Modeling of Li-Air Batteries with Dual Electrolyte

A770 Journal of The Electrochemical Society, 159 (6) A770-A780 (2012)0013-4651/2012/159(6)/A770/11/$28.00 © The Electrochemical Society

Modeling of Li-Air Batteries with Dual ElectrolyteP. Andrei,a,z J. P. Zheng,a,b M. Hendrickson,c and E. J. Plichtac

aDepartment of Electrical and Computer Engineering, Florida A&M University and Florida State University,Tallahassee, Florida 32310, USAbCenter for Advanced Power Systems, Florida State University, Tallahassee, Florida 32310, USAcU.S. Army CERDEC, Fort Monmouth, New Jersey 07703, USA

Li-air batteries with organic electrolyte at the anode and aqueous electrolyte at the cathode (dual electrolyte systems) are modeledusing the mass transport and drift-diffusion equations of the electrolyte during the discharge of the cells. Two regimes of operationare analyzed: (1) when the concentration of the electrolyte is smaller than the concentration of saturation of Li+OH− in water, and(2) when the electrolyte concentration reaches saturation and the reaction product is deposited at the cathode. Numerical simulationsare performed to evaluate the dependence of the specific capacity, energy and power densities on the geometrical and materialparameters during the two regimes of operation. It is shown that the energy density and specific capacity can be improved byincreasing the solubility and the diffusion coefficient of oxygen in the cathode, but they are not much affected by adding a uniformlydistributed catalyst in the cathode. The power density can be increased by 10% by increasing the solubility factor, the oxygendiffusion coefficient, or the reaction rate. The limiting factors for the low power density of these batteries are the low values of theoxygen diffusion coefficient in the cathode and the relatively high separator/anode and separator/cathode interface resistances.© 2012 The Electrochemical Society. [DOI: 10.1149/2.010206jes] All rights reserved.

Manuscript submitted September 19, 2011; revised manuscript received January 10, 2012. Published April 6, 2012.

There is a constant interest in developing battery cells with highenergy density storage. This interest is driven not only by industriesrequiring high electric power such as the electric car industry whorequires fundamental improvements in the current Li-ion technologyin order to extend the driving range of current electric vehicles, but alsoby many low-power electronics industries which require light, high-capacity energy storage devices. Li-air batteries have the potentialto become one of the best candidates for both industries because oftheir high theoretical energy density, which is almost one order ofmagnitude larger than that of Li-ion batteries.1–3 The high capacityof Li-air batteries is partly due to the fact that the active material isnot stored in the battery but is taken from the atmosphere. Computer-aided-design can help designing Li-air batteries in order to improvethe power density and cyclability of these batteries.

Depending on the type of the electrolyte that they use, Li-air batter-ies can be divided in Li-air batteries with organic electrolyte and Li-airbatteries with dual electrolyte (i.e. organic electrolyte at the anode andaqueous electrolyte at the cathode). In the case of Li-air batteries withorganic electrolyte2 the discharge product deposits on the surface ofthe carbon, while in the case of the batteries with dual electrolyte4, 5

the discharge product is soluble in the water at the cathode. This factmakes Li-air batteries with dual electrolyte very attractive becauseone can use air electrodes having a fuel cell structure to continuouslyeliminate the discharge product from the cathode. In this article wefocus on the modeling of Li-air batteries with dual electrolyte, and de-velop a physics-based framework for the simulation of these batteries.The model is based on the mass transport equations of the materialsinvolved in the system and can be easily discretized on a finite-elementgrid and used for the design process.

Li-air batteries with aqueous electrolyte are usually made of a Limetallic anode with organic electrolyte, a solid separator such as alithium-ion conducting glass-ceramic (LIC-GC), and a porous carboncathode filled with aqueous electrolyte (see Fig. 1). External air isallowed to penetrate inside the pores of the cathode, diffuse throughthe electrolyte, and react with the Li+ ions coming from the anode.Depending on the concentration of the aqueous electrolyte in thecathode one distinguishes two regimes of operation:

a. Before saturation (see Fig. 2), when the concentration of Li+

and OH− ions is smaller than the concentration of saturation ofLi+OH− in water (cLi,sat ), and

zE-mail: [email protected]

b. After saturation (see Fig. 3), when the concentration of Li+ andOH− ions is equal to the concentration of saturation of Li+OH−

in water (cLi O H,sat )

The chemical reactions can be summarized as follows:

Anode : Li → Li+ + e− [1]

Cathode (before saturation) : O2 + 2H2O + 4e− → 4OH− [2]

Cathode (after saturation) : 4Li+ + O2 + 6H2O + 4e−

→ 4LiOH · H2O(deposit) [3]

Overall (before saturation) : 4Li + O2 + 2H2O → 4Li+ + 4OH−

[4]

Overall (after saturation) : 4Li + O2 + 6H2O

→ 4LiOH · H2O(deposit) [5]

When the concentration of Li+ and OH− ions reaches the concentra-tion of saturation of Li+OH− in water, cLi,sat , the LiOH · H2O depositson the surface of the carbon, thus filling in the pores of the porouscathode and eventually interrupting the flow of the O2 in the cathode(see Fig. 3). When all the pores from the air side of the cathode areblocked, reaction 3 cannot take place anymore and the battery cannotbe further discharged. As in the case of Li-air batteries with organicelectrolyte,3, 6 the formation and deposition of the discharge productin the channels constitute the main reasons for the relatively short lifeof Li-air batteries.

The article is structured as follows. In the first section we sum-marize the model that we use for the simulation of Li-air batteries.This model is similar to the one that we developed for the simula-tion of Li-air batteries with organic electrolyte,6 however it takes intoconsideration the saturation and deposition of the LiOH · H2O dis-charge product at the cathode, as well as the appropriate source termsfor the transport equations. Details about the derivation of transportequations are presented in the Appendix. In the second section of thearticle, we present the numerical algorithm and simulation results. It isshown that the power density can be slightly improved by increasingthe solubility factor, oxygen diffusion coefficient, or using a catalystin the cathode. The energy density and specific capacity of thesebatteries can be improved by increasing the solubility factor and

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http://dx.doi.org/10.1149/2.010206jes

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Journal of The Electrochemical Society, 159 (6) A770-A780 (2012) A771

Figure 1. Li-air battery with aqueous electrolyte at the cathode and organicelectrolyte at the anode showing the region that is discretized and simulated.The ion and electron flows during discharge are shown by arrows.

Figure 2. Modeling of the oxygen diffusion and formation of Li+ and OH−ions in the porous carbon cathode before saturation (cLi O H < cLi,sat ).

Figure 3. Modeling of the oxygen diffusion and LiOH · H2O deposition inthe porous carbon cathode after saturation (cLi O H = cLi,sat ).

oxygen diffusion coefficient; however, they are not much affectedby using a uniformly distributed catalyst in the cathode.

Model

The model presented below is based on the theory of concentratedsolutions7 to describe the transport of Li+ and O2 in the anode pro-tective layer (APL), separator, and cathode electrolyte. We consider abinary monovalent organic electrolyte in the APL and aqueous elec-trolyte in the cathode, with no convection. After the concentration ofLi ions becomes equal to cLi,sat , we assume that LiOH · H2O precipi-tates on the surface of the carbon at the same location where reaction 3took place. We believe that this is a good approximation in the case ofbatteries with no convection because reaction 3 takes place at the sur-face of carbon and there is little probability that the resulting reactionproduct will deposit at a different location. However, if the electrolyteis flowing in the cathode such as in the case of Li-air flow batteries,8, 9

the LiOH · H2O reaction product might deposit at a different locationinside the cathode. In addition, if we coat the surface of the carbon orseparator with different polymers layers, the lithium hydroxide mightnot stick on the surface of the carbon or separator and precipitatesomewhere else in the aqueous compartment.10

The transport equations before and after the saturation of Li ionsare presented in the next subsections. A detailed proof of the transportequations 6, 7, and 18 is presented in the Appendix.

Model equations before Li+OH− saturation.— The electrostaticpotential of Li+ (φLi ), is assumed to satisfy the following drift-diffusion equation

∇ · (κe f f ∇φLi − κD∇ ln cLi ) − 4RC = 0 [6]

where κe f f is the effective electric conductivity of Li+ in the solidor liquid electrolyte, κD is the diffusional conductivity, cLi is the Li+

concentration, and RC is the oxygen conversion rate (i.e. the numberof moles of oxygen consumed per unit time and unit volume), whichvanishes in the APL and separator and is non-zero in the cathode. Theconcentration of the electrolyte is equal to the concentration of Li+

and satisfies7

∂(εcLi )

∂t= ∇ · (Def f ∇cLi ) + 4RC t+

F− i · ∇t+

F[7]

where ε is the porosity, Def f is the effective Li+ diffusion coefficient,t+ is the transference number, F = 96, 487 C/mol is the Faradayconstant, and

i = −κe f f ∇φLi − κD∇ ln cLi [8]

is the current density of the electrolyte. Since the separator is a grass-ceramic, the porosity is not defined inside the separator and parameterε is set to 1 (i.e. it is ignored) in this region. The oxygen concentrationsatisfies the following diffusion equation

∂(εcO2 )

∂t= ∇ · (DO2,e f f ∇cO2 ) − RC

F[9]

where DO2,e f f is the effective diffusion constant of the oxygen.The values of parameters κe f f , κD , Def f , t+, and DO2,e f f in the

above equations depend on the type of the material, tortuosity (throughporosity), and on the Li+ and oxygen concentrations. These parame-ters should be carefully calibrated for each region (separator, APL, andcathode electrolyte) in order to model the battery accurately. Follow-ing what is usually done in the literature, we assume that all effectivequantities can be written in terms of the porosity using Bruggemancorrelations as7

Def f = εβLi −1 DLi [10]

DO2,e f f = εβO2−1 DO2 [11]

κe f f = εβκ [12]


A772 Journal of The Electrochemical Society, 159 (6) A770-A780 (2012)

where DLi , DO2 and κ are the diffusion coefficients of Li+, O2, and theelectric conductivity of Li, respectively. Constants βLi , βO2, and β arethe Bruggeman coefficients for Li+ diffusion, O2 diffusion, and Li+

conductivity, respectively. The diffusional conductivity is consideredas11

κD = 2RT κe f f (t+ − 1)

F

(1 + ∂ f

∂ ln cLi

)[13]

where R = 8.314 J/mol K is the universal gas constant, and T is theabsolute temperature. In our simulations ∂ f

∂ ln cLiis approximated to

zero.Besides equations 1–3 that are solved in all three regions (APL,

separator, and cathode), the following equation is solved only in thecathode region to compute the electrostatic potential of electrons(φ)

∇ · (σe f f ∇φ) + 4RC = 0 [14]

where σe f f is the effective conductivity of electrons in the cathode,which is assumed to be

σe f f = εβe σ [15]

where βe is the Bruggeman coefficient for the electron conductivityand σ is the electron conductivity in the carbon electrode. Before theLi+OH− saturation, the porosity of the cathode remains constant

∂ε

∂t= 0 [16]

while the water is consumed according to

dm H2 O

dt= − Iapp MH2 O

2F[17]

where MH2 O is the molecular weight of H2O, Iapp is the dischargecurrent (expressed in A) and t is the time.

Model equations after Li+OH− saturation.— The electrostatic po-tential of Li+ (φLi ), the oxygen concentration (cO2 ), and the electro-static potential of electrons are assumed to satisfy the same continuityequations as before the Li+OH− saturation: 6, 7, and 14, respectively.However, the equation for the lithium concentration is slightly modi-fied to take into consideration the Li+ and OH− consumption

∂(εcLi )

∂t= ∇ · (Def f ∇cLi ) − 4RC (1 − t+)

F− i · ∇t+

F[18]

where ε is again set to 1 inside the separator region. Another majordifference between the two regimes is that, after saturation, the LiOH ·H2O discharge product decreases the local porosity in the cathode. Tocompute the rate at which the porosity decreases during the dischargeof the battery, we express the total number of moles of LiOH · H2Odeposited per unit volume of the cathode as

1

Vtot

∂ns

∂t= 4RC

F− 1

Vtot

∂nLi O H

∂t[19]

where Vtot is the total volume of the cathode, ns is number of molesof Li O H · H2 O that have been deposited by time t, and nLi O H =cLi,sat (Vtot − VC − Vs) is the total number of moles of Li O H in theelectrolyte. In previous equation VC is the volume of the carbon inthe cathode and Vs is the volume of the deposited layer. The porositychange is

∂ε

∂t= ∂

∂t

(Vtot − VC − Vs

Vtot

)= − 1

Vtot

∂Vs

∂t= − 1

Vtot

∂

∂t

(ms

ρs

)

= − Ms

ρs Vtot

∂ns

∂t[20]

where Ms and ρs are the molecular weight and mass density of de-posited LiOH · H2O. Using the last 2 equations we obtain

∂ε

∂t= −Ms

ρs

(4RC

F− 1

Vtot

∂nLi O H

∂t

)= − Ms

ρs

(4RC

F+ cLi,sat

Vtot

∂Vs

∂t

)

= − Ms

ρs

(4RC

F− cLi,sat

∂ε

∂t

)[21]

which can be solved for the porosity change

∂ε

∂t= −4RC

MsFρs

1 − cLi,sat Ms

ρs

[22]

The concentration of saturation of Li+ in water can be computed byassuming that the solubility of LiOH in water is 12.5 g of LiOH/100 g of water at 25◦C

cLi,sat = 12.5

112.5

ρsat

MLi O H= 5.17 × 10−3 mol/cm3 [23]

where ρsat is the mass density of the electrolyte at saturation (whenthe electrolyte contains 12.5 g of LiOH in 100 g of water) and MLi O H

is the molecular weight and mass density of LiOH.The oxygen consumption rate RC should be computed by taking

into consideration the exact geometry of the cathode, particularly theexact profile of the electrolyte/carbon interface in this region. As itis often done in the literature we consider that the cathode contains alarge number of quasi-cylindrical open-ended pores of average poreradius r̄ p (x). The LiOH · H2O precipitates on the inner surface of thepores (see Fig. 3) and, in this way, decreases the average pore radiusin time. If the shape of the pores is cylindrical, one can show that theporosity can be related to the average pore radius by6, 12

ε = ε0

(r̄ p

r̄ p,0

)2

, [24]

where ε0 and r̄ p,0 are the initial porosity and average pore radius (att = 0). Using the same assumption about the shape of the pores, theoxygen conversion rate can be approximated as6

RC =⎧⎨⎩

2kεO2 cO2

r̄ pcre fO2

×[e

(1−β)FRT ηc − e− βF

RT ηc

]if LC < x < L (cathode)

0, otherwise[25]

where β = 0.5, k is a reaction rate constant, cre fO2

= 1 mol/l is a nor-malization parameter, and ηc is the overpotential at the cathode, whichis given by6

ηc = φLi − φ − Uc − RCρes r̄ p,0

√ε

ε0ln

√ε0

ε[26]

where ρes is the electrical resistivity of the deposited layer and Uc isthe equilibrium potential for reaction 2 at the cathode.

Equations 6–9, 14, and 22, represent a system of partial differentialequations that should be subject to boundary and initial conditionsand solved self-consistently in order to compute the lithium-ion andoxygen concentrations, the electrostatic potentials, and the porosityat each location inside the electrochemical cell as a function of time.The initial and the boundary conditions for the one-dimensional cellsimulated in this article are presented in the next two subsections.

Initial conditions.— Initial conditions are specified for the lithium-ion and oxygen concentrations, porosity, and average pore radius ateach location inside the device. The values of the initial parametersare detailed in Table I. Similar to the case of Li-air batteries withorganic electrolyte, during a short time interval, immediately afterone applies a current through the battery, the battery will consume theoxygen that is already stored in the cathode. This phenomenon is dueto the fact that the initial O2 concentration is relatively high insidethe cathode (and equal to cO2,0), and the diffusion coefficient of O2

is usually much smaller than the diffusion coefficient of Li+. After arelatively short discharge time (typically of the order of seconds) the



Table I. Parameters used in simulations.

Parameter Value Reference

General parameters Molecular weight of Li (MLi) 6.941 g/mol 15Molecular weight of O2 (MO2) 32 g/mol 15Initial pore radius (r̄ p,0) 20 nmCell open-voltage potential (Uc + Ua) 3.4 V 14βLi 2.5βO2 3β 1.5Temperature (T ) 300 K

Anode specific parameters Li ion concentration at x = 0(cLi,0) 10−3 mol/cm3

Li ion conductivity (κ) 11.41 S/cm 16O2 diffusion coefficient (DO2 ) 7 × 10−6 cm2/s 3Li ion diffusion coefficient (DLi ) 3.018 × 10−5 exp (0.357cLi ) 17

Transference number (t+) 0.4492 − 0.4717 cLicre f

+ 0.4106c2

Lic2

re f

− 0.1287c3

Lic3

re f

18

Porosity (ε) 100%APL thickness (L A) 50 nm

Separator specific parameters Li ion conductivity (κ) 0.1 mS/cmO2 diffusion coefficient (DO2 ) 10−5 cm2/sLi diffusion coefficient (DLi ) 3.018 × 10−5 exp (0.357cLi ) 17Transference number (t+) 1Separator/anode interface resistance (RA) 70 �cm2

Separator/cathode interface resistance (RC ) 80 �cm2

Separator thickness (LC − L A) 50 μm

Cathode specific parameters Initial porosity (ε0) 75%Electric conductivity of carbon electrode (σ) 1 S/cmElectric resistivity of LiOH · H2O (ρes ) 0Li ion conductivity (κ in mS/cm) κ (cLi )

= −0.2912c3Li − 48.14c1.5

Li − 188.6cLi ,with cLi expressed in mol/l

16

O2 diffusion coefficient (DO2 ) 2.2 × 10−5 cm2/s 19 and 20External oxygen concentration (cO2,ext in airat 1 atm)

9.46 × 10−6 cm−3

βe 1Solubility factor (s) 0.344982Electrolyte mass density at saturation (ρs ) 1.114 g/cm3 21Li diffusion coefficient (DLi ) 3.018 × 10−5 exp (0.357cLi ) 17Transference number (t+) 0.2 18Reaction rate coefficient (k) 1.5 × 10−8 cm−3s−1 4Molecular weight of LiOH · H2O (ρsat ) 41.96 g/mol 15Mass density of LiOH · H2O (ρs ) 1.51 g/cm3 15 and 22Mass density of carbon (ρC) 2.26 g/cm3 15Cathode thickness (L − LC ) 0.75 mmInitial pore radius (r̄ p,0) 20 nmMass density of the electrolyte at saturation(ρsat )

1.114 g/cm3 23

Initial electrolyte concentration (cLi,0) 10−6 mol/cm3

battery enters a quasi-stationary regime in which O2 is provided by theoutside atmosphere. The time τ required to deplete the initial O2 fromthe cathode can be estimated by neglecting the diffusion current in 9,which gives

ε0cO2 ,0−0

τ= − RC

F = − I4F(L−LC ) , where I is the discharge

current and L − LC is the length of the cathode (see Fig. 1). Hence

τ = 4Fε0cO2,0 (L − LC )

I≈ 1.26

ε0 (L − LC )

I[27]

In the above equation τ is expressed in seconds, L and LC are ex-pressed in cm, and I in A/cm2. The specific capacity SCτ (measuredin mAh/gC) required to deplete the initial O2 from the cathode is

SCτ = FcO2,0ε0

0.9ρC (1 − ε0)≈ 0.155

ε0

1 − ε0[28]

Besides the proportionality constants, equations 27 and 28 arecomparable to the ones for the oxygen depletion time and specificcapacity of Li-air batteries with organic electrolyte.6

Boundary conditions.— Boundary conditions are imposed for theLi+ and oxygen concentrations and potentials, and for the electronpotential at the cathode. The oxygen concentration at the right side ofthe cathode (i.e. at x = L) is assumed constant and can be computedfrom the oxygen solubility s and the external oxygen concentrationcO2,ext (see Table I)

cO2 |x=L = cO2,0 = scO2,ext [29]



The boundary condition at the anode is assumed to be

I cO2|x=0 = 0, [30]

where I cO2= −DO2,e f f ∇cO2 . The boundary condition for the lithium

concentration at the cathode is

I cLi |x=L = 0 [31]

where I cLi = −DLi, e f f ∇cLi , while at the anode the boundary condi-tion is given by the total electrolyte conservation

A∫ L

0cLi (x, t) dx = NLi (t) [32]

where A is the cross-sectional area of the battery and NLi (t) is thetotal number of moles of Li+ that exists in the electrolyte at time t. Innumerical simulations NLi (t) is calculated by summing the numberof moles of Li+ that are being released from the anode and subtractingthe number of moles that are being deposited in the cathode at eachtime step.

The boundary condition for the Li+ transport equation 6 at theright side of or the cathode is

i |x=L = 0 [33]

while at the anode

i |x=0 = RA = I0

(e

(1−β′)FRT ηa − e− β′ F

RT ηa

)[34]

where β′ = 0.5, I0 is a reaction rate constant, and ηa is the overpoten-tial at the anode reaction, which is equal to

ηc = φ (0) − φLi − Ua [35]

where Ua is the equilibrium potential for the lithium reduction andφ (0) is the potential of the Li metal. The values of s, cO2,ext , cLi,0, andI0 are also reported in Table I.

The electrostatic potential of the electrons at the right side ofthe cathode is equal to the applied external potential on the cath-ode electrode. The boundary condition for the electron conductionequation 14 at the cathode-separator interface is

I e|x=LC= 0 [36]

where I e = −σe f f ∇φ. The electron current at the right side of thecathode is equal to the value of the applied current.

Continuous boundary conditions are assumed for all density cur-rent variables at the anode/separator and cathode/separator interfaces(i.e. i |x=L+

C= i |x=L−

C, I cO2

|x=L+A

= I cO2|x=L−

A, etc.). In addition, φLi is

discontinuous at the separator/anode and separator cathode interfaces

φLi |L+A

− φLi |L−A

= RS A I [37]

φLi |L+C

− φLi |L−C

= RSC I [38]

where RS A and RSC are the separator/anode and separator/cathodeinterface resistances measured in �cm2 and I is the discharge currentdensity measured in A/cm2.

The voltage of the cell is calculated as the difference of the electronpotential at the cathode and the potential of the lithium metal at theanode, Vcell = φ (x = L) − φ (0).

Maximum specific capacity.— The maximum specific capacity ofLi-air batteries with dual electrolyte is given by 2 contributions: (1) thespecific capacity before saturation SC1 and (2) the specific capacitygiven by the deposition of LiOH · H2O at the cathode SC2.

(1) The specific capacity before saturation is achieved when thebattery is discharged from the initial state (no electrolyte, i.e.cLi ≈ 0) till cLi = cLi,sat throughout the cathode. If T1 is thedischarge time till saturation, one can write ε0cLi,sat

T1≈ 4RC

F ,

which gives

T1 ≈ FcLi,satε0

4RC= FcLi,satε0 (L − LC )

I= 499

ε0 (L − LC )

I[39]

where I is expressed in A/cm2 and L − LC is the length of thecathode expressed in cm. The corresponding specific capacitySC1 expressed in mAh/gC is given by

SC1 = T1 I

3.6 (1 − ε0) (L − LC ) ρC≈ 61.3ε0

1 − ε0[40]

(2) The specific capacity given by the deposition of LiOH · H2Oat the cathode can be calculated by estimating the total life-time of the battery (T2), which can be computed from 22,�ε

T2≈ − 4RC Ms

Fρs(1 − cLi,sat Ms

ρs)−1, which gives

T2 ≈ ε0 Fρs

4RC Ms

(1 − cLi,sat Ms

ρs

)= 2974

ε0 (L − LC )

I[41]

The corresponding specific capacity is given by

SC2 = T2 I

3.6 (1 − ε0) (L − LC ) ρC≈ 365ε0

1 − ε0[42]

In the last equations T1 and T2 are expressed in seconds.(3) The maximum (total) specific capacity of Li-air batteries with

aqueous electrolyte is

SCmax = SC1 + SC2 = 427ε0

1 − ε0[43]

Eq. 43 shows that the maximum specific capacity of a Li-airbattery with aqueous electrolyte and ε0 = 75% is approximately1,280 mAh/gC.

Numerical implementation.— The model equations presentedabove have been discretized by using finite differences for one-dimensional cells. The nonlinear discretized equations have been im-plemented in RandFlux13 and solved by using the Newton iterativetechnique. The specific capacity of the battery is computed by solvingthe transport equations till the porosity of at least one mesh point be-comes zero. The energy density is computed by integrating I (t)V (t)numerically from 0 to the lifetime of the battery. The power densityis computed by dividing the energy density to the total lifetime of thebattery. The computational time required to compute the power andenergy densities is of the order of a couple of minutes on a standardone-processor computer running at 3 MHz.

Parameter selection.— The reliability of the simulation resultsstrongly depends on the accuracy of measurements of the model-dependent parameters. Hence, we have paid particular attention tocarefully determine the model parameters by performing new experi-ments or using data already published in the literature.

The ionic resistance of the LIC-GC and the interface resistanceswere measured using the electrochemical impedance spectral (EIS)method. The measured LIC-GC is a 50 μm thick membrane with asize of 2.54×2.54 cm2 from Ohara Inc., which was placed and sealedbetween two electrolyte cells. Each electrolyte cell has an open win-dow of 1×1 cm2. A Pt electrode was placed in each electrolyte cell.The detailed configuration of measurement cell is shown in Fig. 4. Twodifferent electrolytes were used with concentrations of 1 M LiOH inH2O and 1 M LiPF6 in propylene carbonate (PC), respectively. TheEIS was measured between two Pt electrodes in a frequency rangefrom 1 Hz to 100 kHz using a Solartron 1250B frequency responseanalyzer controlled by Zplot and Corrware software. Fig. 5 showsthe EIS measured with different electrolytes in the two electrolytecells. The bulk resistance of the LIC-GC membrane and interfacialresistance between the membrane and electrolyte were obtained byfitting EIS using the equivalent electric circuit shown in the inset of



Figure 4. Cell configuration used to measure the bulk resistance of the LIC-GC and the resistances at the two interfaces.

Fig. 2. Resistance Rb represents the total bulk resistance of the LIC-GC membrane and electrolytes. The two parallel combinations ofresistors and capacitors represent the interfacial impedances at thetwo interfaces. The resistance of the electrolyte was measured by re-moving the LIC-GC membrane when the same electrolyte was usedin both electrolyte cells and subtracted from the EIS as shown inFig. 5. The resistance distribution can be summarized as follows: thebulk resistance of LIC-GC membrane was Rb∼50 �-cm2 which isequivalent to a conductivity of 1×10−4 S/cm; the interfacial resis-tances were Rint1∼70 �-cm2 with the aqueous electrolyte (1 M LiOHsolution) and Rint2∼80 �-cm2 with the non-aqueous electrolyte (1MLiPF6 in PC), respectively; therefore, the total cell resistance at lowfrequencies was about 200 �-cm2.

The electric conductivity of the LiOH is represented in Fig. 6 bysymbols. The measured data was interpolated using

κ (cLi ) = −0.2912c3Li − 48.14c1.5

Li − 188.6cLi [44]

where the cLi is measured in mol/l and κ in mS/cm.A summary of the geometrical and model parameters used in simu-

lations as well as the reference from which they are taken are reportedin Table I. The initial porosity of the cathode wes set to 75%. Since

0 1 2 3 4 50

100

200

300

400

Fitting function:

κ(cLi) = A1c3Li + A2c

1.5Li + A3cLi

A1

-0.29127 ±0.04436

A2

-48.14173 ±1.19974

A3

188.64745 ±1.65272

Con

duct

ivit

y, κ

(m

S/c

m)

Li concentration, cLi (mol/l)

Figure 5. EIS measured using (a) 1 M LiOH solution in water in both elec-trolyte cells, (b) 1 M LiPF6 solution in PC in both electrolyte cells, and (c) 1M LiOH solution and 1M LiPF6 in PC in each electrolyte cell.

Figure 6. Lithium conductivity as a function of the lithium concentration inLiOH. The symbols represent the experimental data and the continuous linethe fitting curve.

the thickness of the APL and separator are relatively small, the exactvalues of the diffusion coefficients and transference numbers in theseregions do not affect much the results of the simulations. When areference was not found for these coefficients their values were setsimilar to the values in the cathode. The reaction rate constant k wasdetermined in order to obtain the best agreement between our simula-tions and the discharge characteristics published by Wang and Zhou.4

Due to the lack of available data for the reaction rate coefficient in theliterature, we have considered the same value for k before and afterthe Li+OH− saturation.

Simulation Results

As mentioned in the previous section, during the discharge of theLi-air batteries with dual electrolyte the lithium concentration in thecathode increases from the initial value till saturation, after which itstays constant. This fact is illustrated in the simulations presented inFig. 7, where the maximum concentration of Li+ ions in the cathodeis represented as a function of the specific capacity at a very low dis-charge current. Since the discharge current is constant the maximumconcentration of Li+ ions increases linearly till cLi = cLi,sat . After thispoint the maximum concentration of Li+ ions remains constant till thewhole cathode volume is filled in with LiOH · H2O(deposit). The spe-cific capacity of the battery before saturation SC1 and the total specific

0 200 1000 12000.000

0.001

0.002

0.003

0.004

0.005

0.006

SCmax

Max

imum

Li+

con

cent

rati

on a

t cat

hode

(m

ol/c

m3 )

Specific capacity (mAh/gC)

cLi,sat

SC1

Figure 7. Maximum concentration of Li+ ions as a function of the specificcapacity during a full discharge of the battery. SC1 and SCmax are in agreementwith the theoretical predictions (40) and (43), respectively.



0 200 400 600 800 1000 12000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Min

imum

por

osit

y at

cat

hode

(m

ol/c

m3 )


0.05 mA 0.1 mA 0.2 mA 0.5 mA 1 mA 2 mA

SC1

Figure 8. Minimum porosity at the cathode as a function of the specificcapacity.

capacity SCmax are in full agreement with the theoretical predictions40 and 43, respectively. Almost one seventh of the total capacity ofthe battery is stored in the aqueous electrolyte, which opens newopportunities for designing “flow” rechargeable batteries, where theelectrolyte concentration is less than between cLi,sat .9

To gain more insight into the operation of the batteries we haverepresented the minimum porosity in the cathode as a function ofthe specific capacity for different discharge currents in Fig. 8. Theminimum value of the porosity is usually found at the air side ofthe cathode, where the oxygen concentration has the maximum value.Before Li+ saturation the porosity of the cathode is constant and equalto its initial value 75%. After Li+ saturation the specific capacitydepends on the value of the discharge current: for very low dischargecurrents the specific capacity is approximately equal to the maximumtheoretical value SCmax; for high discharge currents (> 1 mA/cm2) thespecific capacity decreases to SC1.

Fig. 9 presents the cell voltage as a function of the specific ca-pacity for different discharge currents ranging from 0.05 mA/cm2 to1 mA/cm2. The inset shows a detail of these curves at the beginning ofthe discharge process, when the quasi-stationary regime is obtained[see Eq. 27]. During this time, the concentration of the oxygen inthe cathode is relatively high, which explains the slightly high valuesof the cell voltage. The maximum in the inset of Fig. 9 is due tothe fact that the initial concentration of the electrolyte is very small(cLi,0 = 10−6 cm−3), which results in a low electrical conductivity,

0 200 400 600 800 1000 1200

1.6

2.0

2.4

2.8

3.2

0.0 0.2 0.4 0.6 0.8

2.6

2.8

3.0

3.2

3.4

Cel

l vo

ltag

e (V

)

Specific capacity (mAh/g )

Cel

l vol

tage

(V

)


0.05 mA 0.1 mA 0.2 mA 0.5 mA 1 mA 2 mA

Figure 9. Cell output voltage as a function of the specific capacity for differentdischarge currents.

0 300 600 900 1200 15002.4

2.6

2.8

3.0

3.2

3.4

0.0 0.2 0.4 0.6 0.83.0

3.1

3.2

3.3

3.4

Cel

l vol

tage

(V

)

Specific capacity (mAh/g )

(h)

(a)

Cel

l vol

tage

(V

)


Discharge current = 0.1 mA/cm2

(h)(g)(f)(e)(d)

(c)(b)(a)

Figure 10. Cell output voltage as a function of the specific capacity for dif-ferent cathode thicknesses: (a) 33 μm, (b) 65 μm, (c) 0.1 mm, (d) 0.2 mm,(e) 0.4 mm, (f) 0.75 mm, (g) 1 mm, and (h) 3.3 mm.

hence a slightly lower voltage; in our simulations, this maximum dis-appears when the initial concentration of the electrolyte is larger than10−5 cm−3. In practical applications, it might be difficult to observeit because quite often the electrolyte has a small concentration due tovarious salt impurities. The maximum specific capacity of the batterydecreases significantly when the discharge current is increased.

Fig. 10 presents the cell voltage as a function of the specific ca-pacity for different cathode thicknesses. In all the simulations thethickness of the separator and anode separation layer (the organicelectrolyte) were kept constant and equal to 50 μm and 50 nm, re-spectively. For cathode thicknesses much lower than the diffusionlength of the oxygen in the cathode, the specific capacity of the bat-tery tends toward the maximum theoretical value SCmax; however,when the thickness of the cathode increases, the specific capacity ofthe battery decreases. As a design rule, the cathode thickness shouldbe approximately equal to the diffusion length of the oxygen, whichcan be computed by using the same line of reasoning as in6

λ = 4Fε1.5c0

O2DO2

I[45]

Here, factor 4 appears because each molecule of O2 creates 4 electrons.Figs. 11, 12, and 13 are contour plots of the porosity, the oxygen

concentration, and the reaction rate inside the cell as a function ofposition and of the state-of-discharge of the battery. The dischargecurrent in all simulations is 0.1 mA/cm2. The saturation of the lithium

Figure 11. Contour plot of the porosity as a function of the position(x-coordinate) and state-of-discharge of the battery.



Figure 12. Contour plot of the oxygen concentration as a function of theposition (x-coordinate) and state-of-discharge of the battery.

concentration is achieved when the state-of-discharge becomes equalto SC1

SCmax≈ 14%. Initially, the porosity decreases more or less uni-

formly throughout the cathode, but in the last 30% of the dischargecycle the decrease rate accelerates toward the air side of the cathode.Hence, in order to increase the life time of Li-air batteries, it is de-sirable if possible to not fully discharge these batteries but rechargethem as soon as their state-of-discharge gets close to 70%. Otherwise,defects and nonuniformities induced by the reaction product mightdamage the battery irreversibly. The oxygen concentration is almost50% smaller in the left side of the cathode than in the right side duringthe initial 50% discharge period of the battery. However, the morewe discharge the battery, the narrower the oxygen transport channelsbecome, the oxygen cannot penetrate easily inside the cathode, andthe oxygen concentration decreases at the separator side. The reactionrate is almost uniform during the initial discharge period of the battery,however, as the discharge progresses, it increases relatively fast in theregion close to the air side and decreases in the region close to theseparator.

It is apparent from the above discussion that the energy and powerdensities of the Li-air batteries are limited by the relatively low oxygendiffusion rate inside the cathode. Hence, in order to improve the energyand power densities of these batteries we should increase the oxygensolubility14 and diffusion coefficients in the cathode. Figs. 14 and 15

Figure 13. Contour plot of the reaction rate as a function of the position(x-coordinate) and state-of-discharge of the battery.

0.1 0.2 0.3 0.4 0.5 0.60

1000

2000

3000

7.4

7.5

7.6

7.7

7.8

600

800

1000

1200

I = 0.1 mA/cm2Ene

rgy

dens

ity

(Wh/

kg)

Solubility factor

Specific capacity

Energy density Power density

Pow

er d

ensi

ty (

W/K

g)

Spe

cifi

c ca

paci

t y (

mA

h/g)

Figure 14. Energy density, power density, and specific capacity as a functionof the solubility factor of O2.

present the energy density, power density, and the specific capacityof the cell as a function of the oxygen solubility factor and oxygendiffusion coefficient respectively. The energy density, power density,and the specific capacity increase with the values of the solubilityfactor and oxygen diffusion coefficient. However, while the increase ofthe power density with respect to the solubility factor is approximatelylinear, the increase with respect to the oxygen diffusion coefficientseems to saturate when DO2 � 10−5 cm2/s.

It has been suggested before that one could use a catalyst in order toincrease the energy and power densities in the case of Li-air batterieswith organic electrolyte.6 Hence, Fig. 16 presents the energy density,power density, and the specific capacity of our cell as a function ofthe reaction rate coefficient. In these simulations the reaction rateis assumed uniformly constant thought the cathode. We observe thatalthough the energy and power densities are predicted to increase by atmost 10% due to the introduction of the catalyst, the specific capacityof the battery does not increase. The energy density is maximumwhen the reaction rate is approximately equal to 4×107 A/cm2 butthis maximum depends on the characteristics of the battery such asgeometry, discharge current, and other model specific parameters. Thedecrease of the energy density and specific capacity at high valuesof the reaction rate coefficient is due to the fact that the reaction rateincreases significantly in the right side of the cathode and the depositedLiOH · H2O blocks faster the flow of the oxygen.

Due to the potential of the Li-air batteries with dual electrolyte tobe used as a flow battery, we present in Fig. 17 the output character-istics of the cell working only in the “before saturation” regime for alarger variety of the discharge current. It takes approximately 50 min

10-8

10-7

10-6

10-5

10-4

10-3

0

1000

2000

3000

4000

5000

7.0

7.2

7.4

7.6

7.8

0

300

600

900

1200

1500

Ene

rgy

dens

ity

(Wh/

kg)

Oxygen diffusion coefficient (cm2/s)

Pow

er d

ensi

ty (

W/K

g)

I = 0.1 mA/cm2

Spe

cifi

c ca

paci

ty (

mA

h/g)Power density

Energy density

Specificcapacity

Figure 15. Energy density, power density, and specific capacity as a functionof the diffusion coefficient of O2 in the cathode.



10-9

10-8

10-7

10-6

10-5

3600

3800

4000

4200

4400

7.4

7.6

7.8

8.0

1100

1150

1200

1250

1300

1350

Ene

rgy

dens

ity

(Wh/

kg)

Reaction rate coefficient, k (A/cm2)

Pow

er d

ensi

ty (

W/K

g)

I = 0.1 mA/cm2

Spe

cifi

c ca

paci

ty (

mA

h/g)

Power density

Energy density

Specific capacity

Figure 16. Energy density, power density, and specific capacity as a functionof the reaction rate coefficient in the cathode.

for the battery to saturate the electrolyte in the cathode at a dischargecurrent of 10 mA/cm2. As also observed experimentally,4 the outputcharacteristics are significantly affected by the high separator/anodeand separator/cathode interface resistance. The maximum of the powerdensity is obtain for a discharge current of approximately 7 mA/cm2,which is slightly lower than the value obtained by Wang et al.,4

11 mA/cm2 because the total interface resistance in our battery is150 � compared to approximately 100 � in Ref. 4. The interfaceresistance is known to vary significantly with the type of the solventand electrolyte concentration. The initial variation of the I-V curvesat high discharge currents is due again to the fact that the initial con-centration of the electrolyte in simulations is very small (cLi,0 = 10−6

cm−3). In normal applications this effect is usually not observed be-cause the electrolyte might already contain a significant initial amountof lithium ions.

Finally, Figures 18, 19, and 20 present the dependence of the energyand power densities of the battery on the solubility factor, oxygendiffusion coefficient, and reaction rate coefficient, when the dischargecurrent is equal to 7 mA/cm2. To avoid the LiOH · H2O deposition inthe cathode, the battery was discharged only up to the point where thelithium concentration in the electrolyte became equal to the saturationvalue cLi,sat . Hence, the specific capacity in the simulations is constantand equal to SC1 and the power density was obtained from the energydensity divided by T1 defined in Eq. 39). In agreement with the resultsshown in Figures 14–16 we observe again that the power and energydensities of Li-air batteries can be slightly increased by increasingthe solubility factor, the diffusion coefficient of the oxygen, and the

0 30 60 90 120 150 1800

1

2

3

4

0.1 mA/cm2

10-5 mA/cm

2

1 mA/cm2

5 mA/cm2

8 mA/cm2

Cel

l vol

tage

(V

)

Time (s)

Figure 17. Output voltage as a function of time for different discharge cur-rents. state-of-discharge of the battery.

0.1 0.2 0.3 0.4 0.5 0.6250

260

270

280

290

220

230

240

I = 7 mA/cm2Ene

rgy

dens

ity

(Wh/

kg)

Solubility factor

Pow

er d

ensi

ty (

W/K

g)

Figure 18. Energy and power densities as a function of the solubility factorof O2.

10-8

10-7

10-6

10-5

10-4

10-3

250

260

270

280

290

300

310

320

220

230

240

250

260

270

I = 7 mA/cm2

Ene

rgy

dens

ity

(Wh/

kg)

Oxygen diffusion coefficient (cm2/s)

Pow

er d

ensi

ty (

W/K

g)

Figure 19. Energy and power densities as a function of the diffusion coeffi-cient of O2 in the cathode.

10-9

10-8

10-7

10-6

10-5

10-4

280

300

320

340

360

380

240

260

280

300

320

Ene

rgy

dens

ity

(Wh/

kg)

Reaction rate coefficient, k (A/cm2)

I = 7 mA/cm2

Pow

er d

ensi

ty (

W/K

g)

Figure 20. Energy and power densities as a function of the reaction ratecoefficient.



reaction rate constant. In addition, the power and energy densitiessaturate when the reaction rate coefficient is increased, however, itdoes not decrease for high values of this coefficient like in Fig. 16.

Conclusions

The electric properties of Li-air batteries with dual electrolytewere analyzed by using a model based on the mass transport anddrift-diffusion of the electrolyte material. The model uses the theoryof concentrated solutions and takes into consideration the diffusionof the O2, the electric conductivity of electrons in the cathode, thediffusion and conductivity of Li ions in the cell, the reaction rates atthe anode and cathodes, and the saturation and deposition of lithiumhydroxide at the cathode. The major limiting factors for the power den-sity of these batteries are the low values of the oxygen diffusion lengthin the cathode and the high separator/anode and separator/cathode in-terface resistances. The energy density and specific capacity could beincreased by increasing the solubility of the oxygen in the aqueouselectrolyte and the oxygen diffusion coefficient in the cathode.

Acknowledgments

This work was supported by the National Science Foundationunder Engineering Research Center Program no. EEC-0812121 andUS Army CERDEC.

Appendix

The electrolyte concentration cLi in Li-air batteries with dual electrolyte can reachvalues at saturation as high as 15% of the total solvent concentration. Hence, thesebatteries could be modeled using the theory of dilute solutions at low Li+ concentrations,but they should be treated using the theory of concentrated solutions at higher Li+

concentrations. For this reasons we give below a derivation of the electrolyte transportequations 6 and 7 by using first the theory of dilute solutions and, then, the theory ofconcentrated solutions. The functional forms of the resulting transport equations in thetwo derivations are similar, however, the driving forces and the exact expression of thediffusion coefficients are different. The transport equations obtained using the theory ofdilute solutions can be derived as a particular case of the transport equations obtained usingthe theory of concentrated solutions, when cLi is much less than the total concentrationof the solvent.

Throughout the derivations we assume that the concentration of the O2 in the elec-trolyte (cO2 ) is much smaller than the total concentration of the electrolyte (cT ), whichimplies that

cT = c0 + cLi + cO H [46]

where c0 is the concentration of the solvent (water), cO H is the concentration of theanions in the electrolyte; for a binary electrolyte: cO H = cLi . Eq. 46 provides a goodapproximation for Li-air batteries because the O2 usually comes from the air at a muchsmaller concentration than any of the three terms in the right-hand-side of 46. To simplifynotations we will also assume ε = 1 throughout this section.

The mass transport (continuity) equations for the Li+ and OH− ions are

∂cLi

∂t= −∇ · N Li + RLi

F[47]

∂cO H

∂t= −∇ · N O H + RO H

F[48]

where

N Li = cLi vLi [49]

N O H = cO H vO H [50]

are the fluxes of the Li+ and OH− ions respectively and RLi and RO H are the generationrates for these ions. Note that, depending on whether we are before or after Li+ saturationonly one of the two transport equations will actually have a nonzero source term, asshown in Table II. In Li-air batteries with dual electrolyte it is possible to increase thetotal concentration of the electrolyte by creating additional Li+ and OH− ions. This isunlike the case of Li-air batteries with organic electrolyte, where the total number ofanions and cations remains constant during the discharge of the battery. This fact requiresparticular attention because it leads to slightly different transport equations than in thecase of Li-air batteries with organic electrolyte.

The total current density of the electrolyte is

i = F (N Li − N O H ) [51]

Table II. Cathode reaction rates.

Before saturation After saturation(cLi < cLi,sat ) (cLi ≥ cLi,sat )

RO H 4RC > 0 0RLi 0 −4RC < 0

Depending on whether we use the theory of dilute solutions or the theory of concentratedsolutions the Li+ and OH− fluxes have slightly different forms. An useful equation for theconservation of the total current density in Li-air batteries can be derived by subtracting47 and 48 and using 51

∇ · i = RLi − RO H [52]

It will be shown below that Eqs. 6 and 7 are equivalent to the mass transportequations 47 and 48.

Transport equations in dilute solutions

In the framework of the theory of dilute solutions, the Li+ and OH− fluxes are givenas the superposition of the drift and diffusion fluxes

N Li = −uLi FcLi ∇φLi − DLi, e f f ∇cLi [53]

N O H = uO H FcO H ∇φO H − DO H, e f f ∇cO H [54]

where uLi and uO H denote the mobilities, and DLi, e f f and DO H, e f f are the effectivediffusion coefficients of Li+ and OH− ions, respectively. The electrostatic field seen bythe Li+ ions is identical to the electrostatic field seen by the OH− ions since both types ofions move in the same medium and are in the same phase, φLi = φO H . The total currentdensity of the electrolyte becomes

i = −F[(uLi + uO H )FcLi ∇φLi + (DLi, e f f − DO H, e f f )∇cLi ] [55]

and 52 gives

∇ · [(uLi + uO H )FcLi ∇φLi + (DLi, e f f − DOG H, e f f )∇cLi ] = RO H − RLi [56]

The last equation is identical to 6, in which if κe f f = (uLi + uO H )FcLi andκD = (DLi, e f f − DOG H, e f f )cLi . The relationship between the effective electric con-ductivity and diffusion conductivity of the electrolyte could be in principle established byusing the Nernst-Einstein equation, however, this will lead to slightly different expressionsthan in 70, which considers a theory of concentrated solutions.

Equation 7 can be derived by replacing 53 in 47, 54 in 48, adding the resultingequations, and performing a few rearrangements. Indeed, multiplying 47 by uO H and 48by uLi and adding the two equations we obtain

(uLi + uO H )∂cLi

∂t= uO H ∇ · (uLi FcLi ∇φLi + DLi, e f f ∇cLi )

− uLi ∇ · (uO H FcLi ∇φLi − DO H, e f f ∇cLi )

+ uO H RLi + uLi RO H

F[57]

If we divide the last equation by uLi + uO H and use the chain rule, we obtain after afew rearrangements

∂cLi

∂t= ∇ ·

(uO H DLi, e f f − uLi DO H, e f f

uLi + uO H∇cLi

)+ t− RLi + t+ RO H

F

+ ∇t− · (−uO H FcLi ∇φLi − DO H, e f f ∇cLi − uO H FcLi ∇φLi + DO H, e f f ∇cLi )

[58]

where t+ = uLiuLi +uO H

and t− = uO HuLi +uO H

are the cation and anion transference numbers

respectively. Making use of 55 Eq. 58 becomes

∂cLi

∂t= ∇ ·

(uO H DLi, e f f − uLi DO H, e f f

uLi + uO H∇cLi

)+ t− RLi + t+ RO H

F− ∇t+ · i

F

[59]

which can be identified with 7 if Def f = uO H DLi, e f f −uLi DO H, e f fuLi +uO H

.

Transport equations in concentrated solutions

Eqs. 53 and 54 are not accurate when the concentration of the electrolyte is highbecause they does not consider the interaction or friction force between the solutesthemselves. In addition, as discussed in Ref. 7, the driving force for the diffusion fluxshould be an activity gradient, and activity gradients are equal to concentration gradientsonly in dilute solutions.



In the case of binary electrolytes composed of anions, cations and solvent, the multi-component diffusion equations give

cLi ∇μLi = −KLi vLi + KLi,O H (vO H − vLi ) [60]

cO H ∇μO H = −KO H vO H + KO H,Li (vLi − vO H ) [61]

where μLi and μO H are the electrochemical potentials of Li+ and OH−; KLi , KO H ,KLi,O H , and KO H,Li (KLi,O H = KO H,Li ) are the friction (or interaction) coefficientsbetween Li+ and solvent (water), OH− and solvent, Li+ and OH−, and OH− and Li+,respectively; vLi and vO H are the average velocities of the Li+ and OH− ions. Sincethe solvent in Li-air batteries is usually stationary, the velocity of the solvent has beenneglected in 60 and 61. These equations need to be revised when modeling Li-air flowbatteries, because in these batteries the electrolyte is constantly circulating between thecathode and the reservoir.8, 9

By following the analysis presented by Newman7 it is convenient now to introducethe chemical potential of the electrolyte as

μe = μLi + μO H [62]

and use the fact that cLi = cO H . The driving force is the gradient of the chemical potential,which can be expressed as

∇μe = ∇μLi + ∇μO H = −KLi vLi + KLi,O H (vO H − vLi )

cLi

+ −KO H vO H + KO H,Li (vLi − vO H )

cO H

= −KLi vLi − KO H vO H

cLi[63]

Since

KLi = RT cLi c0

cT DLi[64]

KO H = RT cO H c0

cT DO H[65]

where DLi and DO H are the interaction diffusion coefficients between Li+ and solvent,and between OH− and solvent, respectively, equation 63 becomes

cT cLi

RT c0∇μe = cLi vLi

DLi+ cO H vO H

DLi= N Li

DLi+ N O H

DLi[66]

Equations 51 and 66 can be solved for N Li and N O H

N Li = −DcT cLi

2RT c0∇μe + i t+

F[67]

N O H = −DcT cLi

2RT c0∇μe − i t−

F[68]

where t+ = DLiDLi +DO H

and t− = DO HDLi +DO H

are the transference numbers of Li+ and OH−

with respect to the solvent and D = 2DLi DO HDLi +DO H

is the effective diffusion coefficient of theelectrolyte. The transference numbers that appear in 67 and 68 should not be confusedwith the transference numbers that appeared in the framework of the diluted solutions inEq. 58. Finally, notice that equations 67 and 68 are equivalent to 53 and 54 but, in thiscase, the driving force is the gradient of the chemical potential. As noted by Newman7 the

diffusion coefficient D that is usually measured experimentally is based on the gradient ofthe electrolyte concentration and not on the gradient of the chemical potential μe . Hence,

it is more convenient to use D cT cLi2RT c0

∇μe = D(

1 − d ln c0d ln cLi

)∇cLi , which, together with

47 and 67 gives

∂cLi

∂t= ∇ ·

[D

(1 − d ln c0

d ln cLi

)∇cLi

]− ∇ ·

(i t+F

)+ RLi

F=

= ∇ ·[

D

(1 − d ln c0

d ln cLi

)∇cLi

]− i · ∇t+

F+ RLi t−

F+ RO H t+

F[69]

The last equation is identical to the electrolyte balance equation 7 with Def f

= D(1 − d ln c0d ln cLi

). Finally, Eq. 6 results from 52, in which7

i = −κe f f ∇φLi + 2RT

F

(1 + d ln f

d ln c

)(1 − t+) ∇ ln cLi [70]

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