Modeling detailed chemistry and transport for solid-electrolyte-interface (SEI) films in Li–ion batteries

Electrochimica Acta 58 (2011) 33–43

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Modeling detailed chemistry and transport for solid-electrolyte-interface (SEI) films in Li–ion batteries Andrew M. Colclasure a , Kandler A. Smith b , Robert J. Kee a,∗ a b

Engineering Division, Colorado School of Mines, Golden, CO 80401, USA National Renewable Energy Laboratory, Golden, CO 80401, USA

a r t i c l e

i n f o

Article history: Received 14 June 2011 Received in revised form 17 August 2011 Accepted 18 August 2011 Available online 19 September 2011 Keywords: Li–ion battery Solid-electrolyte-interface Computational model

a b s t r a c t The thin solid-electrolyte-interface (SEI) films that grow around electrode particles play important roles in Li–ion battery performance. The objective of the present paper is to develop and apply models of SEI behavior that incorporate detailed chemical kinetics and multicomponent species transport. Species- and charge-conservation equations are derived and solved within the SEI film. The SEI model is coupled at its boundaries with an intercalation model within the electrode particles and with Li–ion transport and chemistry at the interface between the SEI and the electrolyte solution. The results of the model provide new insights concerning the influences of the intercalation fraction and cycling rate on SEI growth rates. The model also provides new insight concerning the influence of the SEI film on reversible potential and interfacial resistance. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction During lithium-ion (Li–ion) battery operation and storage, thin solid-electrolyte-interface (SEI) films grow on the surfaces of the graphite anode particles. Although the SEI films provide a beneficial stabilization function, they also contribute to deleterious internal resistance growth and capacity loss. This paper develops a computational model that enables the incorporation of detailed electrolyte reduction chemistry, coupled with the transport of Li+ in the electrolyte solution and the intercalation of Li within electrode particles. The results from the model provide new quantitative insight concerning the effects of SEI growth on battery performance. Fig. 1 illustrates a Li–ion cell at the porous-electrode scale, highlighting carbon anode particles with SEI films. Using single electrode particles, this paper concentrates specifically on the chemistry and transport processes associated with SEI growth. As developed previously by Colclasure and Kee, the model incorporates non-ideal thermodynamics and detailed chemical kinetics [1]. Chemical kinetics at the interfaces between the SEI and the electrode particle and the electrolyte solution are based upon those developed by Aurbach et al. [2] to describe the two-electron reduction of ethylene carbonate (EC) to form lithium carbonate. Although the present paper is concerned with single electrode particles, the models are structured to be incorporated into membrane electrode

∗ Corresponding author. Tel.: +1 303 273 3379; fax: +1 303 273 3602. E-mail address: [email protected] (R.J. Kee). 0013-4686/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2011.08.067

assembly models such as those developed by Newman and colleagues [1,3–6]. The model is applied to predict how the SEI growth rate depends upon the intercalation fraction and cycling rate. The results show that the concentration and electric-potential profiles within the SEI depend weakly on the current density, resulting in a weak dependence on the cycling rate. However, the concentration and electric-potential profiles depend greatly upon the intercalation fraction, with high intercalation fractions accelerating film growth rate. The model predicts that the film thickness, and hence capacity loss owing to trapped lithium ions, is proportional to the square root of time. The film resistance, which is high at low intercalation fractions, is predicted to be caused primarily by transport within the SEI and thermal chemistry at the SEI–graphite interface. The charge-transfer chemistry at both SEI interfaces is relatively facile and does not contribute significantly to cell polarization. 2. Prior research and literature Surface films form on both cathode and anode particles. However, SEI chemistry on anode particles, especially graphite, has been more extensively studied and is found to be more critical to battery lifetime and performance [7,8]. Aurbach attributes the increased importance of the anode SEI film to thermodynamic stability differences [7]. Graphite electrodes operate with a potential of 0.3–0.0 V relative to a lithium reference electrode (Li/Li+ ). The minimum voltage for the electrochemical stability window of typical organic-based electrolytes is around 1.3 V (Li/Li+ ) [9]. If the electrode potential is reduced below this potential, the electrolyte

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A.M. Colclasure et al. / Electrochimica Acta 58 (2011) 33–43

Fig. 1. Section of a Li–ion cell showing detail of SEI film on carbon anode particles.

solution becomes thermodynamically unstable and will spontaneously reduce. Thus, an anode surface film must form to block rapid reduction of the electrolyte solution. Ploehn et al. [10] modeled SEI film growth on planar graphite surfaces as functions of temperature and concentrations within the surrounding electrolyte solution. In their model, a solvent species from the electrolyte solution diffuses through the SEI and is reduced at the graphite–SEI interface. They assume that the dominant reaction is the irreversible two-electron reduction of EC (C3 H4 O3 ) to form lithium carbonate. Ploehn et al. also assume an excess availability of lithium ions and electrons at the graphite–SEI film interface and that solvent diffusion limits SEI film growth [10]. By further assuming the solvent concentration vanishes at the graphite–SEI interface, Ploehn et al. developed an analytic expression to show that SEI growth under open-circuit conditions is proportional to the square root of time. In cell-level models, SEI chemistry is often neglected or represented simply as a constant resistance [11]. However, incorporating this important chemistry into battery models in a more fundamental manner enables the investigation of SEI resistance, SEI film thickness, and open circuit potential as functions of elapsed time, state of charge (SOC), and cycling conditions. Christensen and Newman developed a detailed chemistry model to predict SEI growth on a planar graphite surface [12]. An irreversible elementary reaction mechanism was developed to describe lithium intercalation and the reduction of EC to form a single-species SEI film (lithium carbonate). Transient charge- and species-conservation equations were solved as functions of position within the film. In addition to predicting concentration and electric-potential profiles within the SEI film, Christensen and Newman investigated the effects of intercalation fraction and transference number within the film on film growth rate. The present model builds upon the Christensen and Newman reaction mechanisms, implemented in the context of single spherical electrode particles. However, the present model represents a significant advancement of the Christensen and Newman model. Because intercalation and electric-potential profiles are determined within the graphite particle, the present model predicts SEI growth throughout charge and discharge cycles, which Christensen and Newman’s model could not. Also, the present model uses the Redlich–Kister equation to represent the non-ideal effects of the lithiated graphite. Further, the present model results

provide new insight about how physical parameters and operating conditions affect SEI growth. Assuming that variations within the electrolyte phase are negligible, electrode function can be represented with a single spherical particle [13]. Such models are usually appropriate for low discharge/charge rates, say below 2 C. Assuming spherical symmetry is clearly a major simplifying assumption, enabling onedimensional models. Three-dimensional X-ray tomography and focused-ion-beam–scanning-electron-microscopy show that the actual electrode microstructure is highly complex, including nonspherical, connected particles with internal pores, cracks, and grain boundaries [14,15]. Such features certainly compromise shrinkingcore lithiation models. Moreover, it is known that the thickness, composition, and structure of the SEI can vary significantly along a graphite particle surface [16]. Nevertheless, results from the relatively simple spherical model provide significant insight at a small fraction of the computational cost associated with geometrically correct three-dimensional models.

3. Model formulation The model solution domain in the present work is restricted to a single graphite particle surrounded by electrolyte solution. The graphite particle, SEI film, and electrolyte solution are modeled as concentric spheres. With this geometric configuration, SEI film growth can be studied with a one-dimensional model. The model considers four phases – electrolyte (E), film surface (Ss ), bulk film (Sb ), and bulk graphite (C6 ). Each species is associated with a phase, with the phase being identified by the label within the parentheses or a subscript for a minor film species not occupying a lattice site.

3.1. Minor species conservation within the SEI Charged-species transport within the SEI is driven by electrochemical potential gradients (i.e., concentration and electricpotential gradients). Because of low minor-species concentrations (i.e., electrons e−b , interstitial lithium ions Li+b , and bulk lithium S

S

vacancies V− (Sb )), dilute solution theory is appropriate. Species conservation may be represented generally as

A.M. Colclasure et al. / Electrochimica Acta 58 (2011) 33–43

∂Ck = ∇ · Jk + ω˙ k , ∂t

(1)

where Jk and ω˙ k represent the molar diffusion fluxes and homogeneous chemical production rates of the kth species. Restricting attention to one-dimensional spherical coordinates, the radial r component of the diffusive flux is

Jk = −Dk

∂Ck zk FCk Dk ∂ − , RT ∂r ∂r

(2)

whereDk and Ck are the diffusion coefficient and molar concentration of species k, respectively, and the electric potential is represented as . Other variables include the Faraday constant F, the gas constant R, and the species charge zk . Substituting Eq. (2) into Eq. (1) yields

∂Ck 1 ∂ = 2 r ∂r ∂t

 r 2 Dk

∂Ck ∂r

 +

1 ∂ r 2 ∂r

 r2

zk F ∂ D C RT k k ∂r

 + ω˙ k .

(3)

The concentration of filled lithium lattice sites CLi(Sb ) is evaluated in

terms of the concentration of lithium lattice sites as CLi(Sb ) = C(Sb ) − CV− (Sb ) . Eq. (3) is a second order parabolic equation, requiring two boundary conditions for solution. The species flux entering and exiting the SEI is established by the heterogeneous chemistry at the film interfaces and film growth rate. At the SEI–electrode interface (4)

where rp is the electrode particle radius and s˙ k,B is the species molar production rate by heterogeneous reaction. At the electrolyte-SEI film interface

 Jk − Ck

∂ı ∂t

3.2. Charge neutrality within SEI The electric-potential profile within the SEI is determined to maintain charge neutrality, K 

zk Ck = 0.

(6)

k=1

It may be noted, however, that Eq. (6) is only strictly correct when the length scale for charge separation is small compared to the thickness of the film [12]. Using the electrical permittivity of lithium carbonate as  = 4.3 × 10−11 C V−1 m−1 and with average lithium-ion and electron concentrations between 1.3 × 10−2 and 1.3 × 10−3 kmol m−3 , the Debye lengths are in the range 2.1 ≤ D ≤ 0.66 nm. Since the SEI is tens of nanometers thick, a small error may be incurred by forcing strict charge neutrality. Enforcing charge neutrality as a strict constraint (Eq. (6)) directly can deleteriously affect numerical convergence in solving the full coupled system of equations. To facilitate computation, the constraint is differentiated to produce a transient differential equation as K  ∂C

k

k=1

∂t

zk = 0.

(7)

Substituting the species-conservation equation (Eq. (1)) converts the charge-neutrality constraint to a differential equation as K  ∂(r 2 Jk )

zk

Jk |r=rp = s˙ k,B ,

35

k=1

∂r

= 0.

(8)

Because the constraint has been differentiated, Eq. (8) is valid only if the initial concentration profiles satisfy charge neutrality (Eq. (6)). 3.3. Film growth rate

 |r=rp +ı = −s˙ k,S ,

(5)

where ı is the SEI thickness and s˙ k,S is the species molar production rate by heterogeneous reaction at the exterior SEI surface. Species diffusion coefficients within the SEI are not well known and therefore are used as adjustable parameters in the model. The diffusion coefficients for interstitial lithium ions and lithium vacancies are assumed to be equal, with the value being assumed such that the overpotential required to move lithium ions through the film is reasonable. Here, the nominal value DLi+ = DV− (Sb ) = Sb

2 × 10−14 m2 s−1 is within the range reported by Christensen and Newman [12]. Because experiments indicate that capacity loss during storage is approximately proportional to the square root of time, it is reasonable to expect that the SEI thickness during storage is roughly proportional to the square root of time, ı ∝ t0.5 [10,17]. This functional relationship is attributed to mass transport limitations. For typical film thicknesses between 10 and 100 nm, mass-transport limitation requires that the diffusion coefficient for either Li+ or e− must be below 1 × 10−18 . However, with DLi+ < 1 × 10−18 , unreSb

alistically high overpotentials would be required to drive lithium ions through the film. Therefore, the electron diffusion coefficient is assumed to be De− = 5 × 10−19 m2 s−1 . The SEI is modeled as an Sb

“ideal film” in the sense that the diffusion resistance for electrons is much higher than the diffusion resistance for lithium ions.

Film growth rate depends upon the heterogeneous production rate of lithium carbonate at the SEI–electrolyte interface. The SEI is assumed to be a dense layer of lithium carbonate. Thus, the SEI thickness ı is determined by solving

∂ı WLi2 CO3 , s˙ = Li2 CO3 Li2 CO3 ,S ∂t

(9)

where s˙ Li2 CO3 ,S is the net production rate of bulk-phase lithium carbonate at the SEI–electrolyte interface. The molecular weight and density of lithium carbonate are WLi2 CO3 = 0.07389 kg mol−1 and

Li2 CO3 = 2.11 × 103 kg m−3 , respectively. Because the film thickness varies with time, it is convenient to introduce a coordinate transformation that enables solution on a fixed computational mesh network [12]. A new independent variable, , is introduced as =

r − rp , ı

(10)

with the transformed species continuity equation (Eq. 3) becoming

∂Ck  ∂ı ∂Ck 1 ∂ = + 2 2 ı ∂t ∂ ı r ∂ ∂t +

1 ∂  2 r 2 ∂



r2



∂C r Dk k ∂



2

zk F ∂ D C RT k k ∂



+ ω˙ k .

(11)

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A.M. Colclasure et al. / Electrochimica Acta 58 (2011) 33–43

3.4. Species conservation within electrode particle The graphite particle is modeled with two phases – lithium vacancies and intercalated lithium. The radial concentration profiles can be determined by solving

∂CLiI (B) ∂t

1 ∂ = 2 r ∂r



2

r DLiI (B)

∂CLiI (B)



,

∂r

(12)

where CLiI (B) is the concentration of intercalated lithium and DLiI (B) is the diffusion coefficient. The solution depends upon the boundary conditions. Radial symmetry requires that the flux of intercalated lithium must vanish at the center of the particle r = 0. The flux of intercalated lithium into the particle is determined as DLiI (B)

∂CLiI (B) ∂r

= s˙ LiI (B),B ,

(13)

where s˙ LiI (B),B is the net chemical production rate of intercalated lithium at the electrode-SEI interface. The electrical conductivity of graphite is between 10 and 100 S m−1 [11,12]. With such high electrical conductivity, the electric potential within the particle B is assumed to be spatially uniform, varying as a function of time alone. The electric potential of the graphite particle is found by solving se−

C6

,B F

− id = 0,

where se−

C6

,B

(14)

is the net production rate at the graphite–SEI film

interface of electrons contained within the bulk graphite phase, and id is a specified electrical current density at the graphite particle surface. The SEI model varies the electrode electric potential such that the specified current is produced. A small false-transient term is added to Eq. (14) to assist numerical stability. The resulting equation is exp(−kB t)

∂B = se− ,B F − id , C6 ∂t

(15)

where kB is a constant that is typically set to kB = 0.1. Thus, the transient term becomes negligible after approximately 100 s and the specified current density is satisfied. The electric potential of the electrode B is measured relative to the fixed electrolyte electric potential E = 0. 3.5. Conservation within the electrolyte solution Species transport (i.e., Li+ and EC) within the electrolyte solution that surrounds the SEI is considerably faster than the

solid-state diffusion inside the graphite particle [1,11]. Therefore, the electrolyte solution is treated as a continuously stirred tank reactor [18]. That is, the spatially uniform concentrations within the electrolyte solution are functions of time alone. The electrolyte solution at its outer radius rE,o is surrounded by a fixed, impermeable enclosure. Thus, species can only enter and exit the electrolyte solution via heterogeneous chemistry at the SEI surface. The species continuity equation within the electrolyte solution is VE

∂Ck,E = s˙ k,S Aelyt,E , ∂t

(16)

where Ck,E is the concentration of species k within the electrolyte solution. Eq. (16) is valid for all solvents, ionic species, and gasphase species (ethane) contained within the electrolyte solution. Because the nanometer-scale SEI thickness is so small, the electrolyte volume is assumed to be nearly constant (i.e., VE is not inside time derivative of left hand side of Eq. (16)). The electrolyte 3 − r 3 ), and the SEI–electrolyte interfacial volume is VE = 4/3 (rE,o E,i 2 . The inner radius of the electrolyte solution is area is AE,SEI = 4 rE,i rE,i = rp + ı. The results in this paper use a particle radius of 5 ␮m and an overall electrolyte thickness is of 50 ␮m. Thus, the electrolyte concentrations are essentially constant. The concentration of solvent and lithium ions are assumed to be CEC(E) = 15.0 kmol m−3 and CLi+ (E) = 1.2 kmol m−3 , respectively.

4. Reaction mechanisms Table 1 lists the reversible elementary reactions making up the graphite SEI mechanism. The forward reaction rate expression kf and anodic symmetry factor ˇa are also listed in Table 1. To preserve microscopic reversibility, the reverse reaction rate kr is calculated based on an equilibrium constant. The cathodic symmetry factor for an elementary charge-transfer reaction is calculated as ˇc = 1 − ˇa . Despite not producing/consuming an electron, Reaction (2) involves charge transfer between the electrolyte phase and SEI surface phase. Thus, a symmetry factor must be specified for Reaction (2). The outer SEI bulk phase and SEI surface are modeled as having the same electric potentials. Therefore, a symmetry factor does not need to be specified for Reactions (4) and (5). The kinetic rate expressions and symmetry factors are taken generally from Christensen and Newman [12]. However, kf for Reactions (4)–(6) are reduced by two orders of magnitude. With this adjustment, model results for film growth better match the experimental results (i.e., film thicknesses of approximately 50–100 nm after 100 days of storage [17]). The mechanism in Table 1 is only intended for modeling film growth at room temperature, hence temperature dependencies (i.e., activation energies) are not used.

Table 1 SEI film growth reaction mechanism for a graphite particle. The reactions are based on Aurbach et al. [2], and the rates expressions kf are derived from Christensen and Newman [12], but with the forward rate constants for Reactions (4)–(6) being reduced by two orders of magnitude compared to the values given by Christensen and Newman. Note that C2 H4 (Ss ) is assumed to occupy two surface sites. Reaction

kf (kmol, m, s)

ˇa

SEI–electrolyte interface 1 2 3 4

C3 H4 O3 (E) + (Ss )  C3 H4 O3 (Ss ) Li+ (Ss )  Li+ (E) + (Ss ) C2 H4 (E) + 2(Ss )  C2 H4 (Ss ) C3 H4 O− (Ss )  C3 H4 O3 (Ss ) + e−b 3

1.0 × 1011 1.0 × 1013 2.5 × 1017 1.0 × 109

N/A 0.5 N/A N/A

5

C2 H4 (Ss ) + CO2− (Ss )  C3 H4 O− (Ss ) + e−b + 2(Ss ) 3 3

1.0 × 109

N/A

CO2− (Ss ) + 2Li (Ss ) + (Sb )  Li2 CO3 (Sb ) + 3(Ss ) 3 Li(Sb ) + (Ss )  V− (Sb ) + Li+ (Ss ) + + Li b + (Ss )  Li (Ss )

1.0 × 1032 7.5 × 10−6 1.0 × 103

N/A N/A N/A

6 7 8

S

S

+

S

Graphite–SEI interface 9

e

1.0 × 10−12

0.5

10

Li(C6 ) + V− (Sb )  Li(Sb ) + e− + (C6 ) C

1.0 × 10−10

0.5

11

Li(C6 )  Li

1.0

0.5

− Sb

 e− C

6

+

Sb

+ e− + (C6 ) C 6

6

A.M. Colclasure et al. / Electrochimica Acta 58 (2011) 33–43

The SEI film is modeled as a single compound (lithium carbonate) with a constant density. However, it is known that the SEI microstructure and composition can be much more complex [7,16,19]. The inner regions (i.e., close to the electrode particle) tend to be more dense and composed of inorganic, lithium-containing species that have low oxidation states and low molecular weights (LiF, Li2 O, Li3 As, Li3 N, etc.) [19]. Closer to the electrolyte solution, the SEI is more porous and composed of organic, high-oxidation-state species. The particular lithium-ion salt used in the electrolyte-phase affects the structure of organic and inorganic layers [16]. The SEI composition and structure change during storage, cycling, and temperature variations. Moreover, small amounts of additives (e.g., vinyl carbonate) or impurities (e.g., water) in the electrolyte phase can significantly influence SEI structure and stability [7,20,21]. Nevertheless, such details are not considered in this paper. The SEI reaction mechanism, which is a subset of the mechanism proposed by Aurbach et al. [2], describes the two-electron reduction of EC to produce lithium carbonate (SEI bulk phase). Reactions (1)–(8) (Table 1) occur at the SEI film-electrolyte interface, and Reactions (9)–(11) proceed at the graphite–SEI film interface. Aurbach’s mechanism also includes reactions that produce bulkphase (CH2 OCO2 Li)2 at high EC concentrations. However, because the present model considers a single SEI bulk phase, these reactions are neglected. It may be noted that Li atoms within the solid lithium carbonate Li(Sb ) are assumed to be uncharged while the Li vacancy V− (Sb ) is assumed to be negatively charged. From the modeling perspective the choice is somewhat arbitrary; the results would be identical if the vacancies were considered to be neutral and the lithium ions were positively charged. Because, the lithium carbonate is a poor electronic conductor the electron e−b concentration S

varies throughout the film thickness. Lithium ions can be consumed by film formation or be transported between the bulk electrode and electrolyte phases, which is the desired intercalation/deintercalation process. Ethylene gas is produced and released into the electrolyte phase as a byproduct of film growth. A charge-transfer reaction moves electrons between the bulk electrode phase and the bulk SEI phase. The electrons are transported through the film and consumed in the production of s 2− s EC anions C3 H4 O− 3 (S ) and carbonate CO3 (S ). Because the reaction mechanism involves species associated with surfaces (interfaces), site conservation equations are solved to determine surface site fractions k occupied by these species, n ∂ k = s˙ k,S , k (n) ∂t

(17)

where n is the available site density for phase n and k (n) is the site-occupancy number for species k on phase n [18]. The net rates of progress for elementary charge-transfer and charge-neutral reactions are evaluated using the theory discussed by Colclasure and Kee [1]. Evaluating the forward and reverse rates for these reactions requires the evaluation of species activities ak . Species within the electrolyte solution and film are modeled using ideal thermodynamics. That is, the activity of an SEI film or electrolyte species is simply proportional to its concentration ak = Ck /Ck ◦ . However, the graphite particle is modeled as a nonideal solution, with properties discussed previously [1,11]. The activity of intercalated lithium or a lithium vacancy is evaluated as ak = Xk k , where Xk is the mole fraction. The activity coefficients k are evaluated using the Redlich–Kister expansion and parameters given by Colclasure and Kee [1]. Evaluating equilibrium constants for the reactions requires standard-state chemical potentials and concentrations. The standard-state concentration for the electrode phase is assumed to be the maximum intercalation concentration Cmax . The

37

Table 2 Standard-state chemical potentials for species involved in the SEI film growth mechanism detailed in Table 1. The ◦ were determined from the reverse rate expressions given by Christensen and Newman [12]. ◦ (kJ mol−1 ) Bulk electrode species Li(C6 ) (C6 ) e− C 6

Bulk film species + Li b S

V− (Sb ) e−b S

Li2 CO3 (Sb ) Li(Sb ) (Sb )

−11.65 0.0 0.0 34.1 29.4 −22.8 0.0 0.0 0.0

SEI surface film species Li+ (Ss ) C3 H4 O3 (Ss ) C3 H4 O− (Ss ) 3 (Ss ) CO2− 3 C2 H4 (Ss ) (Ss )

17.0 338.1 298.9 207.4 50.0 0.0

Electrolyte species C3 H4 O3 (E) Li+ (E) C2 H4 (E)

315.6 0.0 52.5

standard concentration for lithium ions within the electrolyte phase is assumed to be 1.2 kmol m−3 . The standard concentration for EC is assumed to be its pure-phase concentration C◦ EC(E) = 15.0 kmol m−3 . The standard concentration for minor and bulk film species are assumed to be 1 kmol m−3 . Standard-state species chemical potentials are evaluated from the reverse rate expressions given by Newman and Christensen [12]. The known G◦ for each reaction is written in terms of the unknown standard-state chemical potentials, and an iterative algorithm is used to determine the standard-state chemical potentials. In principle, the 11 reactions (Table 1) can be used to determine the standard-state chemical potential for 11 species. However, the system is overdetermined, with only nine independent standard-state chemical potentials. Reactions (2), (7) and (10) can each be used to determine the standard-state chemical potential of an adsorbed lithium ion, yielding ◦ Li(Ss ) = 17 kJ mol−1 , 17 kJ mol−1 , and 11.4 kJ mol−1 . The model assumes ◦ Li(Ss ) = 17 kJ mol−1 . Table 2 lists all the standard-state chemical potentials. In addition to interface reactions, interstitial lithium and lithium vacancies may react homogeneously within the SEI to fill a lithium lattice site Li+b + V− (Sb )  Li(Sb ). S

(18)

According to Newman and Christensen, Reaction (18) remains near partial equilibrium. Thus, in the present model the forward reaction rate expression is set sufficiently high as to maintain the reaction near partial equilibrium (kf,18 = 7.48 × 102 m3 kmol−1 s−1 ). In summary, the model uses 11 heterogeneous reactions, 1 homogeneous reaction, and 3 minor, ionic SEI species. 5. Numerical implementation Spatial derivatives in the conservation equations are discretized using the finite-volume method. The discretized equations form a set of differential algebraic equations (DAE). The SEI model is written in C++and uses the banded DAE solver contained within the SUNDIALS software [22]. The chemically reacting flow software CANTERA is used to evaluate net rates of progress and thermodynamic properties [23]. A CANTERA class is developed that uses

A.M. Colclasure et al. / Electrochimica Acta 58 (2011) 33–43

6. Results 6.1. Open-circuit simulations The single-particle model is used initially to study SEI film growth under open-circuit conditions. Using the mechanism in Table (1), the bulk electrode phase can be the only source of lithium ions for film growth under open-circuit conditions. The overall process occurring at open circuit operation can be represented as 2Li(C6 ) + C3 H4 O3 (E) + (Sb )  Li2 CO3 (Sb ) + C2 H4 (E) + 2(C6 ),

(19)

(Sb )

where the Li2 CO3 production rate causes the film growth. If a lithium ion from the electrolyte phase were to be consumed, then there would be a net consumption of electrons at the SEI–electrode interface as represented by the following global reaction: 2Li+ (E) + 2e− + C3 H4 O3 (E) + (Sb )  Li2 CO3 (Sb ) + C2 H4 (E). C 6

(20)

However, at open circuit (i.e., zero net current) there cannot be a net consumption of electrons. Fig. 2 shows the predicted film growth for different initial intercalation levels during 10-day open-circuit storage simulations. The graphite particle radius is rp = 5 ␮m, and the simulations begin with an initial film thickness of ı = 1 nm. Ideally, the simulations would begin with a vanishingly small film thickness (e.g., 1 × 10−3 nm). However, using a very small initial film thickness invalidates the assumption of charge neutrality. The growth rate depends upon the initial SOC, where 0% SOC corresponds to a uniform intercalated Li mole fraction of XLiI (B) = 0.126 and 100% SOC corresponds

to XLiI (B) = 0.676 [11]. After 10 days the film thickness is between 8 nm and 23 nm, corresponding to capacity loss between 2% and 5%. The SEI growth rate decreases with increasing time, suggesting that film growth is mass-transport limited [10,12]. In this case, electron diffusion within the film is the rate-controlling process. If the electron diffusion coefficient De− is increased by an order of magnitude, Sb

then the film grows considerably faster. The film grows linearly when De− is increased by two orders of magnitude. Increasing the Sb

diffusion coefficient for the lithium species within the film (i.e., DLi+ and DV− (S ) ) does not noticeably change the growth rate. b

Sb

After approximately 6 h, the film thickness becomes proportional to the square root of time (ı = k(t1/2 ), where k is a constant that depends on the initial SOC). Christensen and Newman report that their SEI model predicts film growth is proportional to t0.4 at long times [12]. They state that this relationship is due to a rapid

SEI thickness (nm)

25 Initial SOC = 100% 90%

20

70%

15

30% 10

50% 0%

5 0

10%

0

2

4 6 Time (days)

8

10

Fig. 2. SEI film thickness under open-circuit conditions as a function of time for various initial states of charge. The simulations assume an initial SEI thickness of 1 nm.

0.02 Concentration (kmol m-3)

the Redlich–Kister expansion to represent the nonideal thermodynamic behavior of lithiated graphite [1]. A discharge or charge cycle can be simulated in 2–6 h on a personal computer.

(a)

Initial SOC = 100% 90% 70% 0.01

50% 30% 0% 10%

0 -0.10 Electric potential (V)

38

(b) -0.15 0%

10% 50%

30%

-0.20

70% 90% Initial SOC = 100% -0.25 0

5 10 15 20 Radial position within SEI film (nm)

25

Fig. 3. Concentration profiles of electrons and interstitial lithium ions (Ce− ≈ CLi+ ) Sb

Sb

and electric-potential profiles as functions of radial position within the film r − rp after ten storage days at open circuit for various initial SOCs. The electric potential is measured relative to a lithium metal reference electrode.

decrease in the graphite SOC. Some of the forward rate expressions for film growth used in the model presented here have been reduced compared to those used by Christensen and Newman [12] (see Table 1), resulting in slower film growth. Thus, the graphite SOC for the present model is not reduced as rapidly and the rate of film growth remains proportional to t0.5 . For the fastest film growth case (i.e., initially 100% SOC), the overall SOC is only reduced by 9% (final SOC of 91%). Fig. 3a illustrates predicted electron and interstitial lithium ion concentrations within the SEI film at the end of 10 storage days under open-circuit conditions. The concentration of electrons and interstitial lithium ions is high near the graphite–SEI interface and decreases linearly toward the SEI–electrolyte interface. Consistent with the predictions of Christensen and Newman [12], the concentration of lithium vacancies is orders of magnitude lower than the concentration of interstitial lithium ions. Due to electroneutrality, the concentrations of electrons and the interstitial lithium ions are essentially equal (Ce− ≈ CLi+ ). The interstitial lithium ion flux Sb

Sb

due to the concentration gradient is from the graphite–SEI interface toward the SEI–electrolyte interface. At the SEI–electrolyte interface, electrons and interstitial lithium ions are consumed to produce bulk-phase lithium carbonate. The interstitial lithium ion concentration at the SEI filmelectrode interface increases with an increase in the intercalation fraction (i.e., higher initial SOC), which can be explained in terms of Reaction (11) (Table 1). Because the forward and reverse rates are fast, this reaction remains near partial equilibrium. Thus, an increase in the activity of intercalated lithium at the SEI–electrode interface causes an increase in the activity of interstitial lithium ions at the SEI–electrode interface. The electron and interstitial lithium ion concentration approach zero (C < 3 × 10−4 kmol m−3 ) at the SEI film-electrolyte interface. Thus, an increase in the intercalation fraction causes the concentration gradient to steepen, promoting higher Li–ion fluxes and faster film growth. Fig. 3b shows that the SEI electric potential is lowest near the graphite–SEI interface and increases toward the SEI–electrolyte

A.M. Colclasure et al. / Electrochimica Acta 58 (2011) 33–43

20 SEI thickness (nm)

interface. The electric potential within the film shifts downward as the initial SOC increases, which is a result of lower electrode electric potential as SOC increases (i.e., open-circuit potential decreases at higher SOC). The electric-potential jump across the graphite–SEI interface is 0.34–0.39 V, and the jump across the SEI film-electrolyte interface is 0.11–0.14 V. The voltage difference across the graphite–SEI interface (B − S,i ) decreases with an increase in SOC, producing a higher electron flux into the SEI film via Reaction (9). The voltage difference across the SEI–electrolyte interface (S,o − E ) becomes more negative with increasing SOC, thus tending to impede interstitial lithium ion transfer into the electrolyte solution. The overall electric-potential variation within the film is between 80 and 94 mV and increases with increasing SOC. Because of the relatively low electron diffusion coefficient De− ,

39

C/15

15 10 C/7.5 C/4.6

5

C/15

0 0

0.5

1.0 1.5 2.0 Time0.5(days0.5)

2.5

3.0

Fig. 5. SEI film thickness during the first few cycles as a function of the square root of time for various cycling rates. The intercalation fraction varies approximately between 0.1 and 0.77 during cycling.

Sb

S

approximately the same concentration gradient. Thus, the nearly equal species gradients would result in a much higher flux of interstitial lithium ions compared to electrons. Therefore, the electric potential must impede in the transport of Li+b and assist in the S

transport of e−b . Because electrons are attracted toward a more posS

itive electric potential, the SEI electric potential increases toward the SEI–electrolyte interface. 6.2. Charge and discharge cycling Because the model is fully transient, it provides a capability to evaluate the influence of SEI chemistry and transport during cycling. Fig. 4 illustrates SEI film growth during five discharge and charge cycles at C/15 rates. The graphite particle radius is rp = 5 ␮m, and the initial SEI thickness is ı = 1 nm. During the first cycle, the film grows by 6.5 nm. Over the course of the second and third cycles, the film grows by 3.2 nm and 2.4 nm, respectively. Thus, film growth slows with continued cycling. At a given intercalation fraction, the electron and lithium-ion concentrations at the two SEI film interfaces vary only slightly as functions of time and cycling conditions. Thus, as the film thickens, the concentration gradients decrease causing the molar flux of electrons through the SEI to decrease. Due to the low electron diffusion coefficient, both the concentration gradients and electric-potential gradients adjust to maximize the transport of electrons. The film growth rate is limited by the molar flux of electrons within the SEI (i.e., increasing De− would Sb

significantly increase the growth rate). Therefore, slow electron transport within the SEI film causes the decrease in film growth rate with continued cycling. As was observed under open-circuit conditions, the SEI rate growth increases at higher SOC. During charge, the intercalation fraction rises and accelerates film growth. During discharge, the intercalation fraction decreases and slows film growth. Fig. 5 illustrates film thickness during the first several cycles as a function of the square root of time. The three cycling rates shown are C/15, C/7.5, C/4.6. Although SEI growth during a single cycle is not proportional to the square root of time, over the course of several cycles, the average growth rate is roughly proportional to the square root of time. Moreover, the square-root relationship is only weakly affected by cycling rates. These results are all consistent with the expectation that film growth during cycling and storage is limited by slow electron diffusion [10,12,24]. Smith et al. [25] used high precision coulometric studies to show SEI film growth during cycling is dictated by operating temperature and elapsed time, not the cycling rate. These measurements also show SEI film growth remains proportional to the square root of time during cycling [25]. Fig. 6 illustrates the interstitial lithium ion and electron concentration profiles within the SEI at the end of consecutive C/7.5 charge and discharge cycles. The intercalation fractions at the end of the five charge and discharge cycles are approximately 0.77 and 0.1, respectively. Interestingly, the concentrations are high near the graphite–SEI interface and decrease linearly toward the

Conc. × 103 (kmol m-3) Conc. × 102 (kmol m-3)

the SEI electric potential increases in the direction of the positive radial coordinate r to assist in electron transport (cf., Fig. 3b). The production of a mole of lithium carbonate at open-circuit conditions consumes two moles of intercalated lithium and two moles of bulk-graphite phase electrons. Thus, the fluxes of interstitial lithium ions and electrons toward the SEI film-electrolyte interface are roughly equal. However, the diffusion coefficient for Li+b S is over four orders of magnitude higher than the diffusion coef− ficient for e b . To preserve electroneutrality, both species have

2.5 (a) Charge

2.0 3rd

1.5

4th

1.0

5th

5.0

2nd

1st

0.0 3.0 2.5 2.0 1.5 1.0 5.0 0.0

(b) Discharge

3rd 1st

4th

5th

2nd 0

2

4 6 8 10 Radial position in SEI (nm)

12

Fig. 6. Interstitial lithium ion and electron concentration (CLi+ ≈ Ce− ) as a function Sb

Fig. 4. SEI film thickness during first five cycles with a rate of C/15. After the first charge cycle, the intercalation fraction varies nominally between 0.1 and 0.77 during cycling.

Sb

of radial position within the SEI film r − rp at the end of the first five charge-discharge cycles. The cell is cycled with a rate of C/7.5 between 0.1 < XLiI (B) < 0.77. The film thicknesses during these cycles is illustrated in Fig. 5.

A.M. Colclasure et al. / Electrochimica Acta 58 (2011) 33–43

Electric potential (V)

0.22

-0.13 -0.15 -0.16 -0.18 -0.20 -0.22 -0.23 -0.25

4th 5th

0.20

Electric potential (V)

Electric potential (V)

40

3rd 2nd 1st (a) Charge

0.18

0.14 0.12

1st

-0.14 -0.16 -0.18

0.08 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Intercalation fraction at graphite-SEI interface

4th 3rd

-0.12

2nd

5th

Fig. 8. Graphite electric potential as a function of XLiI (B) at the graphite surface during the first five C/7.5 cycles. The graphite electric potential is measured relative to the electrolyte electric potential E = 0. The film thickness during these cycles is illustrated in Fig. 5.

(b) Discharge 0

2

1st charge 2nd charge 5th charge

0.10

-0.08 -0.10

1st discharge 2nd discharge 5th discharge

0.16

4 6 8 10 Radial position in SEI (nm)

12

Fig. 7. SEI electric potential S as a function of radial position within the SEI film r − rp at the end of the first five charge-discharge cycles. The cell is cycled at a rate of C/7.5 between 0.1 < XLiI (B) < 0.77. The film thicknesses during these cycles are illustrated in Fig. 5.

toward the graphite–SEI interface. Hence, the lithium ion flux from the electric-potential gradient is larger than the flux induced by the concentration gradient. During discharge, the net lithium-ion flux is in the opposite direction, and the flux due to the concentration gradient is larger than the flux induced by the electric potential. 6.3. Voltage losses

SEI–electrolyte interface during both charge and discharge cycles. During discharge, the net lithium ion flux is from the graphite–SEI interface toward the SEI–electrolyte interface, with the lithium ion flux proceeding down the concentration gradient during discharge. In contrast, during charge, the net lithium ion flux is from the SEI–electrolyte interface toward the graphite–SEI interface. The lithium ion flux proceeds up the concentration gradient during charge, sometimes labeled uphill diffusion. Although the interstitial lithium-ion flux can be transported up its chemical-potential Li+ gradient (proportional to concentration gradient), the species Sb

fluxes must always proceed down the electrochemical potential  ˜ k gradient. Thus, during charge, the electrochemical potential of interstitial lithium ions decreases toward the graphite–SEI interface (i.e., the SEI electric potential gradients dominate the concentration gradients). Furthermore, the concentration gradients at the end of the charge cycles are almost an order magnitude higher than the gradients at the end of discharge due to variations in the surface intercalation fraction. Corresponding to the conditions in Fig. 6, Fig. 7 illustrates the electric-potential profiles within the film at the end of the charge and discharge cycles. During the charge and discharge cycles, the electric potential is lowest near the graphite–SEI interface and increases nonlinearly toward the SEI–electrolyte interface. The electron flux is always from the graphite–SEI interface toward the SEI–electrolyte interface. Due to slow electron diffusion, both the electric-potential profile and concentration profile must transport electrons in the same direction. The concentration gradient is negative and moves electrons from the graphite–SEI interface toward the SEI–electrolyte interface. Electrons are attracted to the relatively higher electric potential at the SEI–electrolyte interface. Figs. 6 and 7 illustrate that the concentration and electricpotential profiles within the SEI move interstitial lithium ions in opposite directions. During both charge and discharge cycles, the concentration gradient is always negative, inducing a species flux from the graphite–SEI interface toward the SEI–electrolyte interface. Conversely, the electric-potential gradient is always positive and moves interstitial lithium ions from the SEI–electrolyte interface toward the graphite–SEI interface. The lithium-ion flux induced from low charge/discharge rates (C/15) is a few orders of magnitude greater than the flux induced by SEI growth. Thus, the net flux during charge cycles is from the SEI–electrolyte interface

Fig. 8 illustrates graphite electric potentials at the graphite–SEI interface during the first five C/7.5 cycles. During charging, the electrode potential is initially high due to a low surface intercalation fraction. As lithium is intercalated (i.e., as XLiI (B) increases), the electrode potential decreases. To promote cathodic reactions, the graphite electric potential during charge is lower than the potential during discharge at the same surface XLiI (B) . The electric potentials during both charge and discharge shift upward with cycling because of changes in film thickness and minor-species concentrations within the SEI film. This upwards shift lowers the overall cell potential. The rate of the upward shift reduces as the film growth rates decrease after many cycles. Because the reversible potential cannot be evaluated simply using thermodynamic parameters, electrode performance can be evaluated based on the difference in cell voltage between discharge and charge at a given surface XLiI (B) and cycling rate. This overall voltage loss in moving a lithium ion from the electrolyte solution into the graphite particle surface or vice versa (i.e., a pseudo overpotential) for a particular cycle is defined as total

= E dis − E chg , Edis

(21)

Echg

where and are the electric-potential differences between the electrode and electrolyte for the particular cycle during discharge and charge, respectively. Ideally, the graphite particle would cycle with total = 0, thus operating with maximum thermodynamic efficiency. However, due to slow film diffusion and heterogeneous chemistry, total is nonzero. Fig. 9 shows that total increases considerably with a decrease in the intercalation fraction. The overall voltage loss is composed of the three specific losses as total

=

where

B,S B,S

interface,

+

film

+

S,E ,

(22)

chg

dis − E = EB,S is the voltage loss across the graphite–SEI B,S chg

film dis ES,E

dis − E = Efilm is the voltage loss through the SEI, film chg

and S,E = − ES,E is the voltage loss across the SEI–electrolyte interface. Thus, the voltage loss at an interface or across the SEI is not simply the voltage difference at an interface or across the SEI film. Rather, the loss is the change in the voltage difference between discharge and charge cycles. Although individual voltage

0.04 0.03 4th cycle 5th cycle 3rd cycle

2nd cycle

0.01 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Intercalation fraction at graphite-SEI interface Fig. 9. Overall voltage loss total as a function of the intercalation fraction at the surface of graphite particle during cycles 2–5 at a cycling rate of C/7.5.

variations E may be negative, the voltage losses always positive.

eq

RT ln + F

eq



=

−◦ e−

Sb

F

+

Sb

CLi+ a(C6 )

F



Sb

aLi(C6 )



RT ln Ce− F Sb

.

(23)



,

(24)

where Ce− is the electron concentration at the inner SEI film. The Sb

chemical potential of the electron within the graphite is formulated by assuming its activity is equal to its concentration. The overpotential for the electron transfer is eq

9 = EB,S − E9

eq

ior of 9 , EB,S , and E9 with respect to the intercalation fraction are the same for all discharge and charge cycles. The overpotential for electron transfer is always negative, causing Reaction (9) to proceed in the cathodic (reverse) direction, transferring electrons from the graphite into the SEI. The overpotential becomes more cathodic (negative) as XLiI (B) increases, increasing the electron-transfer rate. The exchange current density for electron transfer is proportional to C 0.5 − , which increases with an increase in XLiI (B) . Thus, as the interSb

-0.003

eq

E9

0.33 0.32 -0.004 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Intercalation fraction at graphite-SEI interface

Fig. 10. Reversible potential and overpotential for the electron-transfer reaction (Reaction (9)) and the potential difference cross the graphite–SEI film interface EB,S during the 4th C/7.5 discharge as a function of the graphite–surface intercalation fraction.

0.020 0.015 0.010

5th cycle 4th cycle

0.005 3rd cycle

0.000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Intercalation fraction at graphite-SEI interface Fig. 11. Voltage loss across the SEI film as a function of XLiI (B) at the graphite surface during cycles 3–5 at charge and discharge rates of C/7.5.

6.3.2. Voltage loss through SEI film Fig. 11 shows the predicted voltage loss across the SEI film film during C/7.5 cycling. Due to changes in film concentrations, film increases with a decrease in the intercalation fraction at the surface of the graphite particle. The voltage loss across the film increases during repeated cycling because of film growth. Interestingly, most of the electric-potential variation through the film does not result in a voltage loss. If all electric-potential variations within the SEI dis would be positive and E chg would be resulted in losses, then Efilm film negative. However, the electric-potential difference across the film is always considerably negative Efilm < − 0.08 V. Thus, film ≈ 0.01 V is small compared to the magnitude of Efilm . 6.3.3. Voltage loss at SEI film-electrolyte interface Fig. 12 shows the predicted voltage loss across the SEI–electrolyte interface S,E for C/7.5 cycles. This voltage loss S,E is not due to slow charge-transfer kinetics at the SEI

(25)

Considering the 4th C/7.5 discharge cycle, Fig. (10) illustrates the relationship between 9 and XLiI (B) . However, the functional behav-

e

EB,S

◦ Li+ + ◦ (C6 ) − ◦ Li(C6 )

The relationship between film growth and XLiI (B) is explained

eq

0.34

η9

0.025

by determining the overpotential for electron transfer between the graphite and film (Reaction (9)). The reversible potential for the electron transfer reaction is E9 =

-0.002

0.37 0.36 0.35

e−C 6

(Eq. (21)) are

6.3.1. Voltage loss at graphite–SEI interface For a given intercalation fraction, the potential difference at the graphite–SEI interface is independent of the imposed current density B,S = 0. Thus, EB,S remains at the reversible potential for a dominating reaction. During cycling, the fluxes of lithium vacancies and electrons within the film are very small compared to the flux of interstitial lithium ions. Thus, the net production rate of interstitial lithium ions at the graphite–SEI interface is much larger than the production rate of electrons and lithium vacancies. Therefore, the potential difference at the graphite–SEI film interface remains at the reversible potential for the interstitial lithium ion reaction (Reaction (11)), which is evaluated as EB,S = (B − S )

-0.001 − Reaction 9: eS b

0.38

Voltage loss, ψ film (V)

0.02

0.40 0.39

calation fraction at the graphite surface rises, both an increase in the cathodic overpotential and the exchange current density for the electron-transfer reaction lead to higher SEI growth rate.

0.024 Voltage loss, ψ S,E (V)

Total voltage loss (V)

0.05

41

Over potential (V)

Electric potential difference (V)

A.M. Colclasure et al. / Electrochimica Acta 58 (2011) 33–43

0.022 0.020 0.018 5th cycle

0.016 0.014 0.012

3rd cycle 4th cycle

0.010 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Intercalation fraction at graphite-SEI interface Fig. 12. Voltage loss across the SEI film-electrolyte interface S,E as a function of XLiI (B) at the graphite surface during cycles 3–5 at charge and discharge rates of C/7.5.

42

A.M. Colclasure et al. / Electrochimica Acta 58 (2011) 33–43

film-electrolyte interface. Instead, S,E is related to changes in the reversible potential of the lithium-ion adsorption reaction (Reaction (2)). These reversible-potential losses are from surface coverage changes due to the slow exchange of lithium ions between the SEI surface and bulk SEI via Reaction (8). The model assumes that the SEI surface and the outermost control volume within the bulk SEI are at the same electric potential. Therefore, Reaction (8) behaves as a thermal reaction and not as a chargetransfer reaction. Similar to film , S,E is considerably higher at low intercalation fractions. The voltage loss at the SEI–electrolyte interface increases slightly with continued cycling due to a lower interstitial lithium-ion concentration at the interface from SEI film growth. If the forward rate expression for Reaction (11) is increased by two orders of magnitude, then S,E is substantially smaller. 6.4. Area-specific resistances The overall voltage loss can be used to evaluate a specific resistance per graphite surface area (m−2 ) as Rtotal =

total chg

iddis − id

=

total

2iddis

.

(26)

Assuming equal charge and discharge rates, the current density chg chg during discharge iddis and charge id are simply related as id =

Area-specific resistance (Ω m 2)

−iddis . Fig. 13 shows predicted area-specific resistances for various cycling rates. The particular cycles shown are selected because they start at nearly the same elapsed time from the initial charge, making the SEI film thicknesses similar. The transport resistances within the SEI film Rfilm are all approximately equal. Interestingly, the resistance at the SEI–electrolyte interface at low intercalation fractions is higher at a rate of C/15 compared to RS,E at rates of C/7.5 and C/4.6. As previously discussed, the SEI–electrolyte interface resistance is due primarily to changes in the equilibrium electricpotential differences associated with surface coverage variations. The total area-specific resistance Rtotal varies between 0.13 -m2 and 0.45 -m2 . There is substantial uncertainty in evaluating the voltage loss caused by the SEI that forms on graphite particles. Srinivasan and Newman used RSEI as a constant fitting parameter in a Newmantype cell model to match experimental data for a natural-graphite half cell [26]. They found that an SEI resistance of RSEI = 0.023 -m2 best represented the voltage discharge curves. In a similar manner, Doyle and Fuentes used an SEI resistance of RSEI = 0.065 -m2 for a graphite anode to match experimental discharge curves for a Sony lithium-ion polymer battery [27]. In both cases, the RSEI

0.5 0.4

C/15 Rtotal C/7.5,C/4.6 Rtotal

0.3 0.2 0.1 0.0

C/7.5,C/4.6 RS,E Rfilm

C/15 RS,E

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Intercalation fraction at graphite-SEI interface Fig. 13. SEI film-electrolyte interface, SEI film, and total resistance as a function of XLiI (B) at the graphite surface for cycling rates of C/15 (3rd cycle), C/7.5 (5th cycle), and C/4.6 (8th cycle). The electrode-SEI film interface resistance RB,S is essentially zero and therefore is not plotted. The SEI film thicknesses are approximately ıapprox11 nm.

values were empirically fitted, not directly measured. The SEI resistance likely depends on graphite type and morphology. The present SEI model predicts that the overall resistance varies significantly with the intercalation fraction. Thus, the SEI resistance should be measured at different intercalation fractions. Newman-type battery models have been modified to account for voltage losses associated with SEI films [11,26,27]. In these models global Butler–Volmer equations for lithium intercalation are modified as i = i0



exp

˛ F a RT



˛F c

( − iRSEI ) − exp −

RT

( − iRSEI )



,

(27)

where RSEI is the total resistance of the SEI. Often, RSEI is simply assumed to be a constant. However, as shown by the present model, RSEI varies considerably with intercalation fraction and also increases slowly during cycling. Thus, the resistance should account for losses associated with transport limitations within the SEI and ion exchange between the bulk phase and SEI surface (Reaction (8)). However, Rtotal and RSEI are not necessarily equivalent. The total resistance Rtotal is based upon the voltage losses from the SEI between discharge cycles and charge cycles with the same current density magnitude. However, the Rtotal is based on the voltage change from open-circuit to discharge/charge. Assuming that the resistance to intercalation is the same during both charge and discharge, then RSEI = Rtotal . In any case, the detailed SEI model could be used to inform the modified Newman model about significant variations in RSEI . However, without significant further modification, a model in the form of Eq. (27) would not predict the changes in the reversible potential or anode SOC as a result of SEI film growth. 7. Summary and conclusions A one-dimensional model of a single graphite particle has been derived and implemented to study the effects of SEI chemistry and transport on the lithium intercalation process. Species- and chargeconservation equations are solved throughout the graphite particle, SEI film, and surrounding electrolyte solution to determine concentration and electric-potential profiles within each phase. The SEI grows with time according to the net production rate from heterogeneous chemistry on the SEI film surface. Reversible elementary reactions are used to model the charge-transfer processes at the graphite–SEI and SEI–electrolyte interfaces. An important advantage of the present model compared to previously published models is that the present model can be used to study SEI film growth and resistance under cycling conditions. For the set of parameters studied, the SEI film growth rate and interfacial resistances depend greatly upon the intercalation fraction, but are only weak functions of cycling conditions. This is because the electric-potential and concentration profiles within the SEI film are strong functions of the intercalation fraction. Due to relatively facile charge-transfer chemistry at the SEI film interfaces, the concentration and electric-potential profiles within the SEI film are nearly independent of the imposed current density (i.e., SEI growth rate during storage is nearly the same as during cycling). An increasing intercalation fraction causes an increase in the electron and interstitial lithium-ion concentration within the SEI film. The higher ion concentrations within the SEI film promote faster SEI film growth and lower the interfacial resistance. The SEI film thickness, regardless of cycling conditions, is roughly proportional to the square root of time, which suggests SEI film growth is limited by slow electron diffusion. The SEI interfacial resistance is caused by species transport limitations within the SEI and slow lithium-ion exchange chemistry at the graphite–SEI interface. The SEI resistance at low SOC can be much greater than the resistance at a high SOC. The model also offers new insight into the effects SEI film properties have on graphite open-circuit potential.

A.M. Colclasure et al. / Electrochimica Acta 58 (2011) 33–43

Acknowledgements This effort was supported by the Office of Naval Research via an RTC grant (N00014-05-1-03339) and by the National Renewable Energy Laboratory under funding from the U.S. Department of Energy, Office of Vehicle Technologies, Energy Storage Program. We gratefully acknowledge many insightful discussions with Prof. David Goodwin (Caltech) concerning the elementary-reaction representation and software implementations of complex electrochemical charge-transfer processes. References [1] A.M. Colclasure, R.J. Kee, Electrochimica Acta 55 (2010) 8960. [2] D. Aurbach, B. Markovsky, I. Weissman, E. Levi, Y. Ein-Eli, Electrochimica Acta 45 (1999) 67. [3] M. Doyle, T.F. Fuller, J. Newman, J. Electrochem. Soc. 140 (1993) 1526. [4] M. Doyle, T.F. Fuller, J. Newman, Electrochimica Acta 39 (1994) 2073. [5] M. Doyle, T.F. Fuller, J. Newman, J. Electrochem. Soc. 141 (1994) 982. [6] M. Doyle, J. Newman, Electrochimica Acta 40 (1995) 2191. [7] D. Aurbach, The role of surface films on electrodes in Li–ion batteries, in: W.A. van Schalkwijk, B. Scrosati (Eds.), Advances in Lithium–Ion Batteries, Kluwer Academic Publishers, 2002, p. 7. [8] D. Aurbach, K. Gamolsky, B. Markovsky, G. Salitra, Y. Gofer, U. Heider, R. Oesten, M. Schmidt, J. Electrochem. Soc. 147 (2000) 1322. [9] J.B. Goodenough, Y. Kim, Chem. Mater. 22 (2010) 587. [10] H.J. Ploehn, P. Ramadass, R.E. White, J. Electrochem. Soc. 151 (2004) A456.

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