Three-dimensional transient modeling of a non-aqueous electrolyte lithium-air battery

G Model EA 26870 No. of Pages 15

Electrochimica Acta xxx (2015) xxx–xxx

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Three-dimensional transient modeling of a non-aqueous electrolyte lithium-air battery Geonhui Gwak, Hyunchul Ju* Department of Mechanical Engineering, Inha University, 100 Inha-ro Nam-Gu, Incheon 402-751, Republic of Korea

A R T I C L E I N F O

A B S T R A C T

Article history: Received 6 September 2015 Received in revised form 4 March 2016 Accepted 7 March 2016 Available online xxx

A three-dimensional transient model is developed for a non-aqueous-electrolyte lithium-air battery to investigate numerically key phenomena during discharging. The proposed model rigorously assesses lithium peroxide (Li2O2) formation and growth in an air electrode, and its complex interactions with electrochemical reactions and species/charge transport. We assume that the porous air electrode mainly consists of sphere-like carbon particles, and hence a spherical film growth model is adopted to simulate the precipitation of Li2O2 on the spherical carbon surfaces. The model is first validated against voltage evolution data measured at different discharging current densities. Good agreement between the simulation results and experimental data is achieved, demonstrating that the model accurately captures voltage decline behaviors due to accumulated Li2O2 in the air electrode. Additional simulations are carried out with different air electrode designs in order to establish the optimization strategy for the air electrode. Detailed simulation results, including multidimensional contours, clearly indicate that the electrode thickness and degree of electrolyte filling are key factors for improving the discharging performance of lithium-air batteries. ã 2016 Elsevier Ltd. All rights reserved.

Keywords: Air electrode design Discharged products Lithium-air battery Non-aqueous electrolyte Numerical model

1. INTRODUCTION The specific energy density of most lithium-ion batteries is
150 Wh kg1 or 650 Wh L1, which is not sufficient for electric vehicles (EVs) to attain a driving range comparable to that of internal combustion engine vehicles, i.e.
700 Wh kg1 [1,2]. However, metal-air batteries have received much attention in recent years for EV applications, since much higher specific energy densities than lithium-ion batteries can be achieved owing to the unlimited supply of oxygen from the surrounding air. Among them, lithium-air batteries exhibit the highest theoretical specific energy density of 11,140 Wh kg1 that is close to that of gasoline fuel (13,000 Wh kg1). Various lithium-air batteries with different electrolytes, electrodes, and cell designs have been developed and presented in the literature [3]. These can be classified into two types: an aqueous electrolyte type and a non-aqueous electrolyte type. Using aqueous electrolytes can mitigate the issue of air electrodes clogging with discharged products (i.e., Li oxides), because these are generally

* Corresponding author. E-mail addresses: [email protected], [email protected] (H. Ju).

soluble in the aqueous electrolyte. However, lithium metal is highly reactive with water and thus, in the aqueous electrolyte types, lithium ion conducting and a water impermeable membrane must be present to prevent undesired reactions between water and the lithium electrode. Hasegawa et al. [4] examined the water stability of the NASICON-type lithium ion-conducting LATP (Li1+x+y AlxTi2xSiyP3yO12) in various types of aqueous solutions. They fabricated a Li-air cell with LATP and aqueous 1 M LiCl, and showed that a stable open circuit voltage of 3.6 V at 25  C could be successfully maintained for one week. Stevens et al. [5] designed a composite air electrode with an anionic polymer electrolyte membrane and applied it to an aqueous lithium hydroxide electrolyte-based lithium-air cell. They showed that using an anionic polymer electrolyte membrane successfully mitigated the precipitation of lithium carbonate and lithium hydroxide inside the air electrode using untreated air and 5 M or saturated lithium hydroxide electrolyte. Puech et al. [6] fabricated a ceramic LATP-AP for aqueous electrolyte-based lithium-air batteries. Although the membranes were thin (40–55 mm) and exhibited good mechanical and watertight properties, the overall ionic resistance of the membrane (55 mm) and coated layer (LiPON thin film of 12 mm) were still high, around 105 V over the area of 1 cm2.

http://dx.doi.org/10.1016/j.electacta.2016.03.040 0013-4686/ ã 2016 Elsevier Ltd. All rights reserved.

Please cite this article in press as: G. Gwak, H. Ju, Three-dimensional transient modeling of a non-aqueous electrolyte lithium-air battery, Electrochim. Acta (2016), http://dx.doi.org/10.1016/j.electacta.2016.03.040

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Nomenclature a C D E E f F H i I j k K n N MW P r R RLi2O2 ReO2 S t t+ T u ve

Ratio of active surface area per unit electrode volume, m2 m3 Molar concentration of species, mol m3 Species diffusivity, m2 s1 Roughness factor Equilibrium potential, V Activity coefficient of LiPF6 salt Faraday’s constant, 96,487C mol1 Henry’s constant Superficial current density, A m2 Current density, A m2 Transfer current density, A m3 Reaction rate constant, m s1 Effective permeability Number of electrons transferred in the electrode reaction Molar flux of species, mol s1 m2 Molecular weight, kg mol1 Pressure, Pa Particle radius, m Universal gas constant, 8.314 J mol1 K1 Film resistance, V m2 Oxygen transport resistance in electrolyte, s m1 Source term in transport equation Time, s Transfer number Temperature, K Superficial velocity, m s1 Velocity of the electrolyte, m s1

Greek symbols b Transfer coefficient d Thickness, m e Volume fraction h Surface overpotential, V k Proton conductivity, S m1 f Phase potential, V r Density, kg m3 s Electronic conductivity, S m1 t Surface effect factor m Viscosity, kg m1 s1

Superscripts e Electrolyte eff Effective g Gas 0 Initial conditions or standard conditions, i.e., 298.15 K and 101.3 kPa (1 atm)

Subscripts Average avg carbon Carbon particle in air electrode e Electrolyte electrode Air electrode g Gas i Species Li+ Lithium ion KC Kozeny-Carman constant Li2O2 Lithium peroxide O2 Oxygen

ref s void 0

Reference Solid, surface Void Initial conditions

It may be difficult to meet all the requirements of a membrane (sufficient lithium conductivity, mechanical strength, and water impermeability) for the aqueous-electrolyte-based lithium-air batteries. Therefore, considerable efforts have been directed toward non-aqueous electrolytes, or mixed electrolytes comprising both aqueous and non-aqueous electrolytes. In 1996, Abraham and Jiang [7] presented the first rechargeable lithium-air battery in which a non-aqueous organic membrane was sandwiched between a thin lithium electrode and a carbon composite air electrode. They successfully showed an open circuit voltage of
3.0 V and a specific energy density range of 250–350 Wh kg1 with good rechargeability over three cycles. Since then, lithium-air batteries with various non-aqueous electrolytes have been studied intensively [8–19]. The major issue with a non-aqueous electrolyte is that the discharged products are not soluble in the electrolyte, and hence continuously accumulate in the air electrode during the discharging process. This product discharge substantially raises resistance to both oxygen and electron transport in the air electrode. Generally, there are two types of discharged products formed during the oxygen reduction reaction (ORR) in nonaqueous electrolytes: lithium peroxide (Li2O2) and lithium carbonate (Li2CO3). While Li2O2 electrochemically decomposes into Li and O2 by an oxygen evolution reaction (OER) during the charging process, Li2CO3 resulting from the decomposition of nonaqueous electrolytes (carbonates, ethers, etc.) or CO2 in the atmospheric air is an irreversible byproduct. Many researchers have conducted experimental and theoretical studies to resolve the issues related to the precipitation of Li2O2 and Li2CO3 in nonaqueous lithium-air batteries [8,9]. Xu et al. [8] analyzed the discharge products in non-aqueous organic carbonate electrolytes for lithium-oxygen batteries using X-ray diffraction and in situ gas chromatography/mass spectroscopy. Their data indicated that Li2CO3 was the main product at the air electrode and was hardly oxidized below 4.6 V versus Li/Li+. In lithium-oxygen batteries with alkyl-carbonate electrolytes and Ketjenblack (KB)-carbon-based air electrodes, the primary discharge product was irreversible Li2CO3. Freunberger et al. [9] applied ether-based electrolytes to a lithium-oxygen battery and showed that the majority of the discharged products were Li2O2 during the first discharge/charge cycle, indicating that the ether-based electrolyte was more stable than the organic carbonate electrolytes. Thereafter, however, the decomposition of the ether-based electrolytes occurred by continuous cycling, and the fraction of Li2O2 produced considerably decreased with increasing cycle number. Hence, identifying appropriate non-aqueous electrolytes to make lithium-oxygen batteries fully rechargeable remains a major challenge. Searching for an optimum air-electrode material and structure is another key factor for improving the capacity and performance of lithium-air batteries. Xiao et al. [10] tested various commercial carbon materials for air electrodes in lithium-air batteries. Among them, KB carbon exhibited higher absorption capability for the electrolyte, and larger volume expansion that facilitated the Li/O2 reaction, and extra volume to hold the discharged products. Consequently, the highest capacity of 1756 mAh g1 was achieved with a KB-based air electrode under a dry air environment. Mirzaeian et al. [11] fabricated an air electrode using activated carbon that were synthesized through polycondensation of resorcinol with formaldehyde. They showed that the cell voltage

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was substantially affected by the morphology of the carbon electrode, and the proper control of pore volume and size was critical to obtain high cell performance. Tran et al. [12] reported a linear relationship between the capacity and average pore diameter of an air electrode with a non-aqueous electrolyte. Li et al. [13] synthesized nitrogen-doped carbon nanotubes (NCNTs) using a floating catalyst chemical vapor deposition method. They showed that the N-CNT electrode exhibited superior electrocatalytic activity for the Li/O2 reaction during discharge, and delivered a specific discharge capacity of 866 mAh g1, roughly 1.5 times higher than that of CNT itself. Zhang et al. [14] fabricated an air electrode using a single-wall carbon nanotube and carbon nanofiber, and showed that the discharge capacity decreased as the air electrode thickness increased. The highest discharge capacity of
2500 mAh g1 was achieved with a thin air electrode of 20-mm thickness at 0.1 mA cm2. The SEM images obtained from a fully discharged cell confirmed the experimental trend in which the discharged products were concentrated on the pores near the air side whereas the pores in the membrane side were almost empty. Various electrochemical and transport processes occurring in lithium-air batteries have been modeled in order to gain a better understanding of key fundamental phenomena and to optimize cell design and operating conditions. Sahapatsombut et al. [15] developed a one-dimensional (1-D) theoretical model for nonaqueous lithium-air batteries in which the Li2O2 was assumed only a discharged product and reversibly oxidized during charging. Good agreement was achieved against the voltage-capacity curves experimentally measured during a single discharge-charge cycling. Later, the formation of Li2CO3 was additionally considered in this model. Cell performance deterioration due to formation of the irreversible Li2CO3 product was predicted during 10 chargedischarge cycles [16]. They also numerically compared cell performances operating with pure oxygen and ambient air, and suggested the use of an oxygen-selective membrane under ambient air conditions that prevents access of CO2 and moisture from the atmosphere, and minimizes evaporation of the electrolyte [17]. Wang [18] approximated the Li2O2 precipitation process in the air electrode using three different film growth modes, i.e. planar, cylindrical, and spherical growth shapes. The calculated results showed that none of these modes accurately captured the experimental voltage degradation behavior during discharge. For better agreement with the experimental data, they suggested an empirical coverage model, i.e., similar to the ice coverage model in PEM fuel cells. Andrei et al. [19] developed a drift-diffusion-based 1-D model for non-aqueous lithium-air batteries and analyzed the effects of solvent and catalyst distribution in the air electrode. In their simulations, the oxygen diffusivity and solubility in the electrolyte were altered to simulate the effect of using different solvents whereas the oxygen reduction kinetics varied along the air electrode thickness for non-uniformly distributed catalysts. The higher discharging performances were predicted with higher oxygen diffusivity and solubility, and the cell energy density was substantially improved with the non-uniform catalyst design. They also suggested partly wetted air electrode design to improve cell discharging performance but relevant simulation results were not presented. Later, they presented a 1-D model for lithium-air batteries with dual electrolyte, i.e. non-aqueous electrolyte in the lithium electrode side and aqueous electrolyte in the air electrode [20]. Microscopic and mesoscopic, first-principle modeling of air electrode was carried to more precisely study the influences of air electrode morphology and structure on Li2O2 growth and cell performance degradation. A good overview in these fields was provided by Franco and Xue [21]. Recently, Xue et al. [22] developed a multiscale model for aqueous lithium-air batteries in

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which the microscopic porous structure effect of air electrode was approximated by using a bimodal pore size distribution function. Different shapes of cell discharging curve were predicted, depending on whether the discharging process was limited by electron tunneling or linear ohmic resistance through the Li2O2 film. Later, they included the effect of solution phase reaction in their multiscale model, for which an escape function was defined and adjusted to describe the transport of O2radicals into large open pores of the air electrode [23]. Li [24] combined a microscopic statistics model with a macroscopic CFD model to analyze microstructural effects of the air electrode on cell discharging performance. He elucidated the tradeoffs between mass transfer resistance and effective reaction surface area as the mean pore size varied. Welland et al. [25] adopted multiscale modeling approach successfully simulated the three stages of LiO2 evolution (i.e. nucleation, growth, and coarsening) in the cathode during discharge. A phase-field model was employed to capture the nanometer-scale deposition process of LiO2 on the carbon surface while species consumption/production and transport processes in the electrolyte were described by a millimeter-scale reactiondiffusion model. Blanquer et al. [26] accounted for both solution phase and thin-film mechanisms in their mesoscopic Li-O2 battery model. The model was applied to the simplified nanoporous domain comprising a spherical pore with two cylindrical channels and resolved the reaction and transport phenomena of the lithium and oxygen species via Kinetic Monte Carlo algorithm. The distribution and evolution of porosity during discharge were predicted under various pore and channel sizes as well as different oxygen and lithium ion concentrations. In this study, we present a 3-D battery model for non-aqueous electrolyte-based lithium-air batteries. The model is first validated against the experimental data measured at different discharging current densities. Next, using the validated model, we investigate the impact of the key design factors of an air electrode on cell discharging behaviors. A major focus is placed on the impact of electrolyte filling and thickness of air electrode. As noted in Zhang et al. [27], the degree of electrolyte filling in the air electrode significantly alters oxygen and lithium ion (Li+) transport through the electrode, which results in different cell voltage drop patterns and discharge capacities. Xia et al. [28] experimentally demonstrated that the partially wetted electrode greatly improved cell discharge capacity, compared to a conventional flooded electrode design. Nevertheless, to our knowledge no numerical work for the partially wetted air electrode design has been reported in the literature. Therefore, we attempt to numerically compare two air electrode designs, for which a Li2O2 film model is developed and incorporated into a comprehensive lithium-air batter model in order to calculate oxygen concentrations in both gas and electrolyte phases. Additionally, in the majority of lithium-air battery designs, the whole area of the air electrode is not likely utilized for an ORR and storage of discharge products. Therefore, the effect of the air electrode thickness is investigated to determine the optimum air-electrode thickness that is able to shorten the oxygen and Li+ transport pathway, and maximize the specific capacity (mAh g1 or mAh cm3) of the lithium-air cell. 2. NUMERICAL MODEL 2.1. Model assumptions The present electrochemical-transport-coupled 3-D model for non-aqueous-based lithium-air batteries is developed by accounting for key electrochemical reactions and transport phenomena in the sub-regions of a cell. The sub-regions are the Li+ conducting membrane, lithium and air electrodes, and current collectors. The assumptions made in developing the model are as follows:

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(1) Li2O2 is only a discharged product that deposits inside the porous air electrode. For instance, Li2CO3 formation due to side reactions with the electrolyte or CO2 in the air is neglected. (2) Li2O2 produced by ORR is deposited as a thin film form in the air electrode. A toroidal Li2O2 formation and Li2O2 particles produced by solution phase reaction are assumed to be negligible, which is reasonable at high discharging current densities and/or low donor number solvents [23,29]. (3) The air electrode is treated as an isotropic and homogeneous porous layer, and is characterized by effective porosity and permeability. (4) Oxygen dissolved in the electrolyte is in equilibrium with the air or oxygen in the gas phase. (5) The lithium-air battery is operated in isothermal conditions so that reversible and irreversible heat generation and resultant heat transfer are neglected.

2.2. Electrochemical reactions As shown in Fig. 1, a lithium metal electrode and porous carbonbased air electrode are separated by a membrane. During discharge, lithium metal is oxidized into Li+ ions and electrons. That is, a lithium oxidation reaction (LOR) as described in Eq. (1) occurs. While the electrons are conducted through an external circuit to reach the air electrode, the Li+ ions transport through the membrane driven by the Li+ concentration gradient as well as the electrolyte phase potential (fe ) gradient. In the air electrode, oxygen meets with the Li+ ions and electrons, and the Li+ ions are reduced with a high degree of complexity, usually involving multiple steps. Based on the assumption (1) written in Section 2.1, the oxygen reduction reaction (ORR) is simplified to a four-electron process, as expressed in Eq. (2).
 LOR in the lithium electrode : Li ! Liþ þ e E0ref ¼ 0:00V ð1Þ

þ

2Li
þ O2 þ 2e  ! Li2 O2 E0 ¼ 2:96Vvs:Li

ORR in the air electrode :

ð2Þ

where E0 denotes the theoretical equilibrium potential for each electrode, and can be calculated by the thermodynamic properties under standard state conditions.

The electrochemical kinetics for the LOR and ORR in Eqs. (1) and (2) can be expressed by the Butler-Volmer equation as represented in Eqs. (3) and (4), respectively.      1b b F h  exp  F h ð3Þ LOR : j ¼ anFk exp RT RT

ORR :

   1b  1b 1b Fh j ¼ anFk C Liþ C O2 exp  RT

ð4Þ

where the transfer current density, j, is determined by the electrochemical kinetics parameters (n, k, and b) as well as the geometric parameter (i.e., a representing the electrochemically active area per unit volume of electrode). The kinetics for LOR is extremely fast, and thus the overpotential for LOR is assumed to be near zero. In contrast, the ORR is sluggish, leading to a large overpotential. Hence, the Butler-Volmer equation for the ORR was simplified to Eq. (4) using the Tafel approximation to simulate the discharging process. The expression of the overpotential, h, for the ORR is given by Eq. (5).

h ¼ fs  fe  E0 þ DhLi2 O2

ð5Þ

In Eq. (5), DhLi2 O2 represents the additional ohmic voltage drop due to Li2O2 film formation and growth on the reaction surface, which can be determined by the local electronic current density, is, across the film, and the overall electric film resistance, RLi2 O2 , which is assumed to be proportional to the Li2O2 film thickness, dLi2 O2 , as shown in Eq. (6).

DhLi2 O2 ¼ is RLi2 O2 ¼ is

dLi2 O2 s Li2 O2

ð6Þ

where s Li2 O2 denotes the electron conductivity through the Li2O2 film. It should be noted that although the magnitude of s Li2 O2 (1012-1011 S cm1) is much smaller than that of Li+ conductivity (1010-109 S cm1) [21], the ORR likely occurs on the outer Li2O2 film surface rather than the inner carbon particle surface because the Li2O2 film is almost impermeable to the oxygen. As the ORR occurs on the spherical carbon particles in the air electrode, the film thickness, dLi2 O2 , can be expressed by Eq. (7). " #  ecarbon þ eLi2 O2 1=3 dLi2 O2 ¼  1 rcarbon ð7Þ

ecarbon

Fig. 1. Schematic of a lithium-air battery during a discharging operation.

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where rcarbon represents the radius of a spherical carbon particle. ecarbon and eLi2 O2 are the volume fractions of carbon particles and Li2O2 inside the air electrode, respectively. Since the discharge product, Li2O2, is insoluble in the electrolyte and gradually accumulates on the active reaction surfaces, the size of the combined carbon and Li2O2 spherical particles continuously increases and agglomerates during discharge, which leads to a reduction in the geometric factor for the electrode, a, in Eq. (4). Eq. (8) can be used based on macroscopic porous media theory.   eLi O t a ¼ a0 1  2 2 ð8Þ

air electrode (ee) and the void fraction for the wetted electrode (evoid). These two electrode configurations are described in Fig. 2. In Eqs. (10) and (11), density is expressed in the same way, meaning that rF is the electrolyte density for the flooded air electrode, and oxygen or air density for the wetted electrode. In addition, superficial velocities are used in Eqs. (10) and (11) in order to automatically ensure mass flux continuity at the interface between the porous and nonporous regions inside a lithium-air battery (e.g. the interface between the air porous electrode and air supply channel regions). In Eq. (10), the solid electrode and the discharge product, Li2O2 are excluded in the control volume. Therefore, Sm stands for the oxygen consumption by the ORR.

where a0 represents the ratio of the active surface area per unit electrode volume at the initial discharging stage. The exponent coefficient, t , can be determined by combining the voltage loss due to the reduction in the active surface area in Eq. (8), and the voltage loss estimated by the spherical film growth model in Eq. (6), which is represented by Eq. (9). Detailed derivation is given in the Appendix. !   ecarbon þ eLi2 O2 1=3 ð1  bÞF is
 rcarbon t¼  1 ð9Þ eLi O ecarbon RTln 1  e2e 2 s Li2 O2

Sm ¼

Su in Eq. (11) denotes the momentum source term for the porous lithium-air battery components devised to recover Darcy’s law under the limiting condition, where the permeability of the porous medium is small and hence the velocity is small.
m ~ u ð13Þ Su ¼  K

2.3. Conservation equations and source/sink terms

K¼

According to the aforementioned assumptions described in Section 2.1, the lithium-air model is based on four principles of conservation: mass, momentum, species, and charge. The mass and momentum conservation equations are given as follows: Mass conservation:

The species conservation equations can be obtained by individual species balance analyses, which precisely accounts for individual species consumption/production in electrochemical reactions and species transport driven by migration, diffusion, and convection. The basic form can be expressed as follows:

@eF rF þ r  ðrF~ u Þ ¼ Sm @t

ð10Þ

Species conservation :

ð11Þ

~ on the right-hand side of Eq. (15) represents the flux of where N i each species via diffusion, migration, and convection. In the present lithium-air battery model, the Li+ and oxygen species in the electrolyte phase, and gaseous oxygen are considered, as shown in Eq. (16)–(18), respectively.

ee

Momentum conservation:

  u u~ u 1 @rF~ rF~ þr ¼ rP þ r  t þ Su eF @t eF

where eF denotes the porosity of a phase, F for the porous components of a lithium-air battery, such as the air electrodes. Therefore, eF is the volume fraction of electrolyte for the flooded

j MO 2F 2

ð12Þ

In Eq. (13), K is the effective permeability of the air electrode, determined using the Kozeny-Carman equation: 4r2p

e3F

ð14Þ

C KC ð1  eF Þ2

@ðei C i Þ ~ þ Si ¼ r  N i @t

~ þ ¼ Defþf rC þ þ ie tþ þ ~ ue C Liþ N Li Li Li F

ð15Þ

ð16Þ

Fig. 2. Schematic illustration of different degrees of electrolyte filling in an air electrode. (a) flooded electrode and (b) wetted electrode.

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Table 1 Geometric and operating conditions. Description

Value

Thickness of the membrane (dmem ) Thickness of the air electrode (delectrode ) Active area of the air electrode (Aelectrode ) Spherical particle size (rcarbon ) Pore radius of carbon air electrode (rp ) Molar concentration of the lithium ion (C Liþ ;0 )

50 mm 375, 750 mm 0.0001 m2 40 nm 10 nm 1000 mol m3

Ambient oxygen concentration (C O2 ;0 ) Operating temperature (T) Operating pressure (P) Porosities for the flooded air electrode, ee , ecarbon , evoid Porosities for the wetted air electrode, ee , ecarbon , evoid

8.584 mol m3 298 K 1 atm 0.73, 0.27, 0.00 0.70, 0.20, 0.10

Eq. (18) should be extended to the air electrode regime. Because the pore radius of the air electrode, rp is sufficiently small (usually around tens of nanometer), the gaseous diffusive transport can be controlled by the Knudsen diffusion effect due to molecule to wall collisions as well as molecular diffusion caused by molecular collisions. The Knudsen diffusion coefficient can be computed according to the kinetic theory of gases as follows: DKO2 ;g ¼

  2 8Ru T 1=2 rp 3 pMO2 ;g

ð21Þ

Therefore, in Eq. (18), the effective diffusivity of oxygen gas, DOef2f;g is obtained by combining both molecular and Knudsen diffusion effects with the effects of porosity and tortuosity of the porous air electrode using the Bruggeman correlation [30]: !1 1 1 þ K ð22Þ DOef2f;g ¼ envoid DO2 ;g DO ;g 2

ef f ~ ~ N O2;e ¼ DO2 ;e rC O2 ;e þ ue C O2 ;e

ð17Þ

ef f ~ ~ N O2;g ¼ DO2 ;g rC O2;g þ ug C O2 ;g

ð18Þ

In Eqs. (16) and (17), ~ ue represents the velocity of the electrolyte. ef f ef f + DLi þ and DO ;e denote the effective diffusion coefficient of Li and O2 2

in the electrolyte phase, respectively. For the porous air electrode, these values need to be modified from their intrinsic value (i.e., as listed in Table 1) using the Bruggeman correlation [30] to account for the effects of porosity and tortuosity in the porous region as shown in Eqs. (19) and (20). ef f n DLi þ ¼ ee DLiþ

ð19Þ

DOef2f;e ¼ ene DO2 ;e

ð20Þ

where, ee is the volume fraction of electrolyte in the air electrode when the air electrode is fully flooded by the electrolyte. For the fully flooded electrode design, the gaseous oxygen transport equation, Eq. (18) is solved only for the gas channel region. On the other hand, for the partially flooded air electrode (see Fig. 2b), the gaseous oxygen transport is dominant in the electrode and hence

where, evoid represents the volume fraction of the void in the air electrode. It should be noted that, based on assumption (3) in Section 2.1, the oxygen concentration in the electrolyte, C O2 ;e , is in thermodynamic equilibrium with the oxygen concentration in the air, C O2 ;g . Therefore, C O2 ;e and C O2 ;g are correlated using Henry’s constant as shown in Eq. (23). C O2 ;e ¼

C O2 ;g H

ð23Þ

We assumed the magnitude of ee decreases as the volume fraction of Li2O2 increases based on the following equation.

ee ¼ 1  ecarbon  eLi2 O2  evoid

ð24Þ

According to Eq. (2), Li2O2 is produced by the ORR, and the rate of increasing eLi2 O2 in the air electrode during discharge can be calculated by Eq. (25).

@eLi2 O2 j  MW Li2 O2 ¼ 2F rLi2 O2 @t

ð25Þ

In Eq. (16), t+ and ie are the transference number for the Li+, and ionic current density through the electrolyte phase, respectively. The expression of ie is obtained based on concentrated solution

Fig. 3. Schematic of the resistance network on a sphere-like carbon particle in an air electrode.

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theory as shown in Eq. (26).   2RT kef f @lnf rlnC Liþ ð1  t þ Þ 1 þ ie ¼ kef f rfe  F @lnCLiþ

ð26Þ

In Eq. (26), fe denotes the electrolyte phase potential, and the ionic conductivity, keff , should be an effective one for the porous electrode via the Bruggeman correlation [30] as represented by Eq. (27).

kef f ¼ ene k

ð27Þ

The second term on the right-hand side of Eq. (15), Si, represents the source/sink terms of individual species, i, due to the electrochemical reactions that occur during discharge of the lithium-air cell, i.e., the LOR and ORR in Eqs. (1) and (2), respectively. For the Li+ and oxygen species in the electrolyte, that can be expressed as Eqs. (28) and (29), respectively. SLiþ ¼

2j F

SO2 ;e ¼

ð28Þ

j 2F

ð29Þ

Charge transport due to the electrochemical reactions (LOR and ORR) in lithium-air batteries is governed by the principle of the conservation of charge, as shown in Eq. (30), ie ¼ r  ~ is ¼ j r ~

ð30Þ

ie , is given by Eq. (26), whereas the electronic current The term, ~ density through the solid phase,~ i , is simply expressed by Eq. (31).

7

It should be noted that Eq. (32) only accounts for the electronic ohmic voltage drop through the initial air electrode in which insoluble Li2O2 precipitates are not present. The additional electronic voltage drop due to Li2O2 film formation and growth is represented by Eq. (6). 2.4. Film model for wetted air electrode If the porous air electrode is fully filled with electrolyte without allowing any gas inside the pores, as schematically shown in Fig. 2a, this type of electrode is called a ‘flooded electrode,’ wherein the oxygen for an ORR is dissolved in the electrolyte and then transported through it to reach the active reaction surface. Since oxygen transport through the electrolyte phase is far more difficult than through the gas phase, the discharging performance of the lithium-air cell is likely to be limited by the oxygen transport in the electrode. If the air electrode is partially filled with electrolyte (i.e. wetted electrode in Fig. 2b), the air electrode can operate optimally, ensuring a higher and uniform distribution of oxygen throughout the electrode. To simulate the lithium-air battery with the wetted electrode, the effect of the electrolyte film should be additionally taken into account in the model. As shown in Fig. 3, the electronic ohmic voltage drop due to the Li2O2 film, DhLi2 O2 , is already considered in Eq. (6). Therefore, the influence of oxygen transport resistance through the electrolyte film should be newly modeled and incorporated into the present lithium-air battery model. The oxygen transport resistance, ReO2 , can be expressed as a function of the electrolyte film thickness, de , and the oxygen diffusivity in the electrolyte phase, DeO2 , as shown in Eq. (33).

s

~ is ¼ s

ef f

rfs

ð31Þ

In Eq. (31), fs denotes the solid phase potential, and the electronic conductivity, s ef f , should be an effective one for the porous electrode via the Bruggeman correlation [30] as represented by Eq. (32).

s ef f ¼ encarbon s

ð32Þ

where ecarbon denotes the volume fraction of a solid matrix in the porous air electrode.

ReO2 ¼

de

DeO2

ð33Þ

The conservation of oxygen in the air electrode requires that the consumption rate of oxygen by the ORR is equal to the rate of oxygen diffusion through the electrolyte film, which is represented by Eq. (34). C O ;e  C O2 ;g =H EDO2 ;e  j ¼E 2 ¼ C O2 ;e  C O2 ;g =H 2Fa de RO2 ;e

ð34Þ

Fig. 4. Computational domain and mesh configuration for a single lithium-air battery cell.

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where j and E denote the local transport current density in the air electrode and a roughness factor representing the electrochemically active area per unit geometric area, respectively. In this model, the kinetic expression for an ORR, i.e. Eq. (4), should be modified in terms of the oxygen concentration on the interfacial surface between the electrolyte and Li2O2 films, C O2 ;e as shown in Eq. (35).    1b  1b 1b Fh C O2 ;e exp  ð35Þ j ¼ anFk C Liþ RT By combining Eqs. (34) and (35), C O2 ;e can be estimated from the interfacial oxygen concentration in the gas phase, C O2 ;g . If we define the following parameters as shown in Eqs. (36) and (37),    1b 1b 1 ð36Þ Fh exp  D ¼ nFk C Liþ 2F RT

G¼

EDO2 ;e

de

ð37Þ

then, Eq. (34) can be rewritten as Eqs. (38) and (39).   1b C O ;g f C O2 ;e ¼ D C O2 ;e  GC O2 ;e þ G 2 ¼ 0 H

ð38Þ

 b df ¼ Dð1  bÞ C O2 ;e G dC O2 ;e

ð39Þ

In this study, the solution to Eq. (38), CO2 ;e , is obtained using the Newton-Raphson method. It should be noted that because the film model considers the effect of oxygen transport through the electrolyte phase, Eq. (17) is not used for the wetted electrode geometry. 2.5. Numerical implementation The 3-D transient lithium-air battery model described in the preceding subsections was implemented numerically into a commercial CFD program, ANSYS-FLUENT, based on its user-

defined functions (UDFs). Fig. 4 schematically shows the 3-D geometry with a mesh configuration. Detailed cell dimensions and operating conditions are listed in Tables 1. In addition, kinetics, physiochemical, and transport parameters used for the lithium-air battery simulations are given in Table 2. The convergence criteria were set to 106 for equation residuals. The grid and time step sizes are inversely proportional to the spatial and temporal gradients of variables calculated by the lithium-air battery simulations, and thus were properly chosen based on several evaluation profiles during discharge. The maximum time step size to ensure adequate time resolution for non-aqueous-electrolyte lithium-air battery simulations is 0.5 s. 3. RESULTS AND DISCUSSION The 3-D lithium-air model described in Section 2 was validated against the discharge voltage evolution curves measured by Read et al. [36]. As seen in Fig. 5, the simulated voltage curves are generally in good agreement with the measured voltage curves at the discharging current densities of 0.1 and 0.2 mA cm2. These results indicate that the voltage decline behaviors due to cumulative discharged product (Li2O2) in the air electrode are correctly captured by the present 3-D lithium-air model. In both the simulation and experiment, the cell voltage gradually decreases because insoluble Li2O2 produced by the ORR continuously builds up in the air electrode. As the cell voltage becomes lower than 2.5 V, the voltage decline rate accelerates, which implies that a significant amount of Li2O2 has accumulated in the pores of the air electrode, and oxygen transport to local reaction sites has been severely limited. The rate of Li2O2 accumulation in the air electrode is faster with faster discharging, which leads to a severe oxygen transport limitation and a larger voltage loss at the higher discharging rate. Consequently, the higher current density case (0.2 mA cm2) exhibits shorter discharging capability. Fig. 6 shows the Li+ and oxygen concentration distributions in the air electrode at different elapsed times of cell discharging. The Li+ concentration is high at the membrane interface and gradually decreases toward the outside due to the ORR occurring in the

Table 2 Kinetics, physiochemical, and transport properties for PTFE/Super P air cathodes flooded with 1 M LiPF6 PC:DME electrolyte. Description

Value

Reference

Initial specific surface area per unit volume of air electrode (a0 ) Density of carbon Density of Li2O2 (rLi2 O2 ) Molecular weight of Li2O2 (MW Li2 O2 ) Molecular weight of O2 (MW O2 ) Henry’s constant for O2 (H) Lithium ion diffusion coefficient (DLiþ )

1333.33 m2 m3 2260 kg m3 2140 kg m3 0.04588 kg mol1 0.032 kg mol1 0.38 2.11 109 m2 s1 1.81 105 m2 s1

[31] [32] [32] [32] [32] [15] [15] [32]

1.83  109 m2 s1

[32]

0.01 S m1 1.141 S m1 100 S m1 0.43 1.03 5.9  1014 m s1 0.5 96485 C mol1 8.314 J mol1 K1 1 109 S m1 1.5 5.55 2.38

[32] [32] [19] [33] [33] [32] [32]

Oxygen diffusivity in gas phase (DgO2 ) Oxygen diffusivity in electrolyte phase (DeO2 ) Ion conductivity of membrane (kmem ) Ion conductivity of electrolyte (ke ) Electronic conductivity of porous carbon electrode (s ) Transference number of Li+ (tþ ) @lnf =@lnCLiþ Kinetic rate constant in the air electrode (k) Transfer coefficient (b) Faraday constant (F) Universal gas constant (R) Electronic conductivity of Li2O2 film s Li2 O2 Tortuosity of electrode (n) Kozeny-Carman constant: porous electrode (C KC ) Roughness factor (E)

[21] [30] [34] [35]

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3.0

Cell potential/V

2.5

2.0

2

Experimental(I=0.1mA/cm ) 2 Experimental(I=0.2mA/cm ) 2 Simulation(I=0.1mA/cm ) 2 Simulation(I=0.2mA/cm )

1.5

1.0 0

200

400

600

800

1000

-1

Specific capacity/mAh g

Fig. 5. Comparison of the simulated (lines) and measured (symbols) discharge curves at different current densities (0.1 and 0.2 mA cm2).

Fig. 6. Contours of (a) lithium ion (Li+) and (b) oxygen concentration in an air electrode at the discharge current density of 0.2 mA cm2.

electrode. As seen in Fig. 6a, the concentration gradient for Li+ across the air electrode is not significant, showing minimum values in the 991–994 mol m3 range near the outer electrode region. In contrast, severe mass-transfer limitation is observed in the oxygen

concentration distributions in the air electrode. Even at the early discharging stage of 30.728 mAh g1, the oxygen concentration in the electrode near the membrane is quite low (around 0.406 mol m3), which is caused by a very small oxygen diffusivity in the

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Fig. 7. Volume fraction of (a) Li2O2 and (b) electrolyte in an air electrode at the discharge current density of 0.2 mA cm2.

electrolyte phase (see Table 2). As discharging proceeds, the depletion of oxygen in the electrolyte of the air electrode is exacerbated, mainly due to the continuous accumulation of Li2O2 in the electrode. Therefore, the discharge performance of the Li+ cell is limited by oxygen transport through the air electrode rather than Li+ transport. The volume fractions of Li2O2, eLi2 O2 , and electrolyte, ee, in the air electrode are plotted in Fig. 7. The higher fraction of Li2O2 is observed in the outer electrode region facing either ambient air or oxygen supply channels, meaning a higher rate of the ORR there. This trend is because Li+ transport in the electrode is much easier than oxygen transport. As discharging proceeds, the volume fraction of Li2O2 near the outer electrode region increases up to 0.206, which significantly raises the oxygen transport resistance, and hinders oxygen access to the reaction sites inside the air electrode. The volume fraction of electrolyte in the electrode exhibits the opposite trend. A lower fraction of electrolyte is observed near the outer electrode region where a higher amount of Li2O2 accumulated. As discussed above, the major limiting factor in discharging operations is oxygen transport in the air electrode. Figs. 6 and 7 clearly show that almost half of the air electrode was not effectively used during the discharging process due to the severe limitation of oxygen transport. Therefore, the air electrode thickness of 750 mm may be too thick under the given cell design and operating conditions. Hence, the electrode thickness was halved to 375 mm for numerical comparison purposes. In addition, as illustrated in Section 2.4 and Fig. 2, oxygen transport through the electrode can

be more efficient if the air electrode is not fully filled with electrolyte (wetted electrode). Therefore, an additional case is defined under wetted electrode conditions (see Table 1). To simulate this case, it is necessary to couple the film model described in the section 2.4 with the comprehensive lithium-air model. Fig. 8 shows the effects of different air electrode designs on voltage evolution during discharge. Particularly, as seen in Fig. 8b, the film model developed for a wetted air electrode design accurately captured the experimental trend reported by Xia et al. [28]. The voltage drop is successfully mitigated by modifying the air electrode design into a thinner electrode or partially filling with electrolyte, indicating that the design changes effectively reduce the oxygen transport resistance through the air electrode. Improvement that is more substantial is achieved by designing partial filling of electrolyte into the electrode, which implies that the wetted electrode design might be crucial to facilitate oxygen transport in the air electrode, and finally increase the performance and specific capacity of lithium-air batteries. Fig. 9 displays the contours for oxygen concentration and volume fraction of Li2O2 in the halved air electrode at five different discharging stages. It is clear that, as the thickness of the air electrode is reduced by half, the utilization of the air electrode is greatly improved. Unlike Figs. 6b and 7a for the case of the 750-mm electrode thickness where the oxygen concentration and Li2O2 fraction in the air electrode are mainly concentrated near the outer electrode region, the oxygen and Li2O2 profiles in the halved electrode (375 mm), however, are spread out along the electrode thickness. This result indicates that the electrode surface near the

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3.0

Cell potential/V

2.5

2.0

1.5

δelectrode=750μm δelectrode=375μm 1.0 0

100

200

300

400

500

600

700

800

900

1000

-1

Specific capacity/mAh g (a) 3.0

Cell potential/V

2.5

2.0

1.5

Exp. of Xia et al.[28] (flooded electrode) Exp. of Xia et al.[28] (wetted electrode) Sim. (flooded electrode) Sim. with the film model (wetted electrode)

1.0 0

1000

2000

3000

4000

-1

Specific capacity/mAh g (b)

Fig. 8. Comparison of discharge curves for (a) different air electrode thicknesses at 0.2 mA cm2 and (b) for different degree of electrolyte filling at 0.1 mA cm2.

membrane interface is also well utilized for an ORR. For instance, when the discharge reaches the cut-off voltage (around 2.0 V), the Li2O2 fraction near the membrane interface is only 0.053 for the 750-mm-thick electrode (Fig. 7a) but much greater with the 375mm-thick electrode (i.e. around 0.171). Figs. 10 and 11 compare the contours of oxygen concentration and volume fraction for Li2O2 for the flooded and wetted air electrodes, respectively. It is seen that the level of oxygen concentration in the wetted electrode is much greater (around 8.37–8.56 mol m3), owing to more efficient gas-phase oxygen diffusion through the electrode. As a result, the distribution of Li2O2 volume fraction in the wetted electrodes is completely different from those in the flooded electrodes. As shown in Fig. 11, the peak of the Li2O2 volume fraction in the flooded electrodes occurs at the outer electrode region. In contrast, highly uniform distributions of the Li2O2 volume fraction are observed in the wetted electrodes, which is indicative that both Li+ and oxygen transport in the electrode are equally effective and consequently, the ORR occurs uniformly throughout the air electrode. So far, all the simulations including the model validation cases against the experimental data were carried out based on single cell geometries without oxygen gas channels wherein the porous air

electrodes were directly exposed to the ambient air and hence oxygen was almost uniformly dissolved into the electrolyte. As a result, all key distributions for Li+, oxygen, volume fractions of Li2O2 and electrolyte were simply 1-D, only varying along the cell thickness. However, a practical lithium-air battery stack design requires gas channels for oxygen supply, which induces fully 3-D distributions of these key variables. Therefore, the present 3-D model is applied to the computational geometry with the oxygen supply channels and the multidimensional effects for the lithiumair battery stack design are elucidated. Fig. 12 compares discharge curves of lithium-air batteries with and without oxygen supply channels. The cell geometry with the oxygen supply channels shows the lower discharge capacity (537.918 mAh g1) compared to that without the channels (584.562 mAh g1). Fig. 13 is the view of oxygen concentration contours in the air electrode with the oxygen supply channels at four different discharge stages. A comparison between Fig. 13 and Fig. 6b clearly indicates that severer oxygen depletion occurs near the electrode region underneath the current collector ribs because the ribs impose more severe oxygen transport limitation. Fig. 14 shows the volume fraction of Li2O2 in the air electrode with the oxygen supply channels at the same four discharge stages. The Li2O2 fraction

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Fig. 9. Contours of (a) oxygen concentration and (b) volume fraction of Li2O2 for a flooded air electrode with a thickness of 375 mm at 0.2 mA cm2.

Fig. 10. (a) the oxygen concentration (C eO2 ) in the electrolyte phase of the flooded air electrode and (b) gaseous oxygen concentration (C gO2 ) in the wetted air electrode. Both cases were simulated at the discharging current density of 0.1 mA cm2.

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Fig. 11. Comparison of volume fraction of Li2O2 for (a) the flooded air electrode and (b) wetted air electrode. The discharging current density of 0.1 mA cm2 was applied to both cases.

3.0

Cell potential/V

2.5

2.0

1.5

Without oxygen supply channel With oxygen supply channel 1.0 0

100

200

300

400

500

Specific capacity/mAh g Fig. 12. Comparison of discharge curves at 0.2 mA cm

2

600

700

800

-1

with and without oxygen supply channels. Both cases are based on a flooded air electrode design.

under the ribs is very low, implying that the electrode region under the ribs is not effectively used. Therefore, the lower discharge capacity of the cell geometry with the oxygen supply channels can be ascribed to more severe oxygen transport limitation caused by the ribs. 4. CONCLUSIONS In this study, a 3-D transient lithium-air model was developed by rigorously accounting for the electrochemical reactions during discharge (LOR and ORR) and the resultant species/charge transport through various components of a lithium-air cell. A major focus was on accurate predictions of Li2O2 formation/growth inside the air electrode, and the ensuing voltage losses due to activation, ohmic, and concentration polarizations. First, the model was validated against the voltage evolution curves measured at the discharging current densities of 0.1 and 0.2 mA cm2. The

simulation results agreed well with the experimental data wherein the model accurately captured voltage decline behaviors due to accumulated Li2O2 in the air electrode. Besides a comparison of predicted and measured voltage evolution curves, detailed profiles of Li+, oxygen, Li2O2, and electrolyte fraction in the air electrode clearly showed that the air electrode surfaces near the membrane interface were not utilized for an ORR during the entire discharge process. This fact indicated that the discharge performance was mainly limited by oxygen transport through the air electrode. To predict the discharge behavior of a partially wetted air electrode design, the film model was developed and coupled with the 3-D lithium-air model. The coupled model successfully captured the experimental trend wherein the partially wetted air electrode considerably outperformed the flooded air electrode. The simulation results obtained from the different air electrode designs highlighted that the optimization of air electrode thickness and/or degree of electrolyte filling were crucial for improving the

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Fig. 13. Oxygen concentration contours in the air electrode with oxygen channels.

Fig. 14. Contours of Li2O2 fraction in the air electrode with oxygen channels.

utilization of the air electrode and, finally, to achieve the high discharging performance and specific capacity of lithium-air batteries. ACKNOWLEDGEMENT Financial support of this study by INHA UNIVERSITY (INHA52026-01) is gratefully acknowledged. The authors also would like to thank TAESUNG S&E, INC. for providing technical support for use of ANSYS FLUENT for lithium-air battery simulations. Appendix. The exponent coefficient, t , in Eq. (9) can be derived by assuming that the discharge products are deposited on the spherical carbon particles as a film form. Using Eqs. (5) and (8), the Butler-Volmer equation for ORR can be expressed in more detail:

  eLi O t  1b  1b j ¼  1  2 2 nFk C Liþ C O2

ee



 b 0 exp 1 RT F fs  fe  E þ DhLi2 O2 Then, the parts of ohmic loss through the Li2O2 film and reduction in the geometric factor, a due to the agglomeration of spherical particles can be separated as:

(A.2)

where j0 represents the initial transfer current density in the absence of Li2O2 film. Equating j and j0 leads to:   eLi O t RT ln 1  2 2 DhLi2 O2 ¼ ðA:3Þ ð1  bÞF ee

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Finally, we can obtain the expression of the exponent coefficient, t , by combing Eq. (A.3) and Eqs. (6) and (7) as follows:   eLi O t RT ln 1  2 2 DhLi2 O2 ¼ ð1  b"ÞF ee #   ecarbon þ eLi2 O2 1=3 is ¼  1 rcarbon ðA:4Þ

s Li2 O2

t¼

ecarbon

ð1  bÞF i
 s eLi O RTln 1  e2e 2 s Li2 O2



ecarbon þ eLi2 O2 ecarbon

1=3

!  1 rcarbon

ðA:5Þ

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