S.K?bitzetal./JournalofPowerSources239(2013)572e583577
[a]
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[b]
0.950.9Cact/CBOT0.850.80.750.70.65
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Full Cycle 25°CFull Cycle 40°CFull Cycle 60°C50% SOC CV 25°C50% SOC CV 40°C50% SOC CV 60°C
21.91.81.71.61.51.41.31.21.11
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200300t/days
400
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Ract/RBOTFull Cycle 25°CFull Cycle 40°CFull Cycle 60°C50% SOC CV 25°C50% SOC CV 40°C50% SOC CV 60°C
Fig.8.ActualcapacityCact(a)anddischargeresistanceRact(b)vs.timeforcyclelifetestsat1C-rateatdifferenttemperatures.Dashedlinesshowthecyclelifetests.SelectedcalendarlifetestsequaltothemeanSOCofthecycletestsaredepictedforcomparisonanddrawnascontinuouslines.Allvaluesarenormalizedtoconditionsatbeginoftest(BOT).
correspondingmeanSOC.AtT?25??C,afterapproximately400daysofcycling,thecapacitydeclineisaboutfactor3.86highercomparedtotheCVcalendarlifetestatthecorrespondingmeanSOCof50%.ForT?40??Cafterthesametimeperiodthisfactoris2.6andforT?60??Cafter170daysitisfactor2.3.ResistancesinFig.8(b)showacomplexdependencyontem-perature.TheRPTafter400daysofcyclingatT?25??Cshowsonly6%increaseofresistancewhilethecorrespondingcalendarlifetestat50%SOCCVshowsanincreaseofabout20ˉter400days.AtthesametimeandatT?40??Ccycleliferesistanceisapproximately25%higherthanthecalendarliferesistance.AtT?60??Cthetestendsalreadyafter200daysofcycling.Atthattimethecycleliferesistanceisalsoapproximately25%higherthaninthecorre-spondingcalendarlifetest.Itisremarkable,thattheresistanceincreaseatT?25??Cissmallerthaninthecorrespondingcalendarlifetest,sincethecapacityfadeisstronger.ThisbehaviorpointstoanadditionaleffectthathastotakeplacebesidesaSEIformationwhichisenhancedbyhightemperatureandhighvoltages.Post-mortemanalysis[1]resultsoncyclinghighlightapossibleexpla-nationforthebehavioroftheresistancecurvesinFig.8(b).Ontheonehandtheelectrodevolumeincreasesduringcyclelifetests,ontheotherhandthemassincreasesespeciallyathightemperature,whichisascribedtoSEIdeposition.Whereasthe?rsteffectin-creasesporositythelatterdecreasesit.Whatisexperimentallyobservedisalargerporosityincreaseatlowtemperaturethanathightemperature.Athighertemperatures,thedepositionreactioncounteractsthevolumeexpansion,leadingtoadecreaseofporositythustohigherresistancevalues.3.4.AnalysisoftemperaturedependenciesFormorein-depthanalysisoftemperatureimpactsonaging,anArrheniusplot(Fig.9)isanalyzedforbothcapacitydeclineandresistanceriseovertimeasdoneintheworkofLiawetal.[18]foragraphitej1.2MLiPF6ethylenecarbonate(EC):ethylmethylcar-bonate(EMC)(w:w?3:7)jLixNi0.8Co0.15Al0.05O2cell.Thoughequippedwithadifferentcathodematerial,theircellconsistsofTable2Cyclelifetestsdiscussedinthiswork.FullcycleswereperformedatconstantcurrentinbetweenEODVandEOCV.AllothertestswerecycledatconstantcurrentandaDODbasedonthenominalcapacity(seeSection2).RangeTemperatureC-rateFullcycle25??C1Fullcycle40??C1Fullcycle60??C1100e50%SOC40??C190e100%SOC40??C145e55%SOC40??C10e10%SOC40??C1similarelectrolyteandanodematerials.Thebasisforthisanalysisisthewell-knownArrheniusequationgiveninequation(1):EaA?A0$expàRT????(1)ItdescribestherateAofathermallyactivatedprocessdependingonA0,whichcorrespondsdirectlytotheprocess,theactivationenergyEarepresentingtheenergybarrierforthethermalactivationprocess,theuniversalgasconstantRandtheabsolutetemperatureTindegreesKelvin.EaisderivedvialinearregressionfromtheslopeofanArrheniusplot.Itissupposedtobelinkedtotheratedeterminingstepinthethermallyactivatedprocesschainyieldingtocelldegradation.Fig.9(a)depictsthelogarithmicca-pacitydeclinenormalizedtotheactualcapacityatBOLversusreciprocaltemperatureatcertaintimeintervalsindays.Datapointsaretakenfromthecalendarlifematrixat50%SOCCVandcorre-spondingtemperatures.Everydotrepresentstheaveragecapacitylossoftwoorthreecells.SincetheRPTsdidnotoccuratthesametimeforalltests,linearinterpolationwasusedwhennecessary.TheactivationenergiesEaresultingfromlinearregressionandtheco-ef?cientofdeterminationR2oftheregressionforcapacitylossandresistanceincreaseareshowninTable3.Theregressionworksquitewellforthecapacitydecline.Fortheresistancevaluessomeimpreciseregressioncurvesarecalculated.Especiallyinthebeginningtheregressioncurves?tquitewell.NeartotheEOTregressioncurve?tsbecomeworse.Possiblythisisindicatingthatdifferentagingprocessesareoccurring,butthisphenomenonmightalsobebasedonmeasurementerrors.Thelatterisquitelikely,becausethechangeinslopeinFig.9(a)beginsafter210dayscorrespondingtoanunexpectedpronouncedresistancedecreaseatT?40??C,whichcanbeseeninFig.4(b).Anywaymostcurves?tquitewell,sotheassumptionofArrheniustypebehaviorappearsjusti?edforbothresistanceincreaseandcapacitydecline.Fig.10displaysthedifferentEavaluesresultingatthedifferenttimestakenfromthe(interpolated)RPTs.ForthecapacitylossameanactivationenergyofEa?43.6kJmolà1hasbeenderived.Thesevaluesareatleastinpartsnearlyequaltothevaluesdeter-minedbyLiawetal.NamelytheC/1capacityfromBOTuntil13weeksoftest(Eabetween55kJmolà1and42kJmolà1)and,andtheC/25capacityinbetween17and21weeksofaging(43kJmolà1,?rstthreeRPTarearound30kJmolà1).IncontrasttotheworkofLiawetal.[18],nosigni?cantincreaseinEacanbeobservedovertime.Followingtheirargumentation,thismeansthereisonlyasingle-stageprocessforcapacitydeclineincaseofthiscell.Theresistanceincreaseleadstomeanactivationenergyof578S.K?bitzetal./JournalofPowerSources239(2013)572e583
[a]
2.95-1-1.5
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1000T-1/K-13.153.25
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4284126168210252294336378420
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4284126168210252294336378420
ln(Closs/CBOT)-2-2.5-3-3.5-4-4.5
ln(Rincrease/RBOT)-1-1.5-2-2.5-3
Fig.9.Logarithmiccapacityloss(a)andresistanceincrease(b)vs.inverseambienttemperature.Setofcurvesshowdifferentpointsoftesttimeindays.Allvaluesarenormalizedtoconditionsatbeginoftest(BOT).Cellsusedforthistestweretakenfromthecalendarlifematrixat50%SOCCVandcorrespondingtemperatures.
Ea?36.9kJmolà1.ThevaluedeterminedbyLiawetal.isEa?32.2kJmolà1after12weeksofaging.HoweverLiawetal.determinedthedependenciesat80%SOCandwithadifferentcathodematerial.FurthermorethecellsatT?25??CintheworkofLiawetal.wereexcludedfromanalysisbecauseofuntypicalbehavior.Thesedifferencesandofcoursethedifferencesintheusedcellmaterialsaresupposedtohaveapartintheobserveddeviationsintheactivationenergies.AnywaythedeterminedArrheniusparametersforcellagingareatleastinpartsquitesimilaranddonotcontradictthehypothesisofSEIdepositionreactionsontheanodeasmainagingmechanism.3.5.AnalysisofvoltagedependenciesForafurtheranalysisofthevoltagein?uenceoncellaging,thechartsinFigs.11and12areintroduced.Theydepictthecapacityloss(a)andresistanceincrease(b)vs.cellvoltageforT?40??CandT?60??Crespectively.ThesetofcurvesrepresentdifferenttimesatorinbetweentheRPTs.Linearinterpolationwasusedforcalcula-tionofparametersiftimeoftheRPTwasdifferent.At80%SOCandT?40??C,inFig.11(a),theOCcurvewasusedasanapproximationfortheCVcurve.Bothtestsbehavequitesimilarconcerningthecapacitydecrease(seeFig.5(a))andthiswaymoredatapointsareavailable,sincetheruntimeoftheOCtestislonger.Allotherdepicteddatainboth?guresistakenfromCVtests.Fig.11(a)showsthatbetweenSOCof50%,80%and90%,thedifferenceincapacitydeclineissmall.E.g.after236daysintest,5.6%ofcapacityhasbeenlostat50%SOC,8.6%at80%SOCand8.1%at90%SOC.Incontrasttothis,18.7%ofthecapacityhasbeenlostat100%SOCatthattime.InFig.11(b)between20%and80%SOCno5045Ea/kJ mol-14035302540120200280t/daysCapacityResistanceCapacity averageResistance average360Fig.10.ActivationenergyEaofprocessescorrelatedtocapacitydeclineandresistanceincreaseplottedvs.testtimeindays.
Table3ResultsfromArrheniusplotanalysisobtainedvialinearregression.TheR2valuesdescribethequalityoflinearregressionofthelogarithmiccapacitylossvs.inversetemperature.Days4284126168210252294336378420R2capacity0.98180.99720.99620.99740.9990.99830.99730.99660.99610.9989Activationenergy(Ea/kJmolà1)38.254941.962445.554345.962843.634842.952743.311743.900544.502145.9983R2resistance0.96390.92410.95890.980.93250.91310.97280.93850.8630.8924Activationenergy(Ea/kJmolà1)40.396435.327938.189538.360736.648936.014637.996936.770933.94734.872clearimpactoftheSOConresistanceincreasecanbeseen.Thedegradationstartstoincreaseslightlyat90%SOCandgrowsrapidlyat100%SOC.TheagingbehaviorillustratedinFig.12(a)and(b)isbasicallyidentical,butdegradationishigherbecauseofArrheniustypebehavior.SincethescatteringoftheresistancesislowerthanatT?40??CandtheoveralldegradationofthetestatT?60??Candishigher,atrendinresistanceincreaseisvisible.Averyinterestingpointistheagingbehaviorat20%SOC.AtT?40??Citshowsanincreaseofthecapacitynotonlyinthebeginning,butalsoduringthe100%SOCOCtest(Fig.5(a)),whichhasbeenstoredtwiceatlowerSOCasdescribedinSection3.2.Sincethereisadischargeandachargebeforeeverycapacitytest,thiscapacityincreaseeffecthastolastforsometime.Ontheotherhand,lookingatthe100%SOCOCcurve,itappearsthatthegainincapacitytendstobeextinctafteranotherstorageperiodof100%SOC.Atthe?rstlookthiseffectofcapacityincreaseat20%SOCisnotobservableatT?60??C.InanycasetheagingcurveinFig.6looksunusualatthistemperature,sincethecapacitydeclineafterBOTislowerthanattheotherSOC.ItissupposedthatalsointhistestatT?60??CatendencyforcapacityincreaseexistsduetothelowSOC,butisoverlappedbythehighcapacitydecreasecausedbythehighertemperature.TocheckthisconnectionbetweentheobservedcurveshapeatT?60??CandtheeffectatT?40??Citisassumed,that50%SOCand20%SOCshowanidenticalagingcon-cerningcapacitydeclineatbothtemperatures.HencethedifferencebetweenthetestsisthegainofcapacityduetothestorageatlowSOC.UsinglinearinterpolationthisdifferenceiscalculatedforthetestsatT?60??C.AfterwardthiscalculateddifferenceisaddedasgaintotheT?40??C50%SOCCVtest.TheresultisthecurveS.K?bitzetal./JournalofPowerSources239(2013)572e583579
[a]
0.250.2Closs/CBOT0.150.10.050-0.05
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SOC: 20%
50%
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testtime/days133174236278321376
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3.73.9cell voltage/V
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SOC: 20%
50%
80% 90% 100%
T= 40°C
4889133174236278321376
testtime/days3.73.9cell voltage/V
4.1
Fig.11.Capacityloss(a)andresistanceincrease(b)vs.cellvoltageatT?40??C.Setofcurvesshowdifferentpointsoftesttimeindays.Allvaluesarenormalizedtoconditionsatbeginoftest(BOT).Linesafter174daysareinterrupted,becausethe80%SOCtestwasaborted.Furthermoreafter272daysthe90%SOCdataends,becausethetestwasstartedlater(seeSection3.2).
depictedinFig.13as“Calculated50e20”.Theremainingerrorbetweenmeasuredandcalculatedcurveisrathersmall,indicatingthatthisprocessmightindeedbetemperatureindependent.Thereasonforthiseffectisunknowntotheauthors,nohintsinliter-atureforthiseffectareknowntotheauthorseither.Itisharderto?ndafunctionforthecapacitydeclineandtheresistanceincreaseindependencyofthevoltagecomparedtothetemperature.Inordertogiveaccesstotheagingparameterse.g.formodelingandalsoforfurtherclari?cationofthevoltagede-pendency,Table3givesanoverviewofthevoltageimpactonlossofcapacityandresistancegain.Valueswerecalculatedbasedonthelastsetofcurves,wherealltestswereavailable.Thisis84daysforT?60??C.Nearestvalueat89dayswaschosenforT?40??C.Since20%SOCatT?40??Cgainedcapacity,thevalueswererelatedto50%SOC.Inliteratureoftenanexponentialfunctionissuggestedforvoltagedependenciesofelectrochemicalprocesses[3].Inthiscasethereisnoevidenceforanexponentialbehavior.Anearlystepwisevoltagedependency,asexpectediftheanodeisthemainvoltagedependentagingfactor[13],isalsoanoptionforinterpretationofthecollecteddatainthiswork.ThegraphitepotentialagainstLi/Litispartlystepwiseinshapeandshowsonlysmalldecreaseinvoltageafterreachingthe2LstagewithongoinglithiationaboveLi0.2C6[19].Thismaycauseonlysmalldifferencesintheanodepotentialin?uencingtheaging,thoughtheSOCandalsocellvoltagechangedsigni?cantly[13].Anyway,eveniftheanodesideisthemainreasonforcapacityloss,thecathodesideisalsolikelytoaccounttoagingperformance.EspeciallyathighSOCandhightemperaturesBrousselyetal.anticipateelectrolyteoxidationatthecathodeleadingparticularlytoresistanceincrease[13].Infact,thepost-mortemanalysisofthepresentcells[1]showsdistinctca-pacitylossforthecathodematerialforhighSOCandhightem-perature,whereSOChasahigherimpactthantemperature.Sinceslightgasevolutionunderhightemperatureandhighvoltageconditionshasbeenobserved,itcannotbeexcludedyetthatthedrasticagingat100%SOCandhightemperaturesisalsoduetocathodesidephenomena.AstheOCtestsat100%SOCandT?40??Cshow,thedegradationsigni?cantlydecreasesifthecellvoltageisallowedtofallabout100mVduetoself-dischargeinbetweentheRPT.Thisbehaviormighthinttoanadditionalagingeffectoccur-ringat100%SOC,butcouldalsobeaveryfastexponentialaccel-erationoftheformerlydescribedagingeffectsattheanode[20].Finallythedifferentprocesseshavetobeannouncedagaininlaterworks,takingalsointoaccounttheelectrochemicalimpedancespectraandtheresultsofthepost-mortemanalysis.3.6.AlternationestorageatvaryingcellvoltageInapplicationsandundertypicaloperatingconditionsstoragevoltageswillvarydependingonspeci?cloadpro?les.Motivatedbythisfact,astoragetestwithvaryingcellvoltagesatCVconditionswasstartedatT?40??C.Fig.14showstheappliedSOCtestpro?le.[a]
0.60.5Closs/CBOT0.40.30.20.103.5
3.73.9cell voltage/V
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SOC: 20%
50%
80% 90% 100%
[b]T= 60°C
42841261682102522943363784204SOC: 20% 50?% 90% 100%T= 60°C3.5342842.512621682101.525212943360.537804203.53.73.94.1cell voltage/Vtesttime/daysFig.12.Capacityloss(a)andresistanceincrease(b)vs.cellvoltageatT?60??C.Setofcurvesshowdifferentpointsoftesttimeindays.Allvaluesarenormalizedtoconditionsatbeginoftest(BOT).
Rincrease/RBOTtesttime/days580S.K?bitzetal./JournalofPowerSources239(2013)572e583
1.051Cact/CBOT0.950.90.850.80.75
0
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200300t/days
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50% SOC CV 40°C20% SOC CV 40°C50% SOC CV 60°C20% SOC CV 60°CCalculated 50-20
Fig.13.Analysisoflossesat20%SOC.Thedifferenceofactualcapacitybetween50%SOCand20%SOCatT?60??Cisaddedtothe50%SOCatT?40??Ccapacity.The“Calculated50e20”curveistheresult.
test.Takingintoaccountthesmalldependencyofresistancein-creaseonSOC(seeSection3.5),alsoscatteringofvaluesmightbeaproblemforinterpretationofthecurve.Similarto90%SOC,BOTforthealternationtestwassixmonthslaterthantheothertest,whichmightexplainthestrongcapacitydeclineatBOTcomparedtothe50%SOCCVtestintheplotduetoearlierstorageconditions.Howeverjustasthe90%SOCtests,acomparisontoinitialvaluesdidnotshowsigni?cantcapacitydeclineorresistanceincreaseatBOT.Thoughthecomplexbehaviorateachtimestepofthistestcannotbeexplainedbyasimplemodel,amethodisdiscussedinthefollowingtoapproximatetheoverallagingofthealternationtest.Totakeintoaccountthevoltagedependency,thevaluesderivedinSection3.5areusedtodeterminetheaverage(avg)weightedSOCduringthealternationtest.ThisweightedaverageSOCissupposedtocorrespondtotheSOCofacalendarlifeCVtest,showingthesamegradeofaging.TheresultingaverageweightedSOCiscalculatedviaequation(2).FromFig.15(a)itcanbeobserved,thatwhenevergoingfromhighertolowervoltageacapacityrecoverytakesplace,similartothe100%OCtestatT?40??C.Goingfromlowertohighervoltage,capacitydeclineacceleratesagainandthealreadyknowncapacityrecoveryeffectsseemtovanish.TheresistanceinFig.15(b)showsnocleardependencyonSOC.20%SOCseemstobesomehowbene?cialtoresistanceincrease,allotherSOCleadtoresistanceincrease.Nevertheless,theoverallresistanceofthetestcomparedtootherstoragetestsappearsquitelow,especiallyatthebeginningofthePSOCavg;weighted?iaeSOCT$SOCi$tiPaeSOCT$tii(2)1009080SOC/p60504030200100200t/days300400SOC avgSOC avg, weighted capacitySOC avg, weighted resistanceSOC test profileFig.14.Alternationtestpro?leandcalculatedresultingarithmeticaverageSOCandaverageweightedSOCforresistanceincreaseandcapacityloss.
a(SOC)istakenfromTable4atT?40??CandtiistheperiodoftimeatSOCi.ResultingaverageweightedSOCisshowninFig.14forweightingwithcapacityandresistanceparameters.Alsothearithmeticmeanvalueisshown,correspondingtoana(SOC)?1.SincethisrathersimpleapproximationisnotsupposedtotakeintoaccounttheeffectofcapacityincreaseatlowSOC,a(20%SOC/50%SOC)issettoone.Calculationpredictsacapacityfadeat400dayscorrespondingtoatestat80%SOCCV.Fig.15(a)showsthecom-parisonof80%OCcapacity,whichissimilarto80%CVmentionedbefore,andalternationbehavingthesameway.Thoughignoringformercellhistory,after400daysbothtestsshownearlythesamecapacitydecline.SincethecapacitygainatlowSOCdoesnotseemtobepermanentathigherSOC,thecapacitypredictionusingaweightedSOCisquitecorrectatdifferenttimepointswithhigherpreviousSOC.Theobservedcapacitygainphenomenonmightbeanerrorsourceforonlinecalculationofactualcapacitiesinbatteryman-agementsystems,sincethegainachievedat214daysisinpartsstillpersistentevenafter22daysofstorageat100%SOC.Forlong-termlifetimeprediction,withapro?leincludinghigherSOC,thesimpleapproximationderivedinthischaptercouldbesuf?cient.Inanycase,theresistancecurvesbehaviorcouldnotbepredictedwithFig.15.NormalizedactualcapacityCact(a)andresistanceRact(b)vs.timeforcalendarlifescenariosatT?40??C.AlternationtestwithCVatchangingSOC.AppliedSOCpro?levaluesbetweenRPTsareshowninthediagrams.
S.K?bitzetal./JournalofPowerSources239(2013)572e583581
Table4Agingfactorsforcapacityfadeandresistancerise.Forcalculationthelastsetofvalueswasused,whereallvoltageswerestillintest.E.g.atestat100%SOCCVandT?40??Chasa3.8timeshighercapacitylossthanthe50%SOCCVtestatthesametemperatureandtime.TemperatureClossRincreaseClossRincrease40C40??C60??C60??C??RelatedtoSOC50 %SOC?100%CV3.82.09.38SOC?90%CV21.631.2SOC?80%CV1.82131.52SOC?50%CV11.22.01.4SOC?20%CVe111[a]10,950,9[b]50% SOCCVdata 25°Cfit 25°C, sqrtfit 25°C, sqrt+lineardata 40°Cfit 40°C, sqrtfit 40°C, sqrt+lineardata 50°Cfit 50°C, sqrtfit 50°C, sqrt+lineardata 60°Cfit 60°C, sqrtfit 60°C, sqrt+linear2,22Ract/RBOT50% SOCCVdata 25°Cfit 25°C, sqrtfit 25°C, sqrt+lineardata 40°Cfit 40°C, sqrtfit 40°C, sqrt+lineardata 50°Cfit 50°C, sqrtfit 50°C sqrt+lineardata 60°Cfit 60°C, sqrtfit 60°C, sqrt+linearCact/CBOT0,850,80,750,70,650100200300400500600700t/days1,81,61,41,210100200300400500600700t/daysFig.16.ActualcapacityCact(a)andresistanceRact(b)overtimeforcellsstoredat50%SOCatdifferenttemperatures.Thedotsshowthemeasurementdatafortheeachcellintest,thesolidlinesthe?ttingresultusingasinglesquareroottimebehavior(eq.(3))andthedashedlinesdepictthe?ttingresultusingacombinationofsquarerootandlinearbehaviorforthetimedependency(eq.(4)).
thismethod.Takingintoaccountthecurvesat100%SOCfromtheprevioussections,theriseofresistancewasexpectedtobehigh.IncontrasttheresistanceinFig.15(b)duringthe100%SOCperiodsdoesonlyleadtoanatthemostmoderateresistancerise.Onereasonmightbethatsuchasimplemodelignoringtheprevioushistorycannotbeusedinhere.Alsoitcannotbeexcludedthatpartsofthecurvesbehavioranddifferencetotheagingcurvesnexttothealternationtestisduetoscatteringofthevalues.3.7.RegressionfunctionandcalendarlifetimeextrapolationBecausestronghintspointtotheaforementionedprocessesontheanodesidebeingthestrongestagingeffect,thisistakenasworkinghypothesisinthefollowing.Inliteraturedifferenttheoriescanbefoundaboutthe?rstprincipleprocessesofSEIgrowthduringaging.Asmentionedintheprevioussection,Brousselyetal.describeinRef.[4]theSEIformingattheSEIsurface/electrolyteinterfaceandthereforeidentifytheelectronicconductivityoftheSEIastheratelimitingfactorforSEIformation.Ploehnetal.[17]ontheotherhandidentifythesolventdiffusionprocessastheratelimitingfactor.However,mathematicaldescriptionsofboththe-oriesleadtotheresultthatSEIformationhastofollowasquarerootoftimedependency.Thecapacityfadeandresistanceincreasemeasuredinthisworksupportthistheory.SimilartotheworkofEckeretal.[3]inthefollowingafunctionwithsquarerootoftimedependency,aswellasacombinationofasquarerootandlinearfunctionhavebeen?ttedtothemeasuredagingresults:?ttingresultsforcapacityfadeandresistanceincreaseforcellsstoredat50%SOCatdifferenttemperatures.InTable5theresulting?ttingparametersandthecorrespondingcorrelationcoef?cientsR2forcapacityfadearegivenexemplarily.Itcanbeseen,thatasquarerootoftimefunctiondescribestheagingofthecellsquitewell.Itcanalsobeseen,thatanadditionallineartermdoesnotimprovethe?ttingresultssigni?cantly.Valuesforthelinear?ttingparameterb2of10à4to10à14show,thatthelinearcontributiontoagingisquitesmallandcanbeneglectedinordertoreducethenumberoffreeparameters.Similarresultscanbefoundfortheresistance.Thereforethesquarerootfunctionwillbeusedinthefollowing.Basedonthe?ttingresultslifetimeextrapolationscanbecalculatedforthevarioustestconditions.Fig.17showscapacitylifetimein(a),de?nedasthetime,wherecapacityreaches70%ofinitialcapacityandresistivelifetimein(b)de?nedasthetime,wheretheresistancereaches200%ofinitialresistanceovertem-peratureatdifferentSOC.AtamoderatetemperatureofT?40??Cthecapacitylifetimerangesfrom2to17yearsdependingonSOC.Acellstoredatconstantvoltageof4.18VatT?40??Cforexampleisonlyexpectedtolive2years.Resistivelifetimeiscomparabletocapacitylifetimeforthisspecialde?nition.AtT?40??Cresistivelifetimevariesbetween2and18yearsfordifferentSOC.AtverymoderateconditionslikeT?25??Cand50%SOCveryhighlifetimescanbereachedaccordingtothesquarerootextrapolation.Capacityp??AetT?1?a1$tAinitp??AetT?1?a2$t?b2$tAinit(3)Table5Fittingresultforcapacityfadeforcellsstoredat50%SOCanddifferenttemperatures.Forthe?ttingeqs.(3)and(4)wereused.The?ttingparametersaswellasthecorrespondingcorrelationcoef?cientsR2areshown.T[??C]25405060a10.00170.00380.00610.0109R20.97940.98850.99790.9802a20.00160.00380.00570.0080b23.86.72.241.78????10à610à710à510à4R20.98130.98850.99940.9993(4)Adenotescellcapacityorresistance,respectivelyanda1,a2andb2areconstantsdeterminedbythe?ttingprocess.Fig.16showsthe
