ¼ g 0 ; g < g 0 and g > g 0 .

(44)

which represents the probability that an island has the mean dimension between ðr=xÞ and ðr=xÞ þ dðr=xÞ (see Fig. 5).

X. Zhan et al. / Electrochimica Acta 173 (2015) 736–742

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From Eq. (51), the rate of growth is given by dR 1 ¼ ht2 ; v ¼ R dt

(52)

where h ¼ ðla=8Þ1=2 . Combining u ¼ t=T with x2 ðu Þ ¼ u=2, the degree of supersaturation at time t is given as

DðtÞ ¼

D0

xðuÞ

¼ vt2 ; 1

(53)

where v ¼ ð2a=lÞ1=2 . Substituting Eq. (53) in Eq. (52) gives the relationship between the rate of growth and the degree of supersaturation as

h

l

v ¼ D¼ D R v 4

Fig. 5. Probability p

pffiffiffi  pffiffiffi z; g 0 as a function of disk dimension z.

pffiffiffi According to dz=dc ¼ z  ð z  1Þg 0 , one has pffiffiffi z1¼

Zz0 0

" # pffiffiffi  ec z  1 zec z0   dz ¼ j ¼ 0: g0 0 g z; g 0

It follows that pffiffiffi r ¼ z x ¼ xðuÞ:

(45)

(46)

For g 0 ¼ 2du =dx2 =4, there is 1 x2 ðuÞ ¼ u: 2

(47)

From Eq. (46), one obtains z¼

z r2 =x2 r2 R2 =Rc0 2 R2 ¼ ¼ ¼ ¼ : z r2 =x2 r2 R2 =R 2 R2 c0

(48)


   2 With f R2 ; u dR2 ¼ F z; g 0 dz and dz ¼ dR2 =R , the function of f ðR2 ; uÞ can be expressed as !
 R2 1 2 f R ; u ¼ nðuÞp 2 2 ; R R

(49)

with nðuÞ ¼ m

Q0 R

2

¼ bu

1

1 2 R ¼ lat 2 Here, m  0:29 and b  0:58Q 0 ðlaÞ1 .

(50)

(51)

(54)

It means that the rate of growth is proportional to the degree of supersaturation if the mass transfer coefficient l is considered as constant. This relationship is qualitatively in accord with the statement made by Pinson et al. [18] in the single particle model of SEI formation. Lifshitz et al. [23] suggested that other secondary effects, including strain energy, have no effect on the qualitative conclusions in analyzing the diffusion driven process. They also suggested that one possible way to consider the effect of elastic strain on diffusion rate is to use effective values for C 1, a, and diffusion coefficient D (or l). However, such modifications will not change the asymptotic behavior and their stability. It needs to be pointed out that the modified LSW analysis is focused on 2D lateral growth of SEI islands and the results are significantly different from those obtained in the classical LSW theory. However, one similarity is observed for the determination of critical radius, and large islands (R > Rc ) tend to coarsen via capturing more matter from small islands (R < Rc ) and the supersaturated solution. From the probability distribution function of island dimensions (see Fig. 5), one notes that the maximum island size is twice as large as the critical size (rmax ¼ 2x). This result suggests that the islands cannot grow unlimitedly. Correspondingly, the 3D approximation in classical LSW theory gives rmax ¼ 1:5x. There exist certain size restrictions in the lateral growth of the SEI islands. No SEI islands with infinite radius will form simply through the mass transport between islands and solution, and local mass transport will occur between islands to reduce surface energies for continuous growth of islands. Also, both the mean dimension R and critical dimension x increase asymptotically with a time dependence of t1=2 while the degree of supersaturation and number of islands per unit area decrease asymptotically as t1=2 and t1 respectively. The above analysis provides a kinetic process for the formation of the first single SEI layer. Further growth of SEI on the formed layer will proceed through mass transport and deposition in the direction parallel to the normal of the island surface, i.e. the direction of the lithium migration to form two- or multi-layered structure. Further study is needed to understand the growth of SEI layers. 4. Conclusion The formation and growth of SEI disk-like islands were analyzed by using the theories of thermodynamics and kinetics, in which the thickness of SEI islands remain unchanged during growth. The result reveals that the driving force for the nucleation of a SEI island is controlled by the decrease of the chemical potential and

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the nucleation occurs under the condition of hDm þ g 1 þ g 2  g 3 þ hMe2 < 0. For the sum of the interface free energy change and mismatch strain energy change being negative, the nucleation of SEI islands can occur even for unsaturated NES (Dm > 0). Increasing the driving force (or decreasing the energy barrier) results in the formation of SEI islands with smaller critical radius which assists the formation of more SEI islands. For the lateral growth of the SEI islands with a fixed thickness, the 2D growth driven by diffusion was analyzed via a modified LSW theory. The result reveals that large islands (R > Rc ) grow at the expense of small ones. Generally, the SEI islands grow fastest at the beginning when the supersaturation degree stays at its maximum. The growth rate of SEI islands is proportional with the degree of supersaturation. It is impossible to grow SEI islands of infinite radius simply through the mass transport between islands and solution since the maximum radius of SEI islands is only twice the critical dimension. Local mass transport will occur between islands to reduce surface energies for continuous growth of islands. In summary, under specific conditions, decomposition products of NES components can precipitate on the surface of active materials and finally form a single SEI layer with a thickness of h after heterogeneous nucleation and lateral growth of SEI islands. References [1] E. Peled, the electrochemical-behavior of alkali and alkaline-earth metals in non-aqueous battery systems - the solid electrolyte interphase model, Journal of the Electrochemical Society 126 (1979) 2047–2051. [2] P. Arora, R.E. White, M. Doyle, Capacity fade mechanisms and side reactions in lithium-ion batteries, Journal of the Electrochemical Society 145 (1998) 3647–3667. [3] P. Verma, P. Maire, P. Novak, A review of the features and analyses of the solid electrolyte interphase in Li-ion batteries, Electrochimica Acta 55 (2010) 6332–6341. [4] J.T. Lee, N. Nitta, J. Benson, A. Magasinski, T.F. Fuller, G. Yushin, Comparative study of the solid electrolyte interphase on graphite in full Li-ion battery cells using X-ray photoelectron spectroscopy, secondary ion mass spectrometry, and electron microscopy, Carbon 52 (2013) 388–397. [5] H. Bryngelsson, M. Stjerndahl, T. Gustafsson, K. Edstrom, How dynamic is the SEI? Journal of Power Sources 174 (2007) 970–975. [6] G.R. Zhuang, Y.F. Chen, P.N. Ross, The reaction of lithium with dimethyl carbonate and diethyl carbonate in ultrahigh vacuum studies by X-ray photoemission spectroscopy, Langmuir 15 (1999) 1470–1479. [7] H.H. Lee, C.C. Wan, Y.Y. Wang, Thermal stability of the solid electrolyte interface on carbon electrodes of lithium batteries, Journal of The Electrochemical Society 151 (2004) A542–A547. [8] O. Matsuoka, A. Hiwara, T. Omi, M. Toriida, T. Hayashi, C. Tanaka, Y. Saito, T. Ishida, H. Tan, S.S. Ono, S. Yamamoto, Ultra-thin passivating film induced by vinylene carbonate on highly oriented pyrolytic graphite negative electrode in lithium-ion cell, Journal of Power Sources 108 (2002) 128–138. [9] S.-K. Jeong, M. Inaba, R. Mogi, Y. Iriyama, T. Abe, Z. Ogumi, Surface film formation on a graphite negative electrode in lithium-ion batteries: atomic force microscopy study on the effects of film-forming additives in propylene carbonate solutions, Langmuir 17 (2001) 8281–8286.

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