Comparison between computer simulations and … between computer simulations and experimental data for ... The motivation is to model lithium-ion polymer ... Parameter Li C Li Mn Ox

Ž .Journal of Power Sources 88 2000 219–231www.elsevier.comrlocaterjpowsour

Comparison between computer simulations and experimental data forhigh-rate discharges of plastic lithium-ion batteries

Pankaj Arora a,1, Marc Doyle b, Antoni S. Gozdz c, Ralph E. White a,), John Newman d

a Center for Electrochemical Engineering, Department of Chemical Engineering, UniÕersity of South Carolina, Columbia, SC 29208, USAb DuPont Central Research and DeÕelopment, Experimental Station, Wilmington, DE 19880-0262, USA

c ( )Telcordia Technologies formerly Bellcore , Red Bank, NJ 07701, USAd Department of Chemical Engineering, UniÕersity of California, Berkeley, CA 94720, USA

Received 9 August 1999; accepted 6 December 1999

Abstract

Computer simulations are compared with experimental data for Bellcore PLION w cells using the graphiter1 M LiPF in EC:DMC6Ž .2:1 rLiMn O system. The motivation is to model lithium-ion polymer cells having higher active material loadings and competitive2 4

energy densities and specific energies to liquid lithium-ion batteries. Cells with different electrode thickness, initial salt concentrations,and higher active material loadings were examined using the mathematical model to understand better the transport processes in theplasticized polymer electrolyte system. A better description of the ionic conductivity is employed based on new conductivity data.

Improvements in the agreement between the simulations and experimental data are obtained by using the contact resistance at the2 Žcurrent collectorrelectrode interface as an adjustable parameter for different cells, whose values vary from 20 to 35 V cm based on

.separator area . The contact resistance is believed to originate at the mesh current collector interfaces. Reducing the salt diffusioncoefficient by a factor of two or more at the higher discharge rates was necessary to obtain better agreement with the experimental data.Based on the experimental data and model predictions from this study, it can be concluded that the solution-phase diffusion limitations arethe major limiting factor during high-rate discharges. q 2000 Elsevier Science S.A. All rights reserved.

Keywords: Diffusion limitations; Contact resistance; Variable diffusion coefficient; Mathematical model; Plastic lithium-ion batteries; Gelled polymerelectrolytes

1. Introduction

Since their introduction in 1990 by Sony, the produc-tion of lithium-ion rechargeable batteries worldwide hasincreased steadily, and new applications in consumer elec-tronic devices have developed. A steady stream of ad-vances in electrode and electrolyte materials and manufac-turing technology has led to improved cell performance,especially energy density, cycle life, and low- and high-temperature performance. To bring lithium-ion batteriesinto high-power applications, improvements in the funda-mental understanding of the rate-limiting phenomenaduring high-rate charge and discharge are necessary. Math-ematical modeling and computer simulations represent

) Corresponding author. Fax: q1-803-777-8265.Ž .E-mail address: [email protected] R.E. White .

1 Present address: Celgard Inc., 13800 South Lakes Drive, Charlotte,NC 28210, USA.

powerful tools for gaining this fundamental understandingin a rapid and cost-effective manner.

Computer simulations have been applied to the Sonyw x wlithium-ion cell 1,2 and Bellcore PLION lithium-ion

w xcell 1,3 . The mathematical description is based on amacroscopic model of the full cell sandwich, which, forthe Bellcore system, may consist of a lithium manganese

Ž .oxide spinel positive electrode, a plasticized polymerelectrolyte, such as 1 M LiPF in a 2:1 mixture of6

ECrDMC in a copolymer matrix of polyvinylidene fluo-Ž Ž ..ride–hexafluoropropylene p VdF–HFP , and a graphite

Ž . w xMCMB 2528 negative electrode 3,4 . The equationsused in the mathematical model are discussed in detail

w xelsewhere 1,2 . Direct comparisons between simulationsand experimental data showed that a satisfactory descrip-tion of the system was possible using macroscopic batterymodeling, and several conclusions regarding battery per-formance and rate limitations were made. Optimization ofthe battery structure including electrode porosities and

0378-7753r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved.Ž .PII: S0378-7753 99 00527-3

( )P. Arora et al.rJournal of Power Sources 88 2000 219–231220

thickness for various applications was pursued using com-puter-generated Ragone plots.

Adequate fitting of all the experimental data availablefor the Bellcore PLION w battery using the same set ofparameters was possible only when additional resistancewas added to the system in the form of film resistances on

w xthe electrode particles or current collectors 3 . High-ratedischarges were limited primarily by transport in the solu-tion phase of the porous electrode. Diffusion limitations inthe solid insertion particles existed during discharge condi-tions but were minor compared to other limitations. Fur-ther work on the Bellcore system utilizing experimentaland simulated impedance data implicated contact resis-tance on the current collector interfaces as the source of

w xthe additional high-frequency impedance in this system 5 .Improvements in cell energy density and rate capability

have been made on the Bellcore PLION w system byoptimizing the electrode compositions and thickness, whiledecreasing the volume fraction of the inactive componentsin the electrodes. The development of more effective sur-face treatments on the metal mesh current collectors hashelped reduce the contact resistance while retaining highcycle life. These improvements lead to better performancebut create additional challenges for battery modeling.

The description of the solution-phase transport pro-cesses in the gelled polymer electrolyte in terms of abinary electrolyte is increasingly hard to justify. It is

Žprobable that salt transport in the separator region com-.posed of polymer, inorganic fillers, solvents, and salt is

different from salt transport in the porous electrodes due tothe different constitutions of the various regions of the cell.In particular, the volume fractions of the polymer compo-nent in the electrodes when optimized are sufficient to

enable lamination of the cell layers in flexible prismaticbattery designs, but are insufficient to absorb fully all theliquid electrolytic solutions present. Fundamental informa-tion about the role of the polymer component as it impactstransport processes in the porous electrodes does not yetexist. Because these processes may dominate high-ratecharging and discharging of these cells, more work isneeded in this important area.

The main goal of the present work is to comparecomputer simulations with experimental data for BellcorePLION w cells having higher active material loadings and arange of electrode thickness. Transport limitations at highrates exist, but the agreement with experimental data be-comes much poorer as the electrodes become thicker andmore highly loaded. The reasons for these discrepanciesare explored based on the fundamental description of salttransport in the plasticized polymer electrolyte andlithium-ion transport in the solid phase of the porouscomposite electrodes.

2. Description of the system

Bellcore plastic lithium-ion cells consist of porous com-posite positive and negative electrodes and a plasticizedelectrolyte, which, in the present study, was composed of 1M LiPF in 2:1 vrv mixture of ECrDMC, silanized6

Ž . Žfumed silica-filled p VdF–HFP three parts of Kynavw Ž wFLEX 2801, two parts of silica Cab-O-Sil TS-530,. ŽCABOT , and five parts of dibutyl phthalate DBP,.Aldrich by weight — pre-extraction and activation com-.position copolymer matrix. The composite electrodes used

in this work were a mixture of active insertion material

Ž .Fig. 1. SEM micrograph of PLIONe cell LiMn O rgraphite: bicell configuration . All active components are embedded in the chemically inert2 4

plasticized polymer matrix.

( )P. Arora et al.rJournal of Power Sources 88 2000 219–231 221

Ž Ž . ŽMCMB 2528 Osaka Gas or LiMn O prepared in-house2 4..by Bellcore , polymer binder, nonaqueous electrolyte, and

conductive filler additive. The current collectors are ex-panded metal mesh made of aluminum or copper. Detailsof the cell fabrication process and current collector treat-

w xments have been discussed elsewhere 4,6–8 .In this study, six different cells are used to compare the

model predictions with experimental data. The cells differin their electrode thickness and initial salt concentrations.The cells with three different electrode thickness will bereferred to as thin, medium, and thick cells, and the cellswith different initial salt concentrations will be referred aslow c0, medium c0, and high c0. The thickness of thepositive and negative electrode were varied from 140 to205 mm and from 80 to 115 mm, respectively. The separa-tor thickness was held constant at 76 mm. The cells withthree different electrode thickness had an initial salt con-centration of 1 M, while the cells with three different

Ž 0 .initial salt concentrations c s0.25, 0.5, and 1.25 M hadŽ .electrode thickness of 154 positive electrode and 110 mm

Ž .negative electrode and separator thickness of 100 mm.2 Ž 2 .The cells under consideration were 77.42 cm 12 in.

Ž .cells with a bicell design AlNCNSNANCuNANSNCNAl .Ž .A scanning electron microscopy SEM cross-section of a

w Ž .PLION cell LiMn O rgraphite: bicell configuration is2 4

Table 1Design-adjustable parameters

Li C Li Mn Ox 6 y 2 4

ParameterŽ .d mm

Thin 83.5 144.4Medium 99.0 179.3Thick 114.7 204.4

Ž .d mm 13.6 16.0ccŽ .R mm 12.5 8.5s

0 3Ž .c molrdms

Thin 20.71 3.92Medium 21.50 3.92Thick 20.00 3.92

´ 0.360 0.416liqy1 aŽ .a cm 1226.4 1305.9

ValuesŽ .d mm 76.2s

´ 0.593ls

´ 0.266ps

´ 0.141SiO 2Ž .T 8C 21.0

0 3Ž .c molrdm 1.000t 0.363q

2 y6Ž .D cm rs 3.35=100

pElectrodes 5.2Separator 2.4

3Ž .r grcm 1.320liq3Ž .r grcm 1.750p3Ž .r grcm 2.2f

a Ž .as3 1y´ y´ y´ rR .liq p f s

Table 2Parameters for the electrodes

Parameter Li C Li Mn Ox 6 y 2 4

2 y10 y9Ž .D cm rs 2.0=10 1.0=10sŽ .s Srcm 1.0 0.0380

2Ž .i mArcm 0.688 0.416o3Ž .c molrdm 30.54 22.86t

3Ž .r grcm 2.20 4.14

shown in Fig. 1. These have one Cu current collector inthe center and two Al current collectors at the outside withtwo layers of anode, cathode, and separator sandwichedbetween the outer pair of Al current collectors. Thedesign-adjustable parameters which include film thickness,volume fractions, particle sizes, salt concentrations, cell

Žtemperature, etc., for the first three cells thin, medium and.thick are given in Table 1. The specific surface area per

unit volume of each electrode was estimated by assumingthat the electrode particles are spherical and uniform insize.

The open-circuit potentials for MCMB 2528 andLiMn O electrodes as a function of state of charge have2 4

w xbeen measured previously by Doyle et al. 3 and are usedas such. Few kinetic data are available for the insertionreactions, which are often characterized by reasonably highexchange current densities. The values used in this work

w xare the same as those used previously 3 . The density ofeach material and exchange current density of both elec-trodes are given in Tables 1 and 2.

3. Polymer vs. liquid lithium-ion systems

For plastic lithium-ion batteries to compete with liquidlithium-ion cells, the excellent energy density of the lattercommercial cells must be met or exceeded by the newpolymer-type cells. This is an extremely challenging featbecause energy densities for liquid lithium-ion cells haveimproved continuously since their introduction and nowexceed 130 W hrkg and 300 W hrl in some medium-sizedprismatic cells. For polymer-type cells to compete, theactive material loadings in the electrodes must be maxi-mized, and thickness of the electrodes must be increased tolevels that are greater than are possible with liquid cells.These changes, along with the lightweight packaging, al-low polymer cells to compete on an energy basis even withthe thicker separators and lower active material loadingsthat are used in these cells. In fact, the major advantagefrom an energy standpoint of the polymer-type cells istheir lightweight packaging, which brings significant im-provements in specific energy although only modest im-provements in energy density when compared to prismaticliquid lithium-ion cells on a per-cell basis.

For plastic lithium-ion battery systems, increases inactive material loading have important consequences for


modeling efforts. Until now, the models in the literaturefor gelled polymer lithium batteries have treated the poly-mer electrolyte as a homogenous single-phase system.However, as polymer content in the porous electrodes isdecreased, the single-phase gelled polymer electrolytemodel becomes less realistic. Instead, a two-phase systemprobably exists, consisting of partially swollen polymerelectrolyte in contact with electrode particles and relativelyless free liquid electrolyte within the pores. We expect themathematical model to be less successful in predicting thecharge–discharge behavior of these current systems be-cause of the two-phase nature of the polymer electrolytewhen the cell is driven at high rates.

4. Transport properties

Knowledge of solution-phase transport property data isessential for mathematical modeling of any electrochemi-cal system. This becomes especially important for theextended charge and discharge of lithium-ion cells becauseof large concentration gradients that can develop in thecell. Using constant values for the electrolyte conductivityand salt diffusion coefficient in the solution phase is notrealistic as these parameters vary strongly with salt con-centration. For plasticized polymer electrolyte-based cells,conductivity data on the liquid electrolytic solution as afunction of salt concentration are not sufficient for aproper description of the ionic conductivity because the

Ž . Žpresence of the p VdF–HFP polymer and filler silica in.separator and carbon black in electrodes affects the trans-

w xport processes in the separator and porous electrodes 3 .While the literature is starting to contain more detailedtransport property data on gelled polymer electrolyte sys-tems, we are still far from having complete datasets on

w xsystems of industrial relevance 9,10 .The salt transport in the separator region of these

lithium-ion polymer cells will probably be different fromthe salt transport in the porous electrodes because of thedifferent constituents present in the two and because of thedifferent pore morphologies of the films. For the systemstudied here, the separator region consists of

Ž .ECrDMCrLiPF and p VdF–HFP mixed with silanized6

fumed silica as the porous matrix. The electrodes, on theother hand, are composed of primarily active material anda small quantity of carbon black held together withŽ .p VdF–HFP and swollen with ECrDMCrLiPF . Thus, to6

have a good description of ionic conductivity, correctionsto the conductivity expression for the neat electrolytesolution ECrDMCrLiPF should be made for each region6

to account for the polymer and other components presentin the system.

We have measured and incorporated a new conductivityexpression as a function of salt concentration into themathematical model. Both solution and film conductivitymeasurements were made for 2:1 EC:DMC electrolyte

Ž .with LiPF salt in p VdF–HFP -based separators. The6

conductivity measurements of the salt solutions were donewith a VWR Scientific conductivity probe. Ionic conduc-tivities of the polymer films were determined fromomplex impedance spectra measured using an EG&G

Ž .Princeton Applied Research model 5315 two-channelpreamplifier in combination with a Wavetek 50 MHz

Ž .pulserfunction generator model 165 . The four-electrodeconductivity cell consists of two parallel Pt wires and twostainless steel blades immobilized in a Teflonw cell sepa-rated by a fixed distance. The four-electrode configurationeffectively eliminates the interfacial impedance’s from themeasurement of conductivity. Separator films composed of

w Ž .Kynar FLEX 2801 Elf-Atochem America , silanizedŽ w .fumed silica Cab-O-Sil , TS530 , and dibutyl phthalate

Ž . Ž .DBP Aldrich in a 3:2:4 mass ratio were obtained fromBellcore. The DBP was extracted from the separator using

Ž .anhydrous diethyl ether EM Industries for 1 h followedby drying under vacuum at 608C for 1 h before immersingthe polymer films into the electrolyte. All the measure-ments were done inside a nitrogen-purged glove box.

The conductivity of a 2:1 vrv mixture of ECrDMC at218C was fit to the following function of salt concentra-tion:

k s2.905=10y4 q2.32702=10y2 cy1.82683liq

=10y2 c2 q5.1708=10y3c3 y4.977=10y4 c4

1Ž .

where k is the solution conductivity in millisiemens perliq

centimeter and c is the salt concentration in molar. Theabove expression is valid over a wide range of salt concen-

Ž .tration 0–4 M as shown in Fig. 2. This range is suffi-cient, as the maximum concentration observed in these

Ž .cells from simulated concentration profiles at very highrates is about 3.8 M.

ŽFig. 2 also shows the average conductivity of p VdF–.HFP –SiO composite films swollen in 2:1 ECrDMC as a2

Ž .function of salt LiPF concentration. The conductivity6

increases with salt concentration, reaches a maximum, andthen decreases just as with the liquid electrolytic solutions.

Fig. 2. Relationship between solution and film conductivity for LiPF in a6Ž .mixture of 2:1 ECrDMC in a copolymer matrix of p VdF–HFP .


The salt concentration varied from 0.01 to 2.25 M for themeasurements. Even for the highest concentration, the saltdissolved completely in the solvent mixtures. The conduc-

Ž .tivity of the p VdF–HFP films reaches a maximum of 3.0Ž .mSrcm for the 1.0 M salt concentration ´ s0.593 . Thel

Žconductivity of a 2:1 vrv mixture of ECrDMC–p VdF–.HFP film at 218C was fit to the following function of the

salt concentration:

k s1.436=10y4 q1.18731=10y2 cy1.95424film

=10y2 cq1.55293=10y2 c3 y6.4712=10y3c4

q1.3259=10y3c5 y1.048=10y4 c6 2Ž .The above fit is valid over the range of 0–4.0 M and isshown in Fig. 2. The conductivity expression given in Eq.2 is suitable for describing the separator region. A correc-tion should be applied before using it in the electroderegion to account for the reduced solution-phase volume.

The film conductivity can also be expressed in terms ofthe liquid conductivity:

k sk ´ p 3Ž .film liq

where ´ is the volume fraction of the electrolyte, and p isŽ .the constant whose value depends on the p VdF–HFP and

silica contents in the different regions. The p value for theseparator region was obtained by comparing the liquidconductivity data to film conductivity data using Eq. 3. pwas found to be 2.4 as shown in Fig. 2. The p value forthe electrode regions was determined by fitting thecharge–discharge data with model predictions for all threetypes of cells and was found to be about 5.2. The p valueobtained for the electrodes is different primarily because ofthe different quantity of polymer present and differentmorphology and pore size distribution. The different pvalues in the two cases suggest that a higher resistance toionic transport exists in the porous electrodes than pre-dicted based on the separator conductivity. This resistancecould have a variety of origins as discussed later.

The diffusion coefficient of salt in the solution phase isused as a parameter to fit the experimental data. Thediffusion coefficient in the porous media can also beexpressed by a relationship similar to Eq. 3:

D s´ py1 ´ D 4Ž . Ž .eff 0

where p is kept the same as that used for the conductivityŽin all three regions ps2.4 in separator and 5.2 in the two

.porous electrodes . ´ is the volume fraction of the solutionphase, and its value changes depending on the region. A

Ž 0 .constant transference number t of 0.363 was used to fitqw xthe experimental data for all thickness and rates 3 . The

transference number used in the simulations was chosen togive the best agreement with the discharge curves of thin,medium, and thick cells. While the transference number isprobably dependent on salt concentration, as shown re-

Ž .cently for lithium bis trifluoromethanesulfonyl imide inŽ . w xECrDMC in a p VDF–HFP matrix 9 , the concentration

dependence is rather weak and does not appear to be ofmajor importance for describing the current data.

Additional transport properties include the electronicconductivity and solid-phase lithium diffusion coefficientfor each of the electrodes as reported in Table 2. Theelectronic conductivity in the electrodes was corrected forporosity and tortuosity by using a Bruggeman exponent of

w x1.5 1 .

5. Simulation results

One of the inputs required for the mathematical modelŽ 0 0.is the initial state of charge x and y of each electrode.

Unfortunately, lithium stoichiometry is not an easy quan-tity to measure experimentally because of side reactionsoccurring on the negative electrode during the formationperiod. The values of x 0 and y0 are determined by fittinga low-rate discharge curve to model predictions. Fig. 3shows the comparison between simulation results and ex-

Ž .perimental data at low rates -Cr5 . As shown in Fig. 3,very good agreement is obtained between experimentaldata and model predictions. The initial states of chargeobtained by fitting the experimental data at small rates arereported in Table 3. The value of 0.171 was obtained fory0, while values of 0.678, 0.704, and 0.655 were obtainedfor x 0 for the thin, medium, and thick cells, respectively.These same initial states of charge were used in thesimulations at all discharge rates for a given cell. Thevalues of x 0 are different for each cell, possibly becauseof different amounts of irreversible capacity loss in thecarbon negative electrode during the formation period. Thecells used in this study were formed by cycling 5–10 timesbefore collecting the data used here. This was done toensure that no significant capacity fade is observed duringthe later rate tests.

For optimum performance of lithium-ion cells, the ratioof the lithium-ion capacities of the two host materials

Fig. 3. Experimental and simulated discharge curves for PLION w cells atlow rates. The C rates for thin, medium, and thick cells are 2.312, 2.906,and 3.229 mArcm2, respectively. The markers represent the experimen-tal data and solid lines, the simulation results.


Table 3Comparison of thin, medium, and thick cell data

Parameter Thin Medium Thick0ax 0.678 0.704 0.6550ay 0.1713 0.1713 0.1713

2Ž .C rate mArcm 2.312 2.906 3.229Ž .Mass ratio g 2.356 2.467 2.428actual

b,cŽ .Excess capacity % 22.2 18.6 19.92 aŽ .R V cm 30.27 28.56 34.05c

a Fitted.b Ž .Based on the theoretical capacity of positive 148 mA hrg and

Ž .negative 372 mA hrg electrodes.c Ž . wŽExcess capacity % s theoretical mass ratio y actual mass. Ž .xratio r theoretical mass ratio =100%.

should be balanced. Capacity balancing refers to the opti-mization of the mass loading in the two electrodes to

Ž .achieve the maximum capacity or energy from the bat-tery under conditions of steady cycling. The condition forbalanced capacities in a lithium-ion cell can be written in

w xterms of a ratio g of active masses in the electrodes 3 :

m d ´ r D xCq q q q yg s s ; g s 5Ž .actual theoreticalm d ´ r D yCy y y y q

The actual mass ratios calculated for the thin, medium,and thick cells are 2.356, 2.467, and 2.428, respectively.The theoretical mass ratio calculated on the basis of the

Ž .theoretical capacity of the positive 148 mA hrg andŽ .negative electrodes 372 mA hrg is 3.03 when using

D xs1.0 and D ys0.83. This leads to the conclusion thatan excess of 22.2%, 18.6%, and 19.9% capacity exists inthe negative electrode of these experimental cells. Consid-ering also the irreversible capacity on the negative elec-

Ž .trode which can be 6–8% for MCMB2528 graphite , it isapparent that a wide safety margin has been designed inthese cells to prevent accidental lithium deposition on thegraphite during rapid charging. In packaged cells whereother safety features would exist, these experimental cellscould be designed differently to provide even higher en-ergy densities by using a larger mass ratio closer to thetheoretical value. The mass ratios and the percentage ex-cess capacity for the thin, medium, and thick cells arereported in Table 3.

5.1. Solution-phase transport processes

In Fig. 4, we examine the cell potential during galvano-static discharge of thin cells at different discharge ratesŽ .Cr6, Cr5, Cr2, C, 1.5C, 2C, 2.5C, and 3C . The cell

Žpotential is plotted as a function of cell capacity mA2 .hrcm . The markers are experimental data while the solid

lines represent the model predictions. The C rate for thethin cell is ca. 2.312 mArcm2, and the electrolyte used inall cells is the same 1 M LiPF in 2:1 vrv mixture of6

ECrDMC. The simulated discharge curves without anyŽ .contact resistance not shown in Fig. 4 do not show good

Fig. 4. Experimental and simulated discharge curves for thin cells withŽ 2 .contact resistance R s30.27 V cm and constant salt diffusion coeffi-c

cient. The markers represent the experimental data and solid lines, thesimulation results.

agreement with the experimental data. The simulationsagree well with the experimental data at low rates, but thediscrepancy between model predictions and experimentaldata increases at higher rates as shown in Fig. 4. Thediffusion limitations in the porous electrodes become moresevere at higher rates, which is clear from the reduction ofcapacity observed at higher rates. The capacity of the thincell at the lowest rate of 0.387 mArcm2 was found to be

Ž 2 .ca. 178 mA h cell areas77.42 cm . The charging ratebetween experimental discharge curves was 1.29 mArcm2,and the cutoff potentials for charging and discharging were4.2 and 3.0 V, respectively.

Similar behavior was observed for the medium andthick cells, which are identical in design to thin cellsexcept for the electrode thickness reported in Table 1. Theagreement between model predictions and experimentaldata was reasonably good at low rates, but starts deteriorat-ing at high rates. Figs. 5 and 6 show the potential of

Žmedium and thick cells as a function of capacity mA2 .hrcm when discharged at different rates. The medium

Fig. 5. Experimental and simulated discharge curves for medium cellsŽ 2 .with contact resistance R s25.81 V cm and constant salt diffusionc

coefficient. The markers represent the experimental data and solid lines,the simulation results.


Fig. 6. Experimental and simulated discharge curves for thick cells withŽ 2 .contact resistance R s30.96 V cm and constant salt diffusion coeffi-c

cient. The markers represent the experimental data and solid lines, thesimulation results.

cells were discharged at the Cr7.5, Cr5, Cr2, C, 1.5C,2C, and 2.5C rates, and the thick cells were discharged atthe Cr8.5, Cr5, Cr2, C, 1.5C, and 2C rates. The C ratesfor the medium and thick cells are 2.906 and 3.229mArcm2, respectively. The capacity of the medium andthick cells at the lowest rate was ca. 224 and 248 mA h,respectively. The model simulations underpredict the ex-perimental data at some discharge rates for all three experi-mental cells because capacity fade was still occurringduring these cycles. The transference number and saltdiffusion coefficient in the solution phase were used asadjustable parameters to fit the experimental dischargecurves for the three cells. A constant transference numberŽ 0 . Žt s0.363 and constant salt diffusion coefficient D sq 0

y6 2 .3.35=10 cm rs were used to fit the experimental dataat all rates simultaneously. This will be discussed in moredetail in a later part of the paper.

Ž .The previous results Figs. 3–6 have demonstrated thatthe agreement between the simulated discharge curves andthe experimental data is unsatisfactory as the dischargerate is increased. In addition, this agreement becomes lesssatisfactory as the thickness of the cell is increased. Be-cause extended high-rate discharges may be dominated bysolution-phase transport processes, it is natural to considerwhether our treatment of these processes needs to bemodified or improved to describe best the experimentaldata.

Several areas can be considered for improvement, suchas:

1. Inclusion of salt precipitation;2. Allowance for solvent transport or segregation of the

two solvents;3. Allowance for convective transport;4. Inclusion of pore wall or surface film mass transport

limitations; and5. Use of a variable salt diffusion coefficient.

Each of these areas will be considered separately todetermine if they can describe the experimental findings.The calculated concentration profiles across the thin cell

Ž 2 .are shown in Fig. 7a and b for low 0.387 mArcm andŽ 2 .high 6.936 mArcm rates, respectively. The concentra-

tion gradient across the cell is small during low-ratedischarge as shown in Fig. 7a. During high-rate discharge,the concentration gradient across the cell is very large, anda maximum of 3.8 M salt concentration is calculated in thenegative electrode as shown in Fig. 7b. The high saltconcentration in the negative electrode may lead to saltprecipitation, which may further lead to blocking of poresin the porous composite electrode and temporary loss ofelectrolyte in the cell. However, because the calculated saltconcentration does not exceed 4.0 M in these cells duringthe highest discharge rates, which is below the solubilitylimit for LiPF in 2:1 ECrDMC, the idea of including salt6

precipitation in the model was discarded.ŽAs described earlier, the electrolyte 1 M LiPF in a 2:16

Ž . .mixture of ECrDMC in a p VdF–HFP copolymer matrixis treated as a binary electrolyte in the present model. The

Ž .Fig. 7. a Simulated salt concentration profiles across the thin cell duringŽ 2 .galvanostatic discharge at the Cr6 rate 0.387 mArcm . The separator

region is set off by dashed lines. Times since the beginning of dischargeŽ .are given in minutes. b Simulated salt concentration profiles across the

Ž 2 .thin cell during galvanostatic discharge at the 3C rate 6.936 mArcm .The separator region is set off by dashed lines. Times since the beginningof discharge are given in minutes.


effect of the second solvent and the polymer present in theelectrolyte has not been considered directly. The differentnature of the interactions between the ions and the twosolvents during the passage of current may lead to segrega-tion of the two solvents. Also, the polymer itself may notbehave simply as an inert filler but may instead interactdirectly with the ionic species. In these cases, it is moreappropriate to treat the system as a multicomponent one

Ž q y.with the five species polymer, EC, DMC, Li , and PF6

leading to 10 independent transport properties. As is oftenthe case, insufficient data exist at present for such asophisticated treatment of the transport properties in thegelled polymer electrolyte.

In previous battery models, it has been shown thatneglecting convective mass transport can be a very good

w xapproximation. Pollard and Newman 11 showed that theeffect of including the convective mass transport in theLirFeS battery model was negligible. This phenomenon2

has not been quantified for the lithium-ion system, but weassume that the effect will be negligible here as well. Theother issue of pore wall mass transport limitations has been

w xexamined previously by Doyle et al. 3 who were unableto explain high-rate transport limitations, although thistopic will be revisited again at the end of this paper.

During the low-rate discharge of these cells, the changein solution-phase salt concentration across the cell is very

Ž .small see Fig. 7a and remains close to the initial saltŽ .concentration 1 M . Hence, a constant diffusion coeffi-

cient is adequate to describe the experimental data at lowrates. During high-rate discharge, a large concentrationgradient is observed, as shown in Fig. 7b. The maximumconcentration in the negative electrode is ca. 3.8 M, andthe minimum concentration in the positive electrode is 0.0M. A constant salt diffusion coefficient is not adequate todescribe the experimental data when the change in concen-

tration across the cell is very high. Instead, a variable saltdiffusion coefficient may be needed to describe the experi-mental data at higher rates. We attempt to fit the experi-mental data better by allowing the salt diffusion coefficientto vary with discharge rate for the three experimental cellsŽ .thin, medium, and thick . While this is not a physicalmodel, it allows us to see if it is possible to describe thedata by using a rate-dependent salt diffusion coefficientwithout having actual diffusion coefficient data as a func-tion of salt concentration.

Until now, we have been using the porosity correctionparameter p, the transference number, the salt diffusioncoefficient, and the contact resistance as parameters to fitexperimental data for the three different PLION w cells.The value used for the transference number is 0.363, p is2.4 in the separator and 5.2 in the two porous electrodes,

Ž .and the value of the salt diffusion coefficient D is0

3.35=10y6 cm2rs. The actual value of the diffusionŽ . pcoefficient D is much smaller because of the ´eff

correction applied to each region, as shown in Eq. 4. Thevalues of contact resistance used for all three cells aregiven in Table 3. The source of the contact resistancebased on half-cell impedance measurements on the cells isthe resistance between the current collectors and electrode

w xinterfaces 5 . Next, we introduce a rate-dependent diffu-sion coefficient into the simulations.

Fig. 8 shows comparisons of model predictions withexperimental data for thin, medium, and thick cells atmoderate and high discharge rates using both contact

Ž .resistance R and rate-dependent salt diffusion coeffi-c

cient as parameters. The markers represent the experimen-tal data, and solid lines represent model predictions. Thesimulated discharge curves show good agreement with theexperimental data over the whole range. The ohmic drop inthe simulations can be increased by increasing the value of

Ž . Ž . Ž . Ž Ž .Fig. 8. Experimental and simulated moderate and high-rate discharge curves for a thin, b medium and c thick cells. A contact resistance R s acŽ . Ž . 2 . Ž .30.27, b 25.81 and c 30.96 V cm and rate-dependent salt diffusion coefficient see Fig. 9 were used as parameters to fit the experimental data. The

markers represent the experimental data and solid lines, the simulation results.


p, but at high p values, the solution-phase limitationsbecome excessive. Thus, a tradeoff between ohmic dropand solution-phase diffusion limitations is needed to de-scribe adequately the transport processes in these lithium-ion cells. The values of contact resistance used for the thin,medium, and thick cells are 30.27, 28.56, and 34.05 V

cm2, respectively.Ž .The exact value of the contact resistance R can bec

determined from high-frequency impedance data. Thehigh-frequency resistance is a sum of various resistances inthe cell including the contact and the bulk resistances ofthe electrodes and the separator. The bulk resistancesof the electrodes are usually very small, and the bulkresistance of the separator is a known quantity from theseparator thickness and conductivity. The remainder of thehigh-frequency resistance is roughly equal to the contactresistance on the current collectorrelectrode interfaces.Most of this contact resistance is at the aluminum currentcollector, but some portions of it are typically associatedwith the copper current collector. Half-cell impedance datacan be used to determine the exact values of contactresistance on each current collector separately. In thisstudy, the contact resistance on the two current collectorsis not differentiated and is a sum of the contact resistanceon the two current collectors.

The contact resistance depends critically on the type ofpretreatment given to the two current collectors and it canvary from 10 to 30 V cm2 for typical grid pretreatments.In this work, relatively high values of contact resistanceŽ 230.27, 28.56, and 34.05 V cm for thin, medium and

.thick cells, respectively are used to explain the experimen-tal data in the absence of impedance data for these particu-

Ž .lar cells. The contact resistance can be modified loweredby incorporating the additional resistances into the otherohmic parameter, p, which is the effective ionic conductiv-ity in the separator and porous electrodes. The value of pin separator region was determined to be 2.4 as shown in

Ž .Fig. 2, but the value of p in the porous electrodes s5.2was determined by comparing the model predictions withthe experimental data. Thus, the value of R can be offsetc

partially by increasing or decreasing the value of p in theporous electrodes. The type of current collector pretreat-ment can also explain the high value of R in the experi-c

mental cells used in this study.The agreement between the solid lines and markers is

good for low discharge rates but is not as good for highdischarge rates even with the adjustable parameters dis-cussed above. Considering the case of the thin cells, a

Ž y6 2 .constant diffusion coefficient 3.35=10 cm rs wasused to fit the experimental data obtained at low discharge

Ž 2 .rates 0.387–4.624 mArcm , or from Cr6 to 2C , but atŽ .high rates 2.5C and 3C , the diffusion coefficients usedŽ y6 2 .were lower 2.65 and 1.85=10 cm rs . Thus, a smaller

Ž .diffusion coefficient D is needed to obtain the best fit to0

discharge curves at higher rates. Similar behavior wasobserved for the medium and thick cells.

Fig. 9. Salt diffusion coefficient as a function of discharge rate for thin,medium, and thick lithium-ion cells giving the best fit to the experimentaldata. A lower diffusion coefficient was needed at the 2C rate for thin cellŽ 2 . Ž 2 .C;2.312 mArcm , 1.5C rate for medium cell C;2.906 mArcm ,

Ž 2 .and 1.2C rate for thick cell C;3.229 mArcm .

ŽIt appears that a variable diffusion coefficient rate-de-.pendent should be incorporated in the model for the

proper description of transport in the system. Fig. 9 givesthe values of the optimized salt diffusion coefficient usedto fit the experimental data at different discharge rates forall three type of cells. A constant diffusion coefficient wassufficient to fit the experimental data up to a certain Crate. At higher rates, a lowering of the salt diffusioncoefficient was observed for all three cells as shown inFig. 9. For thin cells, lower diffusion coefficients wereneeded for rates higher than 2.5C, while for medium andthick cells, lower values of diffusion coefficients wereneeded for rates higher than 2C and 1.5C, respectively.Hence, solution-phase diffusion limitations become moreprominent as the thickness of the electrodes increases.

As no experimental data for salt diffusion coefficient asa function of salt concentration are available in the litera-ture, we incorporated several different expressions in themodel to see if any of these functional forms could de-scribe the experimental data. Linear, polynomial, and ex-ponential relationships were tried. We would normallyexpect D to decrease with increasing concentration ow-salt

w xing to ion–ion interactions at high salt concentrations 12 .Ž Ž ..Some of the expressions Ds f c used are shown below:

Linear expressions:

DsD qm 1yc 6Ž . Ž .0

and

Ds3.35=10y10 when c-2.5 M,

DsD myc when c)2.5 M 7Ž . Ž .0

Exponential expression:

DsD exp mc 8Ž . Ž .0

Polynomial expressions:

Dsm f k , where k s f c 9Ž . Ž . Ž .soln soln


and,

DsD a qa cqa c2 qa c3 q . . . . 10Ž .Ž .0 1 2 3 4

Ž . Ž .Each of the above expressions Eqs. 6–10 for DsD cwas used in the computer simulations, but none of themcould fit the whole data set well without any additionaladjustable parameters. As we have tried various functional

Ž .expressions for salt diffusion coefficient Eqs. 6–10 , it isunlikely that experimental data for salt diffusion coeffi-cient as a function of salt concentration will providesatisfactory fits.

5.2. Solid-phase transport processes

High-rate discharges are limited by diffusion processes,which can exist in either the solution or solid phase orboth. In Section 5.1, it was shown that diffusion limitationsin the solution phase are a possible cause for rate limita-tions at high rates, but it is still not clear whether they arethe main limitation. The solid-phase concentration profilesin the particles of both positive and negative electrodesclearly show that diffusion limitations also exist in thesolid phase during discharge at high rates. In fact, it hasbeen shown previously that a properly designed lithium-ioncell should have transport limitations in both phases duringhigh rate use so that both energy density and power

w xdensity in the cell are optimized 3 .The comparison between computer simulations and ex-

perimental data for thin, medium, and thick cells at moder-ate and high discharge rates is shown in Fig. 10. All theparameters used were identical to those in Section 5.1 andare shown in Tables 1 and 2. The model predictions were

Ž y6done with a constant salt diffusion coefficient 3.35=102 .cm rs in the solution phase, a constant solid-phase diffu-

Ž y10 2 .sion coefficient 2=10 cm rs in the negative elec-trode, and various solid-phase diffusion coefficients in thepositive electrode. The model predictions are in goodagreement with the experimental data at all rates.

Fig. 10. Experimental and simulated moderate and high-rate dischargeŽ . Ž . Ž .curves for a thin, b medium and c thick cells. A contact resistance

Ž Ž . Ž . Ž . 2 .R s a 30.27, b 25.81 and c 30.96 V cm and rate-dependentcŽ .solid-phase diffusion coefficient see Fig. 11 in the positive electrode

were used as parameters to fit the experimental data. The value of saltdiffusion coefficient used was 3.35=10y6 cm2rs. The markers representthe experimental data and solid lines, the simulation results.

Fig. 11. Solid-phase diffusion coefficient in the positive electrode as afunction of discharge rate for thin, medium, and thick lithium-ion cellsgiving the best fit to experimental data.

At low rates, a constant solid-phase diffusion coeffi-Ž y9 2 .cient D s1=10 cm rs was sufficient to describesq

the experimental data, but at higher rates, lower diffusionŽ y10 2 .coefficients D F2=10 cm rs were needed. Atsq

low rates, the effect of changing the diffusion coefficient isminor, and good fits can be obtained when D G3–4=sq10y10 cm2rs, but at higher rates, a small change in thediffusion coefficient leads to significant changes in thedischarge profiles. This is because at high-rate discharges,the cell performance is reduced by the diffusion limitationsin the solid phase. Similar behavior was observed for allthree types of cells. Fig. 11 shows the values of soliddiffusion coefficient in the positive electrode used to ob-tain the best fit to the experimental data. The solid-phasediffusion coefficients in the negative electrode were alsovaried to see whether experimental data could be describedbetter with a composition-dependent diffusion coefficient.The fits obtained by varying D were not as good assythose obtained by varying either D or the salt diffusionsqcoefficient, especially for the medium and thick cells.

5.3. Initial salt concentration

The comparison of model predictions with experimentaldata has demonstrated that diffusion limitations existing ineither the solution or solid phase of the composite positiveelectrode can be used to fit the data adequately. It is notyet clear which is the major limitation during high-ratedischarge. One might expect cells with higher or lowerinitial bulk salt concentrations to have a different degree ofsolution-phase diffusion limitations. In particular, lowerinitial salt concentrations should promote a greater degreeof diffusion limitations in the solution phase. On the otherhand, modifications to the electrode active material particlesize would be expected to impact on solid-phase transportlimitations while not having a significant influence onsolution-phase diffusion processes. We decided to attemptthe former approach and modify the initial salt concentra-tion in experimental cells in order to impact solution-phase


diffusion limitations. We reason that this should have amajor effect on the cell’s rate capability if solution-phasediffusion limitations are the primary cause of rate limita-tions in the experimental cells.

Ž 0Cells with three different initial salt concentrations c.s0.25, 0.5, and 1.25 M were used to determine the major

limiting factor during high-rate discharges. All other de-sign and system parameters were the same as shown inTables 1 and 2. Figs. 12–14 show comparisons of modelpredictions with experimental data for cells with three

Ždifferent salt concentrations at various discharge rates 4C,2 .3C, 2C, 1C, Cr2, Cr5, and Cr10, 1C;2.312 mArcm

Žusing both contact resistance R s20.31, 23.63, and 19.16c0 .for cells with c s0.25, 0.5, and 1.25 M, respectively and

salt diffusion coefficient as parameters. The markers repre-sent the experimental data, and solid lines represent modelpredictions. The simulated discharge curves show goodagreement with the experimental data over the whole rangeof discharge rates. A lower salt diffusion coefficient wasagain needed at high rates to describe the experimentaldata. The simulations were also carried out with a rate-de-pendent solid-phase diffusion coefficient for the positiveelectrode. It was found that very good agreement could beobtained for the cell with high initial salt concentrationŽ 0 .c s1.25 M , but the agreement was not good for cells

Ž .with low initial salt concentration 0.25 and 0.5 M , espe-cially during high-rate discharges. It can be concluded thatsolution-phase diffusion limitations are the major limitingfactor during high-rate discharges.

Fig. 15 compares the experimental and predicted dis-charge capacities for cells with three different initial saltconcentrations. The solid symbols represent the experimen-tal discharge capacity, while the open symbols representthe simulated discharge capacity. The model predictionsmatch reasonably well with the experimental data when arate-dependent salt diffusion coefficient is used. The in-flection at high rates is due to the time required to develop

Fig. 12. Experimental and simulated discharge curves for a cell with highŽ 0 . Žinitial salt concentration c s1.25 M . A contact resistance 19.16 V

2 . Ž .cm and rate-dependent salt diffusion coefficient see Fig. 16 were usedto fit the experimental data. The markers represent the experimental dataand solid lines, the simulation results.

Fig. 13. Experimental and simulated discharge curves for a cell with lowŽ 0 . Žinitial salt concentration c s0.5 M . A contact resistance 23.63 V


the salt concentration gradients inside the cell. Fig. 16shows salt diffusion coefficients used to fit the experimen-tal data at different discharge rates for the three cells withdifferent initial salt concentrations. At low rates, a constantsalt diffusion coefficient was again sufficient to describethe experimental data, but at high rates, different valueswere needed. The values used for the three cells at lowrates probably reflect the true concentration dependence ofthe salt diffusion coefficient. At high rates, the solution-phase diffusion limitations start to dominate, and a lowersalt diffusion coefficient is needed to describe the experi-mental data. The solution-phase diffusion limitations aremore severe for cells with the lowest initial salt concentra-

Ž 0 .tion c s0.25 M as reflected in the magnitude of thedecrease in D needed to describe the data.salt

After examining the experimental data and model pre-dictions for cells with three different initial salt concentra-tions, the exact reason for the discrepancies between the

Fig. 14. Experimental and simulated discharge curves for a cell with lowŽ 0 . Žinitial salt concentration c s0.25 M . A contact resistance 20.31 V



model and experiments is still not clear. Earlier, we sug-gested that the description of the transport processes in thegelled polymer electrolyte component of the battery neededto be improved. First, the different compositions in thevarious battery layers need a more detailed description andaccounting in the modeling. For example, while the separa-tor layer is composed of polymer, liquid electrolyte, andfiller such as fumed silica, the electrodes are composed ofpolymer and liquid electrolyte in a different ratio as wellas active materials and carbon black. It is well-known thatthe ratio of polymer to liquid electrolyte in gelled polymerelectrolytes has a major impact on ionic transport proper-ties.

Based on the present observations, we conclude that thetransport processes in the gelled polymer electrolyte areindeed the main cause of rate limitations in these cells.However, the binary electrolyte description, which hasbeen used to date to describe these processes, needs im-provement. We suggest that the physical structure andpossibly the process for making the electrode layers needto be considered in the model description. Based on thepresent formulations being employed, a better descriptionof the physical situation may consist of electrode particlespartially covered by gelled polymer electrolyte with inter-stitial spaces filled with essentially liquid electrolyte. Thegelled polymer electrolyte phase covering the particlesmay be incompletely gelled and may have a lower conduc-tivity than that of the separator layer, for example. Thislayer would present a significant diffusional and ohmicbarrier during operation of the battery that cannot bedescribed adequately using the single-phase gelled poly-mer electrolyte description currently in use.

The physical picture we are describing can be translatedinto a mathematical model by considering transport in the

Ždirection perpendicular to the pore direction which we.shall call the z-direction . This pore wall transport is

Fig. 15. Experimental and simulated discharge capacity as a function ofdischarge rate for cells with three different initial salt concentrationsŽ 0 .c s0.25, 0.5, and 1.25 M . The solid symbols are experimental dataand open symbols are model predictions with contact resistance andrate-dependent salt diffusion coefficient as adjustable parameters.

Fig. 16. Salt diffusion coefficient as a function of discharge rate for cellsŽ 0 .with three different initial salt concentrations c s0.25, 0.5 and 1.25 M

giving the best fit to the experimental data.

usually ignored in favor of the x-direction in most batterymodels due to geometric considerations. However, if thetransport properties are significantly reduced in one direc-tion vs. the other, this assumption needs to be revisited.The simplest procedure for describing the mass transportlimitations in the z-direction is to make a pseudo-steady-state assumption and utilize the mass transfer coefficient

w xapproach 3 . As this was attempted already and was notsuccessful, we assume that a two-dimensional batterymodel is now required to resolve these questions. Thetwo-dimensional model supplemented with transport prop-erty measurements will lead to a better understanding of

Ž .the diffusion limitations both solution and solid phase inthese cells.

6. Conclusion

Ž .The solution and p VdF–HFP -based film conductivityfor LiPF in 2:1 ECrDMC is reported as a function of salt6

concentration. An improved description of ionic conductiv-ity is incorporated into a macroscopic mathematical modelto describe the transport processes in the cell. The com-puter simulations are compared to experimental data forcells having different electrode thickness. Good agreementbetween theory and experiment is obtained by using thecontact resistance at the electrodercurrent collector inter-face as an adjustable parameter for different cells and byusing a rate-dependent salt diffusion coefficient in thesolution phase. Additional resistances present in the exper-imental cells can be attributed to the contact resistancebetween the metal mesh current collectors and the compos-ite electrodes.

At low rates, a single diffusion coefficient was suffi-cient to describe the experimental data, while at high rates,lower values were needed. This highlights the need forvariable transport property data in the mathematical model-


ing of real cells. Values of the salt diffusion coefficient inthe solution phase between 3.35=10y6 and 1.7=10y6

cm2rs provided the best fits to the experimental data overthe range of rates. By allowing the solid-phase diffusioncoefficient in the positive electrode to vary from 1=10y9

to 1=10y10 cm2rs, an adequate fit to the data can also beachieved. Diffusion limitations are more prominent forthick cells than for medium and thin cells. Rate-dependentsalt diffusion coefficients are probably an artifact of tortu-ous and inhomogeneous paths for salt diffusion inside theelectrodergelled polymer regions and reflect the inade-quacy of the present simplified treatment of salt transportbased on a binary electrolyte. During high-rate discharges,the solution-phase limitations are the major limiting factor.

7. List of symbols

w y1 xa specific surface area cmA anode

w 3 xc concentration of lithium in the solid molrdmsw 3 xc concentration of salt molrdm

w xC theoretical capacity mA hrg ; cathodew 2 xD salt diffusion coefficient cm rs

D diffusion coefficient of lithium in the solid elec-sw 2 xtrode particles cm rs

i exchange current density for the insertion processow 2 xmArcm

w xm constant or mass gp Bruggeman porosity parameter

w 2 xR contact resistance V cmcw xR radius of electrode particles mms

S separatort 0 lithium ion transference numberq

w xT temperature 8Cx stoichiometric coefficient of negative electrode

w xLi Cx 6

y stoichiometric coefficient of positive electrodew xLi Mn Oy 2 4

Greekg mass ratio of active material, positive to negative

w xd thickness of cell component i mmi

´ volume fraction of component iiw xk ionic conductivity of electrolyte Srcm

w xs electronic conductivity of solid matrix Srcmw 3 xr density of material grcm

Subscriptsq positive electrodey negative electrodea anodec cathodecc current collectorf conductive fillerliq liquid phase of plasticized electrolytep polymer phase of plasticized electrolytes separatort maximum concentration in intercalation material

Superscripts0 with respect to the solvent or initial condition

Acknowledgements

P.A. and R.E.W. acknowledge the financial supportfrom the Office of Research and Development of theUnited States Central Intelligence Agency under ContractNo. 93-F148100-100.

References

w x Ž .1 T.F. Fuller, M. Doyle, J. Newman, J. Electrochem. Soc. 141 19941.

w x Ž .2 T.F. Fuller, M. Doyle, J. Newman, J. Electrochem. Soc. 141 1994982.

w x3 M. Doyle, J. Newman, A.S. Gozdz, C.N. Schmutz, J.-M. Tarascon,Ž .J. Electrochem. Soc. 143 1996 1890.

w x4 C.N. Schmutz, J.-M. Tarascon, A.S. Gozdz, P.C. Warren, F.K.Shokoohi, Abstract 109, in: The Electrochemical Society ExtendedAbstract, Miami Beach, FL, Oct. 9–14, 1994 Vol. 94-21994, p. 172.

w x5 M. Doyle, A.S. Gozdz, J. Newman, Joint general battery session, in:Electrochemical Society Meeting, Los Angeles, CA, May 8, 1996,1996.

w x6 A.S. Gozdz, J.-M. Tarascon, O.S. Gebizlioglu, C. Schmutz, P.C.Warren, F.K. Shokoohi, Abstract 117, in: The ElectrochemicalSociety Extended Abstract, Miami Beach, FL, Oct. 9–14, 1994 Vol.94-21994, p. 184.

w x7 J.-M. Tarascon, A.S. Gozdz, C. Schmutz, F.K. Shokoohi, P.C.Ž .Warren, Solid State Ionics 86–88 1996 49.

w x8 F.K. Shokoohi, P.C. Warren, S.J. Greaney, J.-M. Tarascon, A.S.Gozdz, G.G. Amatucci, in: 37th Proc. Power Sources Conf., 1996, p.243.

w x Ž .9 H. Dai, T.A. Zawodzinski, J. Electroanal. Chem. 459 1998 111.w x10 L. Christie, A.M. Chrische, C.A. Vincent, Electrochem. Solid State

Ž .Lett. 2 1999 187.w x Ž .11 R. Pollard, J. Newman, J. Electrochem. Soc. 128 1981 491.w x12 H.S. Harned, B.B. Owen, The Physical Chemistry of Electrolytic

Solutions, 3rd edn., Reinhold, New York, 1958.

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