Determination of state of charge-dependent asymmetric Butler–Volmer kinetics for LixCoO2 electrode using GITT measurements

Journal of Power Sources 299 (2015) 156e161

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Determination of state of charge-dependent asymmetric ButlereVolmer kinetics for LixCoO2 electrode using GITT measurements A. Hess a, *, Q. Roode-Gutzmer a, C. Heubner a, b, M. Schneider b, A. Michaelis b, M. Bobeth a, G. Cuniberti a a b

Institute for Materials Science and Max Bergmann Center of Biomaterials, TU Dresden, 01062, Dresden, Germany Fraunhofer IKTS Dresden, Winterbergstr. 28, 01277, Dresden, Germany

h i g h l i g h t s
A semi-empirical model for asymmetric charge/discharge kinetics is presented.
A state-of-charge dependent reaction rate constant is introduced.
A state-of-charge dependent charge transfer coefficient is introduced.
Enhanced agreement between model predictions and GITT experiment.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 March 2015 Received in revised form 25 June 2015 Accepted 26 July 2015 Available online xxx

GITT (Galvanostatic Intermittent Titration Technique) measurements of LixCoO2/Li half-cell voltages were numerically simulated based on Newman's well established electrochemical pseudo 2D model. The measurements revealed differences in the charge transfer kinetics between charging and discharging, which change with the state of charge of LixCoO2. To properly account for these differences in the simulations, SOC-dependent reaction-rate constant together with SOC-dependent charge transfer coefficients were introduced, which were unambiguously determined from the measured IR drops of the GITT pulses during charging and discharging. Furthermore, the SOC-dependence of the chemical Li-ion diffusion coefficient in LixCoO2 was analyzed by fitting the GITT data within the framework of the pseudo 2D model, as well as by means of classical analysis by Weppner and Huggins [W. Weppner and R. A. Huggins, J. Electrochem. Soc. 124 (1977) 1569]. Improvement of the simulation of GITT measurements using SOC-dependent rate constants and charge transfer coefficients compared to SOC-independent values is demonstrated. © 2015 Published by Elsevier B.V.

Keywords: Lithium ion battery Lithium cobalt oxide Galvanostatic intermittent titration technique Numerical simulation ButlereVolmer Charge transfer kinetics

1. Introduction Electroanalytical techniques, as e.g. impedance spectroscopy (EIS), cyclic voltammetry (CV), potentiostatic (PITT) and galvanostatic intermittent titration technique (GITT) are widely used for analyzing the charge transport in lithium ion batteries (LIBs). Detailed knowledge of kinetic parameters such as diffusion coefficients in the battery components as wells as reaction rate constant and charge transfer coefficients at the interfaces between the

* Corresponding author. E-mail address: [email protected] (A. Hess). http://dx.doi.org/10.1016/j.jpowsour.2015.07.080 0378-7753/© 2015 Published by Elsevier B.V.

electrolyte and the active electrode material is crucial for the simulation of the chargeedischarge behavior of LIBs. Such simulations are an important tool for optimizing the battery design and improving the battery management effectively. For instance, the accurate description of the overpotential as a function of the applied current density, the temperature, and the state of charge (SOC) is crucial for the determination of Joule heating and for intelligent thermal management during operation. In the present work, GITT and EIS measurements were employed for a detailed exploration of the charge transport in a LIB cell, in particular the charge transfer at the interface between the electrolyte and the LixCoO2 (LCO) electrode particles. The measurements revealed a strong difference in the interfacial charge transfer kinetics between

A. Hess et al. / Journal of Power Sources 299 (2015) 156e161

charging (Li-deintercalation) and discharging (Li-intercalation). To extract the corresponding kinetic parameters from experimental GITT data, the authors performed extensive numerical simulations of GITT pulses over the wide SOC range of an LCO cathode. The most widely used model for the simulation of electrochemical processes in LIBs with porous insertion electrodes is the pseudo-2D (P2D) model, which goes back to the seminal work by Doyle et al. [1]. To improve the agreement between modeling and experimental findings, several extensions of the original model have been proposed, which mainly concern (i) the charge transfer kinetics at the interface between electrolyte and electrode particles and (ii) the diffusion of lithium within the particles. The charge transfer at the electrode-electrolyte interface is usually described by the ButlereVolmer (BV) equation. An accurate prediction of the electrode potential over a wide range of C-rates, however, seems to be challenging [2e4]. As a consequence, empirical expressions were used in the BV equation to achieve better agreement between model predictions and experimental results. In Ref. [3], Zhang et al. introduced C-rate dependent transfer coefficients and a modified concentration dependence of the exchange current density in order to properly describe discharge profiles of LCO cathodes (cell voltage vs. SOC) also at higher C-rates. Lai and Ciucci [5] derived a modified interfacial charge transfer equation by volume averaging of microscopic equations in porous electrodes (generalized PoissoneNernstePlanck equations). Landstorfer et al. [6] considered the applicability of the classical BV equation for the case of solid electrolyte intercalation batteries and concluded that it could overestimate the anode reaction rate while underestimating the cathode rate. Latz and Zausch [7] presented a derivation of the Liion intercalation rate into the cathode material in which the amplitude of the exchange current is free of singular terms. Such terms are problematic in simulations, where lithium concentrations at the interface of solid and electrolyte reach limiting values. Bazant [8] generalized the BV kinetics in order to describe the elementary charge-transfer step and its coupling to phase transformations in the active material during charging and discharging. Very recent investigations on carbon-coated LixFePO4 porous electrodes showed that the interface kinetics in this case can be well described by the MarcuseHusheChidsey theory, pointing to the importance of the reorganization energy and suggesting that the electron transfer at the carbon-LixFePO4 interface is ratedetermining [9]. A further topic being intensively investigated is the SOCdependence of the chemical diffusion coefficient in the active

157

electrode materials. Recent model developments focus on the occurrence of a phase transformation in the particles during (de) lithiation. These works deal mainly with LixFePO4 [10,11] and LCO [12] electrodes. Mostly, spherical oxide particles with a coreeshell two-phase oxide structure have been considered, where core and shell refer to the lithium-rich or -poor phases [10e13]. However, motivated by the observation of an anisotropic lithium insertion [14], also anisotropic models have been developed [15,16]. In this communication, we report on simulations of GITT measurements within the framework of the P2D model. Within this model, the charge transfer kinetics are described by the BV equation. It was found, however, that the GITT measurements on a LCO/ Li half-cell could not properly be described by using uniform (SOCindependent) reaction rate constant and transfer coefficient. For this reason, we propose here an approach where a SOC dependence of these two parameters is included in the model. The corresponding SOC dependence was uniquely determined by analyzing the IR drops of the GITT pulses, covering the wide SOC range of the LIB cell. Furthermore, the lithium diffusion coefficient within the oxide particles was considered to depend on the lithium concentration. The corresponding values of the apparent diffusion coefficient were empirically derived by fitting our simulations to the measured potential change caused by an applied current pulse. In this way, we achieved considerably better agreement between our simulations and the GITT measurements during charging and discharging. 2. Experimental The experimental investigations were carried out using a composite electrode (MTI Corp.) containing active material LiCoO2 admixed with conductive additive and binder upon a 15 mm thick current collector foil. Two layers of borosilicate glass micro-fiber filter (Whatman) formed the separator of the LIB cell. A metallic lithium wire in the middle of the separator served as reference electrode. The anode of the cell was a graphite electrode. Commercially available LP40 (1 M lithium hexafluorophosphate (LiPF6) in ethylene carbonate: diethylcarbonate (1:1 w/w)) (BASF) was employed as electrolyte. The thicknesses of the electrodes and separator as well as the particle size of the active material were determined by scanning electron microscopy. The assembly of the 3-electrode Swagelok cell was carried out in an argon-filled glove box (MBraun). A multi-channel potentiostat/galvanostat (VMP3) with integrated frequency response analyzer (Biologic) was implemented to conduct the electrochemical measurements. The

Table 1 Physical properties of electrode materials and electrolyte used in this work, where the index 0 refers to initial conditions, superindices (c) and (d) to charge and discharge, respectively, and ~ce ¼ ce =ðmol$m3 Þ. Symbol Design specifications ε Rs as l Lithium concentrations c0 cmax ðcÞ SOC0 ðdÞ SOC0 Kinetic and transport properties

Units

mm

1

m mm

mol m3 mol m3

g tþ D De

seff k

m2 s1 m2 s1 S m1 S m1

Anode

Separator

Cathode

0.35 10 1.77,105 99

0.91

0.45 6 1.45,105 88

1000 23895 0.009 0.8395

1000

684

1.5 2.5 0.363 0.363 1.75,1015 1.5,1010 1.5,1010 100 2:8$1014 ~c4e þ 3:2$1010 ~c3e  1:6,102 ~ce þ 0:1

1000 51598 0.837 0.4533 2.5 0.363 Fit 1.5,1010 10

158

A. Hess et al. / Journal of Power Sources 299 (2015) 156e161

cell was placed in a climatic chamber operating at 25  C to ensure constant ambient temperature. The GITT measurements were performed by applying an electric current density of iapp ¼ 11.05 A m2, which corresponds to a C-rate of about 0.4 C. The duration of each current pulse was 300 s followed by a relaxation period of 3 h. Cut-off potentials for charging and discharging were set to 4.4 and 3.0 V, respectively. All GITT measurements were done after a CCCV (constant current constant voltage) pre-conditioning of the cell. Relevant material and geometrical cell data are presented in Table 1. 3. Model Numerical simulations of GITT measurements were performed within the framework of the P2D electrochemical model by Newman [17]. Since the P2D model is widely used for performance studies of batteries with insertion electrodes [1,18,19], the governing differential equations and boundary conditions are briefly summarized in Table 2. Governing equations and boundary conditions. The index i refers to the negative (i ¼ n) or positive (i ¼ p) electrode, or to the separator region in case of the electrolyte. To describe the effective conductivities (keff) and lithium diffusion coefficients (Deff) in the electrolyte in the different parts of the cell, we used the Bruggeman approximation g

keff;i ¼ k εi i ;

g

Deff ;i ¼ De εi i

(1)

where index i refers to the cell regions (anode, separator, cathode). k and De are the values of the pure electrolyte and the exponents gi are the Bruggeman exponents. The effective conductivity of the solids seff was described analogously. The charge transfer at the interface between the active electrode materials and the electrolyte was modeled by the BV equation. The reaction current density across the interface is given by




Fh Fh  exp  ð1  aÞ i ¼ i0 exp a RT RT

(2)

with the overpotential

h ¼ fs  fe  Ueq

(3)

where fs and fe denote the electrical potential at the interface in the solid material and in the electrolyte, respectively, and Ueq is the

equilibrium open circuit potential. The exchange current density is given by

 a i0 ¼ k cae cs;max  cs c1a s

(4)

where the concentrations in the electrolyte, ce, and in the oxide, cs, refer to the values at the interface. cs,max is the maximum lithium concentration in the oxide. k is the reaction rate constant and a is the anodic charge transfer coefficient. The cathodic charge transfer coefficient is thus given by 1a. Faraday's constant is denoted by F, the gas constant by R, and the temperature by T (cf. also Nomenclature). For solving the system of partial differential equations (PDEs) of the P2D model, we implemented these equations in the PDE module of the software COMSOL Multiphysics 4.3b. The implementation was validated by comparing with experimental data by Doyle et al. [20]. 4. Results and discussion The results of GITT measurements on the LCO/Li half-cell are shown in Fig. 1. Closer inspection of the data shows that the increase of the overpotential immediately after switching on the galvanostatic current, referred to as IR drop, changes with the SOC of the LCO cathode. Furthermore, the IR drop for a given SOC is different for charging (Li-deintercalation) and discharging (Liintercalation). Such behavior of the IR drop has also been found in other studies [3,21,22]. Attempts to simulate the sequence of GITT pulses within the P2D model, using SOC-independent kinetic parameters in the BV equation, failed. This can clearly be seen in the plot in Fig. 1a. In this simulation, the reaction rate constant and anodic charge transfer coefficient were fitted to satisfactorily describe the first few discharge GITT pulses. The diffusion coefficient in lithium oxide was chosen as in Fig. 1c. With these data, the disagreement between the IR drops in the simulation and in the measurements clearly increases with increasing lithium concentration in LCO. The IR drop immediately after switching on the GITT pulse is the sum of the overpotential hohm due to ohmic resistance and the contribution hct from the charge transfer at the electrolyte-LCO interface (IR ¼ hohm þ hct). To separate hct from the measured IR drop, the overpotential hohm was ascertained by determining the ohmic resistance of the LIB cell via separate EIS measurements. Fig. 2 shows the ohmic resistance of the LCO/Li half-cell as a

Table 2 Governing equations and boundary conditions. The index i refers to the negative (i ¼ n) or positive (i ¼ p) electrode, or to the separator region in case of the electrolyte. Governing equations

Boundary conditions  

Electrical potential in electrolyte and solid


v vfe  k vx eff;i vx



 
 2RT 1  t þ v v ln ce keff;i þ ¼ as;i ii F vx vx

v2 f seff 2s ¼ as;i ii vx

v  vx fe 

  v f  seff;n vx s

¼0

x¼0

  v f  ¼ seff ;p vx s

x¼lnþlsep þlp

(9)  

 

v    vx ce  x¼0

(10)  

Mass transport in insertion particle
 vcs;i vcs;i 1 v ¼ 2 r 2 Ds;i ¼0 vt vr r vr

x¼lnþlsep þlp

(8)

Mass balance in electrolyte
 vεi ce v vε ce 1  tþ  as;i ii Deff;i i ¼ vx vt vx F

x¼0

  v f  ¼ vx e

v  vr Cs;i 

(11)

r¼0

  v c ¼ ¼ vx e 

x¼lnþlsep þlp

  v C  ¼ 0; Ds;i vr s;i 

r¼Rs

¼0

¼ iFi

¼ iapp

A. Hess et al. / Journal of Power Sources 299 (2015) 156e161

159

4.2

+

Half-cell voltage vs Li/Li (V)

4.3

4.1 4 3.9 3.8 3.7 3.6 3.5

(a)

90 100 110 120 130 140 150 160 170 180 Time (h)

+

Half-cell voltage vs Li/Li (V)

4.5 4.4 4.3 4.2 4.1 4 3.9

10

20

30

40 50 60 Time (h)

70

80

90

4.2

+

Half-cell voltage vs Li/Li (V)

(b)

0

(c)

4

3.8

3.6

Fig. 2. Ohmic resistance RU, derived from EIS measurements, and corresponding ohmic contribution to the overpotential, hU ¼ IRU as a function of the SOC, x, of the LCO electrode.

(Co3þ, Co4þ), causing a delocalization of electrons in the lattice, which may enhance the electronic conductivity [23]. With hohm(x) from Fig. 2, we were able to calculate the activation overpotential hct from the IR drops in the GITT measurements. The plots of the resulting overpotential hct as a function of SOC in Fig. 3 reveal significant changes with the SOC as well as distinct differences between charge and discharge pulses at identical SOCs. To properly account for this behavior of hct, we propose here to introduce a SOC-dependent rate constant k(x) and transfer coefficient a(x) in the BV equation. A noticeable dependence of the reaction rate constant k and the transfer coefficient a on the value and direction of the local current density (reaction rate) is not expected. In a recent publication of the authors [24], it has been shown that the charge transfer reaction of lithium intercalation in LiFePO4 perfectly obeys the classical BV equation with rate constant k and transfer coefficient a being independent of the current density for given SOC. The asymmetry between charging and discharging is then solely described by the charge transfer coefficent a(x) s 0.5. This assumption enables us to determine the functions k(x) and a(x) unambiguously from the measured IR drops via the following nonlinear system of two BV equations for the case of charging and discharging ðdÞ

i

90 100 110 120 130 140 150 160 170 180 Time (h)

Fig. 1. Measured (red circles) and calculated (black line) half-cell voltages of the LCO/Li half-cell as a function of time: (a) simulation of discharge pulses using SOCindependent kinetic parameters in the ButlereVolmer equation showing increasing deviation of the simulation from the measurements with increasing Li content in LCO. Simulations of charging (b) and discharging (c) pulses show considerably better agreement with measured data by using a SOC-dependent reaction rate constant k(x) and transfer coefficient a(x). The dotted lines serve as a guide to the eye and are the envelope of the simulation data. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

function of SOC, which was obtained from the high-frequency intercept of the impedance. The overpotential hohm was found to depend slightly on SOC, which may arise from changes of the electronic conductivity of the cathode with the lithium content x in LixCoO2. The variable x denotes the degree of lithiation of cobalt oxide and is directly related to the state of charge (SOC) of the battery with x ¼ 1 for the completely discharged state. While Li0.75 CoO2 is semiconducting, Li0.5 CoO2 is metallic and all the substoichiometric species LixCoO2 contain cobalt in different valences

¼ kðxÞ cs;max ð1 

ðdÞ

 exp

ðdÞ

aðxÞ xÞaðxÞ xð1aðxÞÞ ce

 ð1  aðxÞÞ

Fhct ðxÞ RT

!!

Fh ðxÞ exp aðxÞ ct RT

!

; (5)

Fig. 3. Overpotential due to charge-transfer resistance, hct, as a function of the Li concentration in LCO derived from the measured IR drops during charging (circles) and discharging (triangles).

160

ðcÞ

i

A. Hess et al. / Journal of Power Sources 299 (2015) 156e161

¼ kðxÞ cs;max ð1 

ðcÞ

 exp

ðcÞ

aðxÞ xÞaðxÞ xð1aðxÞÞ ce

Fh ðxÞ  ð1  aðxÞÞ ct RT ðdÞ

!!

ðcÞ

Fh ðxÞ exp aðxÞ ct RT

!

: (6)

The overpotentials hct ðxÞ and hct ðxÞ are the derived values in Fig. 3 for discharging and charging, respectively. i(d) and i(c) are the corresponding charge transfer current densities for the electrode particles near the current collector of the cathode. The value of these current densities is determined in a rough approximation as the mean value within the electrode, which is obtained by dividing the applied galvanostatic current by the whole surface area of the electrode particles. The system of equations (5) and (6) was solved for k and a for given concentrations x by using the software Matlab. The results of these calculations are shown in Fig. 4. The charge transfer coefficient a(x) increases with increasing Li concentration, starting with a ¼ 0.51 at x ¼ 0.45 up to about 0.57 at x ¼ 0.8. This implies that the anodic reaction (Li extraction) is favored. In particular, according to transition state theory, the activation barrier for Li insertion is higher than for the reverse reaction. As a consequence, the activation overpotential to realize a certain current (reaction rate) is higher for discharging than for charging. The reaction rate k(x) decreases with increasing Li concentration, where the slope diminishes considerably at concentration x > 0.7. At higher concentration, LCO undergoes a first order phase transition between an almost fully lithiated hexagonal phase Li0.93 CoO2 and a vacancyrich hexagonal phase Li0.75 CoO2. According to the shrinking core model proposed in Ref. [12], the lithium concentration at the surface of the particles does not change within the two phase region.

As a consequence, the exchange current density should be constant within this region. The weak change of k and a in Fig. 4 in the region 0.7 < x < 0.85 could be related to this two-phase region. By using the fitted functions for k(x) and a(x), the exchange current density Eq. (4) was calculated as a function of the Li concentration. Fig. 4c shows a comparison with the case of SOCindependent values a ¼ 0.5 and k ¼ 5,108. The current densities differ up to a factor of about 2 for higher concentration x > 0.7. By including the values of k(x) and a(x), derived above, in our simulations, the IR drops of the simulated GITT pulses agree much better with the measured data (cf. Fig. 1b and c), compared to the case of SOC-independent parameters (Fig. 1a). The calculated GITT pulses during charging and discharging match the experimental data satisfactorily except at the beginning of charging at low SOC (Fig. 1b). In this SOC region, the GITT pulse height predicted by our model is smaller than the measured drops despite the fact that unexpected small values for the apparent diffusion coefficient have been used in the simulation (cf. Fig. 5 for x > 0.75). As mentioned above, LixCoO2 undergoes a first order phase transition at smallSOC. Although the transition is reversible, the kinetics of this transition may be different for charging and discharging as proposed by the shrinking core model [12]. Such effects are not captured by our model and could explain the deviations at low SOC for charging. The curve shape of the half-cell voltage vs time during the GITT pulse of 300 s duration is essentially determined by the diffusion in the oxide particles. In a first step, an apparent diffusion coefficient was derived by means of the GITT analysis proposed by Weppner and Huggins [25], which is based on the analytical solution of a planar diffusion problem

D¼



 4 VM i 2 dE=dx 2  pffiffi ; p F dE d t

. t ≪ R2s D:

(7)

This analysis of the discharging pulses reveals a strong dependence of the diffusion coefficient on the SOC (Fig. 5). Particularly small values were found for concentrations in the range from about x ¼ 0.75 to 0.85. This behavior is compatible with a vacancy diffusion mechanism, where the number of available free sites for lithium hopping decreases with increasing lithium concentration. For a predominant di-vacancy mechanism, as proposed in Ref. [26], the diffusion coefficient would diminish with increasing concentration as (1  x)2. To improve the match of our simulations with the half-cell voltage profiles of the measured GITT pulses, appropriate diffusion coefficients were fitted within the P2D model by means of a trial and error parameter study. The estimated apparent diffusion coefficient as a function of the Li concentration is shown in Fig. 5. It Charge Discharge Discharge Weppner

2 -1

D (m s )

1e-14

1e-15

1e-16

1e-17 0.4 Fig. 4. Calculated anodic transfer coefficient a (a) and reaction rate constant k (b) as a function of the Li concentration in LCO. The dimension of the rate constant is (A/m2) (mol/m3)1a. The fitted curves in (a) and (b) are a guide to the eye and were used to calculate the exchange current density i0, Eq. (4), vs the Li concentration (c) in comparison to the case of SOC-independent reaction rate k ¼ 5,108 and symmetric transfer coefficient a¼0.5 (dashed line). Fit functions: a(x) ¼ 0.5 þ (0.017x  0.008)1/2, kðxÞ ¼ ð4:8x2  8:2x þ 3:6Þ$107 .

0.5

0.6 0.7 x in LixCoO2

0.8

0.9

Fig. 5. Estimated lithium diffusion coefficients within LixCoO2 oxide particles as a function of the Li concentration. The black circles refer to values determined by using the procedure by Weppner and Huggins, Eq. (7). The other values were obtained by fitting the diffusion coefficient within the P2D model to the measured voltage profiles of the GITT pulses. The red squares refer to a fit of the discharge pulses and the red circles to a fit of the charge pulses.

A. Hess et al. / Journal of Power Sources 299 (2015) 156e161

turns out that the diffusion coefficients derived by fitting the discharge pulses are larger than the values derived from the charge pulses. In both cases, particularly small values were obtained for Li concentrations x > 0.75, similar as for the analysis by Weppner and Huggins. The reason for this difference between charging and discharging is presently unknown. It could be related to phase transformations in the oxide during (de)lithiation [18,27]. 5. Conclusion GITT measurements on a LCO/Li half-cell revealed significantly differing IR drops for charging and discharging at a given SOC of the LIB cell. These differences cannot be modeled by using a SOCindependent reaction rate constant and symmetric transfer coefficients (a ¼ 0.5) in the ButlereVolmer equation. To account for the chargeedischarge asymmetry in the IR drops, we have determined a SOC-dependent reaction rate constant k(x) and anodic transfer coefficient a(x) from the measured IR drops and the EIS determination of the ohmic resistance of the LIB cell. These SOCdependent parameters enable a satisfactory simulation of the GITT measurements in a wide SOC range. Thus, introduction of SOC-dependent parameters in the ButlereVolmer kinetics is expected to considerably improve battery simulations in general. Further investigations are needed to substantiate the applicability of the present approach to other applied currents and cathode materials. The measured GITT data further suggest a strong SOC dependence of the apparent diffusion coefficient in the LCO electrode particles. In certain SOC regions, particularly slow relaxation of GITT pulses was observed. Thus, the relaxation phase between GITT pulses has to be chosen sufficiently long to reach a completely relaxed equilibrium state. In summary, combined GITT and EIS analysis of charging and discharging of a LIB cell proved to be an efficient method to elucidate the asymmetric charge transfer kinetics in LIBs. Acknowledgment The authors thank Bohayra Mortazavi and Hongliu Yang for fruitful discussions. This work was funded by the European Union and the Free State of Saxony via the project LTA4ITM (grant no. 100096881). We acknowledge the Center for Information Services and High Performance Computing (ZIH) at TU Dresden for computational resources. References [1] M. Doyle, T.F. Fuller, J. Newman, Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell, J. Electrochem. Soc. 140 (1993) 1526. [2] P. Arora, M. Doyle, A.S. Gozdz, R.E. White, J. Newman, Comparison between computer simulations and experimental data for high-rate discharges of plastic lithium-ion batteries, J. Power Sources 88 (2000) 219. [3] Q. Zhang, Q. Guo, R.E. White, Semi-empirical modeling of charge and discharge profiles for a LiCoO2 electrode, J. Power Sources 165 (2007) 427. [4] Y. Ye, Y. Shi, N. Cai, J. Lee, X. He, Electro-thermal modeling and experimental validation for lithium ion battery, J. Power Sources 199 (2012) 227. [5] W. Lai, F. Ciucci, Mathematical modeling of porous battery electrodes e revisit of Newman's model, Electrochim. Acta 56 (2011) 4369. [6] M. Landstorfer, S. Funken, T. Jacob, An advanced model framework for solid electrolyte intercalation batteries, Phys. Chem. Chem. Phys. 13 (28) (2011) 12817. [7] A. Latz, J. Zausch, Thermodynamic derivation of a ButlereVolmer model for intercalation in Li-ion batteries, Electrochim. Acta 110 (2013) 358. [8] M.Z. Bazant, Theory of chemical kinetics and charge transfer based on nonequilibrium thermodynamics, Acc. Chem. Res. 46 (2013) 1144. [9] P. Bai, M.Z. Bazant, Charge transfer kinetics at the solid-solid interface in porous electrodes, Nat. Commun. 5 (2014) 3585. [10] C. Wang, U. Kasavajjula, P. Arce, A discharge model for phase transformation electrodes: formulation, experimental validation, and analysis, J. Phys. Chem. C 111 (2007) 16656.

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Nomenclature as: specific interfacial area (m2 m3) ce, cs: Li concentrations in the electrolyte and solids (mol m3) De, D: Li diffusion coefficient in the electrolyte and in the solid particles (m2 s1) E: cell voltage (V) F: Faraday's constant (C mol1) i: reaction current density (A m2) iapp: applied current density (A m2) i0: exchange current density (A m2) k: reaction rate constant (A m2 (mol m3)a1) l: thickness of anode, separator, and cathode (m) R: gas constant (J mol1 K1) RU: ohmic resistance (U m2) Rs: radius of solid particles (m) T: temperature (K) t: time (s) tþ: transference number of Li ions Ueq: open-circuit potential (V) VM: molar volume of active material (m3 mol1) x: state of charge Greek letters

a: anodic transfer coefficient g: Bruggeman exponents εi: volume fraction of electrolyte or active material in solid matrix

k: conductivity of the electrolyte (S m1) seff: effective conductivity of the solid phase (S m1) f: electrical potential (V) Subscripts or superscripts c, d: charge, discharge e, s: electrolyte, solid eff: effective value n, p: negative, positive electrode sep: separator loc: local property