research article electrochemical and mechanical failure of

Research ArticleElectrochemical and Mechanical Failure of Graphite-BasedAnode Materials in Li-Ion Batteries for Electric Vehicles

Cheng Lin1 Aihua Tang12 Ningning Wu3 and Jilei Xing1

1National Engineering Laboratory for Electric Vehicles and Collaborative Innovation Center of Electric Vehicles in BeijingSchool of Mechanical Engineering Beijing Institute of Technology Beijing 100081 China2Sichuan Provincial Key Lab of Process Equipment and Control School of Mechanical EngineeringSichuan University of Science amp Engineering Zigong 643000 China3Citic Guoan Mengguli Power Source Technology Co Ltd Beijing 102200 China

Correspondence should be addressed to Aihua Tang tahme163com

Received 5 June 2016 Revised 16 August 2016 Accepted 18 September 2016

Academic Editor David Sebastian

Copyright copy 2016 Cheng Lin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Graphite-based anode materials undergo electrochemical reactions coupling with mechanical degradation during batteryoperation can affect or deteriorate the performance of Li-ion batteries dramatically and even lead to the battery failure in electricvehicle First a single particle model (SPM) based on kinetics of electrochemical reactions was built in this paper Then the Li-ion concentration and evolution of diffusion induced stresses (DISs) within the SPM under galvanostatic operating conditionswere analyzed by utilizing a mathematical method Next evolution of stresses or strains in the SPM together with mechanicaldegradation of anode materials was elaborated in detail Finally in order to verify the hypothesis aforementioned surface andmorphology of the graphite-based anode dismantled from fresh and degraded cells after galvanostatic chargedischarge cyclingwereanalyzed by X-ray diffraction (XRD) field-emission scanning electron microscopy (SEM) and transmission electron microscopy(TEM) The results show that large volume changes of anode materials caused DISs during Li-ion insertion and extraction withinthe active particles The continuous accumulations of DISs brought about mechanical failure of the anode eventually

1 Introduction

Li-ion cells are very compelling candidates for power supplieswith their high-power and energy density and low self-discharge rate They have been widely used in new-energyvehicles such as battery electric vehicles (BEV) hybridelectrical vehicles (HEV) and plug-in hybrid electric vehicles(PHEV) in the past few decades [1] For the health manage-ment study of Li-ion batteries in electrical vehicles knowl-edge about the mechanical degradation of anode materialsunder certain operating conditions is of great significance [2]Mechanical degradation may appear owing to deformationsand stress generation as lithium ions intercalatedeintercalateintofrom the active anode and cathode materials duringcharging and discharging operations over time It is impor-tant to understand and mitigate mechanical electrode degra-dation in Li-ion batteries for EVs since it can acceleratebattery aging and further lead to failure [3]

Unfortunately the mechanical degradation under somespecial working conditions is one of the critical challengesin improving performance and prolonging life span of thecells at present The difficulties existing in measurement ofthe critical properties and phenomena in electrode materialsinstigated extensive research activities on numerical model-ing of Li-ion batteries worldwide in the last few years [3ndash5] Christensen and Newman [6] developed a mathematicalmodel based upon concentrated solution theory and porouselectrode theory that calculate the volume changes andsimulate the distribution profiles of Li-ion concentrationand stress during the repeated lithium insertionextractionprocesses intofrom a spherical electrode particle Doyle et al[7] presented numerical models with a finite-volumemethodand porous electrode theory which predict the transportkinetics and reaction processes in batteries These modelsor methods have shown good agreement with dischargeexperiments to a certain extent however they cannot ideally

Hindawi Publishing CorporationJournal of ChemistryVolume 2016 Article ID 2940437 7 pageshttpdxdoiorg10115520162940437

2 Journal of Chemistry

account for mechanical failure of active materials within theelectrodes in batteries applied in EVs Cheng and Verbrugge[4] recently analyzed the evolution of stress and strain energydue to DIS in a spherical insertion electrode particle undereither galvanostatic or potentiostatic condition Deshpandeand coworkers [8 9] established relationships between sur-face stress surface energy and the magnitude of DISs innanowires and investigated the effects of surface elasticity andsurface energy on the stress evolution in spherical electrodesThey also correlated elastic strain energy DISs and stressdiscontinuities at the phase boundaries of the different phasesthat form during lithiationdelithiation using a core-shellstructural model [10]

The literature on modeling of stress and crack formationwithin active electrode particles is quite extensive Stressesand fracture in electrodes undergoing volume changes werepredicted in a one-dimensional model by Huggins and Nixin [11] Furthermore a terminal particle size below whichparticles are not expected to crack was also predicted in theirmodel Crack formation of electrode particles was modeledwith facture mechanics by Aifantis and Dempsey [12] Otherresearchers [4 13ndash21] noted that the maximum stress withinactive particles is in proportion to the product of particlesize and the concentration gradient The maximum stressincreases with current density radius of spherical particlesand lower Li-diffusivity in the electrode material all of whichlead to steeper concentration gradients between the surfaceand the core They were also verified in literatures [6 9ndash11]

Despite significant advances in the theoretical investiga-tion of stress and strain evolution within electrode activematerials of Li-ion batteries some innovative researches arestill needed to explore the mechanisms of mechanical failureIn this paper the SPM was established in Section 2 Thesolutions of the diffusion equations with initial and boundaryconditions and analytic expression of evolution of DISs forthe galvanostatic control within the SPM were presented inSection 3 In Section 4 a commercial Li-ion battery wascycled in a constant current (CC) chargedischarge processwith a battery cycler in the potential range of 30 Vndash42 V Forcomparison the other fresh battery belonging to the samebatch was disassembled in this paperThe anodes dismantledfrom fresh and degraded batteries were analyzed by XRDSEM and TEM The measurements from XRD reveal thelattice parameter changes and SEM and TEM were appliedto verify the microstructure differences of the graphiticnanoparticles between the fresh and degraded anodes Theseare the main contributions and are helpful to understandthe DISs in this manuscript Besides the complex interplaybetween the origins and evolution ofmechanical degradationand structure changes during electrochemical reaction inelectrode materials were also discussed

2 Single Particle Electrode Model

The single particle model (SPM) was first proposed byHaran et al [22] in determination of the hydrogen diffusioncoefficient within a metal hydride electrode and used tocalculate the Li-ion diffusion of the insertion electrode duringdischargingcharging process [23] A schematic illustration of

electrochemical-based Li-ion battery is depicted in Figure 1The Li-ion battery is considered as three parts negativeelectrode (graphite anode) separator and positive electrode(metallic oxide cathode) as shown in Figure 1 with singlespherical particle representing respective electrode of Li-ionbatteries in the dischargecharge process according to [24ndash26] During charge process Li-ions diffuse to the surface ofmetallic oxide particles in the cathode where they extractthe lithiated particles travel through the electrolyte separatorand are inserted and diffuse in solid phase of graphite par-ticles in the anode Simultaneously electrons emitted fromthe cathode transmit through the external circuit towards theanode During discharge positive Li-ions and electrons flowin the reverse direction see Figure 1

3 Evolution of Stresses within a SPM

For a spherical particle several groups provided evidence forlithium ion diffusion and phase transformations in mechan-ical properties Qi et al [27] showed that material propertieschange substantially upon lithium insertion Deshpande et al[9ndash11 15] considered the effects of elastic properties depen-dent on solute concentration on diffusion induced stressesin single phase systems Furthermore analytic solutions forstresses [3 4 8ndash11 14ndash17 19ndash23] can help elucidate the effectsof lithium ions diffusion on DISs We study DISs within aSPM caused by lithium insertion and extraction in Li-ionbattery electrodes under galvanostatic charging strategies

31 Solid Mechanics of DIS The stresses induced by Li-iondiffusion are considered in a SPM of radius 119877 The analogybetween thermal and DIS is based on the assumption thatthe bulk of the spherical SP is an isotropic linear elastic solidThe expressions of stress-strain relationships are establishedfor the radial and tangential components in the sphericalcoordinate system by employing the analogy [4 8ndash11 20 21]

120576119903=

1

119864(120590119903minus 2]120590

120579) +

1

3Ω119888

120576120579=

1

119864[(1 minus ]) 120590

120579minus ]120590119903] +

1

3Ω119888

(1)

where 119864 is Youngrsquos modulus ] is Poissonrsquos ratio Ω is thepartial molar volume of the solute and 119888 is the molarconcentration Besides the elastic properties are assumed tobe constant and have nothing to do with the concentration 119888

For infinitely small formulation of deformation the radialand tangential strains of the spherically symmetric particleare given by the following

120576119903=

119889120583

119889119903

120576120579=

120583

119903

(2)

where 120583 is the radial displacement As atomic diffusion ismuch slower than rates of elastic deformation in solidstransients are often neglected in solving the problem of solid

Journal of Chemistry 3

Charge

Discharge

r r

Separator

Copper negativecurrent collector

Aluminum positivecurrent collector

ACharge

Discharge

Negativeelectrode

Positiveelectrode

L

x

e

ee

e

electrolyte

Li+

Li+

Li+

minus

120575120588120575n 120575Sep

Li C6 Li MnO2

Cs

x y

Figure 1 Schematic illustration of a Li-ion battery during dischargecharge

mechanics Without any body-force the static mechanicalequilibrium equation within a sphere is listed by [28]

119889120590119903

119889119903+

2

119903(120590119903minus 120590120579) = 0 (3)

The boundary condition 120590119903(119877) = 0 and remains finite at

119903 = 0 and the solutions for radial and tangential stresses arelisted by

120590119903(119903) =

2

9119864Ω (1 minus ])minus1 [119888av (119877) minus 119888av (119903)]

120590120579(119903) =

119864Ω

9(1 minus ])minus1 [2119888av (119877) + 119888av (119903) minus 3119888 (119903)]

(4)

where 119888av(119903) equiv (31199033

) int119903

0

1199032

119888(119903)119889119903 is the average soluteconcentration of the spherical volume within the particle ofradius 119877

One principal shear stress equals zero and the other twoare both (120590

119903minus 120590120579)2 for the spherical symmetry and the

principal shear stress is given by

120590119903(119903) minus 120590

120579(119903)

2=

119864Ω

6(1 minus ]) (119888 (119903) minus 119888av (119903)) (5)

Thus the stresses distributed at any given time andlocation can be obtained under the conditions that theconcentration distribution is known [4]

32 DIS under Galvanostatic Operation Conditions Thestress evolution within the spherical particles under galvano-static condition is studied in this paper since Li-ion batteriesapplied in EVs are mostly charged with a typical constantcurrent constant voltage (CC-CV) charging strategyA simplerelevant equation of Li-ion diffusion within a spherical SP ofradius119877 is listed in the spherical coordinate systemby [29 30]

120597119888

120597119905=

119863

1199032

120597

120597119903(1199032120597119888

120597119903) (6)

where symbol119863 represents constant diffusion coefficientConsidering that the particle parameters of different

batteries will be different (6) would be transformed intodimensionless form as follows

119883 =119903

119877

120591 =119863119905

1198772

119910 =119888 minus 1198880

119888119877minus 1198880

(7)

Then the governing equation (6) in dimensionless form isshown as follows

120597119910

120597120591=

1

1198832

120597

120597119883(1198832120597119910

120597119883) (8)

This condition denotes that the current is a constant andthe ionic flux is invariant at electrode surface The initial and


boundary conditions subjected to galvanostatic control indimensionless form are given by

119910 (0119883) = 1199100 120591 = 0 0 le 119883 le 1

120597119910 (120591 119883)

120597119883

10038161003816100381610038161003816100381610038161003816119883=0

120591 ge 0

120597119910 (120591 119883)

120597119883

10038161003816100381610038161003816100381610038161003816119883=1

=119868

119865 120591 ge 0

(9)

The analytic solution to address the diffusion equation (8)with initial and boundary conditions (see (10)) subjected togalvanostatic control is as follows [29 30]

119910 =119868119877

119865119863 (119888119877minus 1198880)(3120591 +

1

21198832

minus3

10

minus2

119883

infin

sum

119899=1

(120582minus2

119899csc (120582

119899) sin (120582

119899119883) exp (minus120582

2

119899120591)))

(10)

where 120582119899(119899 = 1 2 3 ) are the positive roots of tan(120582

119899) =

119899By utilizing (10) the average concentrations of Li-ions can

be expressed as

119910av (120591 119883) =3119868119877

119865119863 (119888119877minus 1198880)[120591 +

1

10(1198832

minus 1) minus2

1198833

sdot

infin

sum

119899=1

(119899120587)minus4 csc (119899120587)

sdot (sin (119899120587119883) minus (119899120587119883) cos (119899120587119883) exp (minus1198992

1205872

120591))]

119910av (120591 1) =3119868119877120591

119865119863 (119888119877minus 1198880)

(11)

Equation (11) can be used to solve for stresses after (3) issubstituted into it

3120590119903(1 minus ]) 119865119863119864Ω119868119877

=1

5(1 minus 119883

2

) +4

1198833

infin

sum

119899=1

(119899120587)minus4 csc (119899120587)


1205872

120591))

= 120585119868

119903(120591 119883)

3120590120579(1 minus ]) 119865119863119864Ω119868119877

=1

5(1 minus 2119883

2

)

+ 2

infin

sum

119899=1

exp (minus1198992

1205872

120591)

119899120587 sin (119899120587)(119899120587119883)

minus1

(sin (119899120587119883)

minus (119899120587119883)minus2 sin (119899120587119883) minus (119899120587119883)

minus1 cos (119899120587119883))

= 120585119868

120579(120591 119883)

(12)

where 120585119868119903(120591 119883) and 120585

119868

120579(120591 119883) are radial and tangential stresses

respectively in dimensionless form under galvanostatic con-trol The shear stress is determined by the following

3 (120590119903minus 120590120579) (1 minus ]) 119865119863

2119864Ω119868119877=

1198832

10+

3

1198833

infin

sum

119899=1

(119899120587)minus4 csc (119899120587)


1205872

120591))

minus1

119883

infin

sum

119899=1

(119899120587)minus2 csc (119899120587) sin (119899120587119883) exp (minus119899

2

1205872

120591)

(13)

Figure 2 shows the Li-ion concentration and radialtangential and shear stresses of SPM in the insertion processunder galvanostatic condition The algebraic expressionnormalized by (13)119864Ω119868119877(119865119863)

minus1

(1 minus ])minus1 is a function ofposition and time

The current sign in (9) is defined as follows positivecurrent indicates insertion process and negative current indi-cates extraction of Li-ions The Li+ concentration increaseswith position and time as depicted in Figure 2(a) DuringLi-ion insertion process the radial stress appears in theform of compressive stress at the center of the sphere andapproximates to zero close to the sphere surface It increaseswith time near the sphere center and reaches the maximumat the center (see Figure 2(b))

The tangential stress appears as tensile stress near thesphere center and compressive close to the surface Themaximum of tensile tangential stress appears at the centerbefore the Li-ions reach there and decreases monotonicallytowards the surface then it reverses and appears as com-pression stress Afterwards it increases along the directionof sphere surface Moreover the tangential stress increasesin magnitude with time at any location and finally tendsto a steady-state (see Figure 2(c)) The shear stress reachesthe maximum at the location of sphere center decreasestowards the surface for all times and reaches a steady-stateafterwards In addition the shear stress decreases beforedimensionless time 120591 = 119863119905119877

2

= 01 while it increases afterdimensionless time 120591 = 119863119905119877

2

= 02 for all positions withinthe sphere as shown in Figure 2(d) Similar conclusions (iecompression instead of tension) can be drawn for Li-ionextraction process

4 Validation

To demonstrate the complex causality between electrochem-istry and mechanical degradation of anode in lithium ionbatteries we opted to study mesophase-carbon-microbead(MCMB) anode as a representative material that undergoesinteractions and subsequent DISs during electrochemicalcycling The investigated commercial batteries used in thiswork have a normal capacity rating of 87Ah and containcarbonmaterial in anodesThe fresh battery was first chargedat 1 C rate to 42 V and discharged to 30V subsequently 1 Crate means the current is 87A During each cycle the batterywas operated in a constant current (CC) charge mode andthen underwent a CC discharge until its voltage reaches 30 VThe rest time between charge and discharge is two hoursThe


02 03 04 05 06 07 08 09 101rR

1205917 = 04

1205916 = 03

1205915 = 02

1205914 = 01 1205913 = 00574

1205912 = 001

1205911 = 0001

0

02

04

06

08

1FD

1003816 1003816 1003816 1003816C(120591r)minusC

01003816 1003816 1003816 1003816IR

(a) Li-ion concentration

02 03 04 05 06 07 08 09 101rR

3(1

minus120574)FD120590r(EΩIR

) 1205911 = 0001

1205912 = 001

1205913 = 00574

1205914 = 01

1205915 = 04

minus4

minus3

minus2

minus1

0

1

(b) Radial stress

02 03 04 05 06 07 08 09 101rR

1205915 = 04

1205913 = 00574

1205911 = 0001

1205912 = 001

1205914 = 01

minus02

minus01

0

01

02

3(1

minus120574)FD120590120601(EΩIR

)

(c) Tangential stress

02 03 04 05 06 07 08 09 101rR

1205912 = 01

1205911 = 00574

1205913 = 02

1205914 = 04minus5

0

5

10

15

20

(120590rminus120590120601)

[2EΩ(c

Rminusc 0)]

(3(1

minus))

(d) Shear stress

Figure 2 Characteristic profile of SPM in the insertion process under galvanostatic control

stop criterion for the cycling tests is SOH = 80 at 25∘CThe surface and structure morphology of anode materialsdismantled from fresh and cycled cells were investigated byXRD SEM and TEM

The graphitic MCMB is spherical and contains somewhatrandomly oriented graphitic domains It consists of graphenesheets staggered in either an AB (hexagonal the most com-mon form) or ABC (rhombohedral) stacking arrangementUpon insertion of lithium ions during charge the graphitewas lithiated and every fourth layer was filled before thenext layers begin to take up lithium However lithiationmay be initiated at several different sites on the surfaceof a graphite grain and the lithiated layers did not needto correspond to one another at different nucleation sitesHence the expansion of each grain may represent a largefraction of the expansion of a fully lithiated grain [19]During discharge the graphite was delithiated and each grainmay embody the contraction of a delithiated grain Theanode film mechanically failed due to the stress inducedby the lattice parameter change during Li-ion intercala-tiondeintercalation The average grain sizes of nanoparticleson the anode from fresh and degraded cell are comparedby XRD in Figure 3 Thus the volume changes of theparticle during Li-ion insertion and extraction resulted ina strain differential It accumulated continuously and gaverise to stresses within the particle as the cycle increases [9]Subsequently cracks initiated propagated preferentially atgrain boundaries of the particle and resulted in mechanical

times104

2120579 (∘)

times103

FreshDegraded

Enlarged

(111

)

(311

)

(400

)

545250484644

20 30 40 5010

2120579 (∘)

0

1

2

3

4

5

6

7

Inte

nsity

(au

)

0

2

4

6

8

10

12

Inte

nsity

(au

)

Figure 3 XRD spectra of graphite-based anode from fresh anddegraded cell

failure of the anode materials They were also verified by theSEM and the high-resolution TEM micrographs (Figure 4)with an average particle size of 100 nm which is in good


TEM

(a) Fresh graphite-based anode

TEM

(b) Degraded graphite-based anode

Figure 4 SEM images of (a) fresh graphite-based anode before cycling tests and (b) degraded graphite-based anode after 500 cycles

agreement with the results obtained by XRD [13] TEMimages of lithiated anode display smooth edges of graphite(Figure 4(b)) with a very thin solid electrolyte interphase(SEI) filmThe crack and fracture could be observed on cycledanode from the TEM images (Figure 4(b)) The red arrowsin Figure 4(b) point to the cracked nanoparticles visible onthe graphitic layer surface Besides the edge of graphite sheetstructure began to mellow and some circular particles withdiameter of 100sim150 nm appeared on surface of the sheetwhich can be associated with the lithium plating and sidereaction products produced at the reactions between anodesurface and the electrolyteThey also can be observed in SEMand TEM images in Figure 4(b) By a unique combinationanalysis of SEM TEM and XRD the results suggest thatthe mechanical failure of anode material can be caused byaccumulated DISs within active particles The stresses areinitiated by repeated volume changes while the Li-ions insertand extract the host particles

5 Conclusions

This paper studied the evolution of stresses in a graphite-based anode of Li-ion batteries for EVs considering solidmechanics diffusion theory and electrochemical interfacialkinetics under galvanostatic condition The profiles of Li-ionconcentration and radial tangential and shear stresses inthe SPM were analyzed and presented in the insertion anddeinsertion process of the lithium ions in the particles Fur-thermore the evolution of phase structure and morphologyfor materials with anode demonstrates that a combinationof XRD SEM and TEM can track the cause and effect ofelectrochemical and mechanical failure processes in a Li-ion battery for EVs The experimental and analytical resultsshow that large volume changes of anode materials occurduring Li-ion insertion and extraction within the activeparticlesThe accumulated changes lead to stresses within theactive particles as the battery cycles and they further lead tomechanical failure of the anode

Competing Interests

The authors declare no competing interests regarding thepublication of this article

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (51575044) and Sichuan ProvincialKey Lab of Process Equipment and Control Foundation(GK201603)

References

[1] H Rahimi-Eichi F Baronti and M-Y Chow ldquoOnline adaptiveparameter identification and state-of-charge coestimation forlithium-polymer battery cellsrdquo IEEE Transactions on IndustrialElectronics vol 61 no 4 pp 2053ndash2061 2014

[2] L Zhang L Wang C Lyu J Li and J Zheng ldquoNon-destructiveanalysis of degradation mechanisms in cycle-aged graphiteLiCoO

2batteriesrdquo Energies vol 7 no 10 pp 6282ndash6305 2014

[3] R N Kuzmin D S Maximov N P Savenkova and A VShobukhov ldquoMathematical modeling of hysteresis in porouselectrodesrdquo Journal of Mathematical Chemistry vol 50 no 9pp 2471ndash2477 2012

[4] Y-T Cheng and M W Verbrugge ldquoEvolution of stress within aspherical insertion electrode particle under potentiostatic andgalvanostatic operationrdquo Journal of Power Sources vol 190 no2 pp 453ndash460 2009

[5] B Stiaszny J C Ziegler E E Krauszlig M Zhang J P Schmidtand E Ivers-Tiffee ldquoElectrochemical characterization and post-mortem analysis of aged LiMn

2O4-NMCgraphite lithium ion

batteries part II calendar agingrdquo Journal of Power Sources vol258 pp 61ndash75 2014

[6] J Christensen and JNewman ldquoStress generation and fracture inlithium insertion materialsrdquo Journal of Solid State Electrochem-istry vol 10 no 5 pp 293ndash319 2006

[7] M Doyle T F Fuller and J Newman ldquoModeling of galvano-static charge and discharge of the lithiumpolymerinsertioncellrdquo Journal of the Electrochemical Society vol 140 no 6 pp1526ndash1533 1993

[8] R Deshpande Y-T Cheng and M W Verbrugge ldquoModelingdiffusion-induced stress in nanowire electrode structuresrdquo Jour-nal of Power Sources vol 195 no 15 pp 5081ndash5088 2010

[9] R Deshpande Y Qi andY-T Cheng ldquoEffects of concentration-dependent elastic modulus on diffusion-induced stresses forbattery applicationsrdquo Journal of the Electrochemical Society vol157 no 8 pp A967ndashA971 2010


[10] R Deshpande Y-T ChengMW Verbrugge and A TimmonsldquoDiffusion induced stresses and strain energy in a phase-transforming spherical electrode particlerdquo Journal of the Elec-trochemical Society vol 158 no 6 pp A718ndashA724 2011

[11] R A Huggins andW D Nix ldquoDecrepitationmodel for capacityloss during cycling of alloys in rechargeable electrochemicalsystemsrdquo Ionics vol 6 no 1-2 pp 57ndash63 2000

[12] K E Aifantis and J P Dempsey ldquoStable crack growth in nanos-tructured Li-batteriesrdquo Journal of Power Sources vol 143 no1-2 pp 203ndash211 2005

[13] M Ebner F Marone M Stampanoni and V Wood ldquoVisual-ization and quantification of electrochemical and mechanicaldegradation in Li ion batteriesrdquo Science vol 342 no 6159 pp716ndash720 2013

[14] C Lin A Tang H Mu WWang and C Wang ldquoAging mecha-nisms of electrode materials in lithium-ion batteries for electricvehiclesrdquo Journal of Chemistry vol 2015 Article ID 10467311 pages 2015

[15] S J Harris R D Deshpande Y Qi I Dutta and Y-T ChengldquoMesopores inside electrode particles can change the Li-iontransport mechanism and diffusion-induced stressrdquo Journal ofMaterials Research vol 25 no 8 pp 1433ndash1440 2010

[16] W H Woodford Y-M Chiang and W C Carter ldquolsquoElectro-chemical shockrsquo of intercalation electrodes a fracture mechan-ics analysisrdquo Journal of the Electrochemical Society vol 157 no10 pp A1052ndashA1059 2010

[17] S K Soni B W Sheldon X Xiao A F Bower and M WVerbrugge ldquoDiffusion mediated lithiation stresses in Si Thinfilm electrodesrdquo Journal of the Electrochemical Society vol 159no 9 pp A1520ndashA1527 2012

[18] R T Purkayastha and R M McMeeking ldquoAn integrated 2-Dmodel of a lithium ion battery the effect of material parametersand morphology on storage particle stressrdquo ComputationalMechanics vol 50 no 2 pp 209ndash227 2012

[19] J Christensen ldquoModeling diffusion-induced stress in Li-ioncells with porous electrodesrdquo Journal of the ElectrochemicalSociety vol 157 no 3 pp A366ndashA380 2010

[20] Y-T Cheng and M W Verbrugge ldquoDiffusion-induced stressinterfacial charge transfer and criteria for avoiding crackinitiation of electrode particlesrdquo Journal of the ElectrochemicalSociety vol 157 no 4 pp A508ndashA516 2010

[21] Y-T Cheng and M W Verbrugge ldquoApplication of Hasselmanrsquoscrack propagation model to insertion electrodesrdquo Electrochem-ical and Solid-State Letters vol 13 no 9 pp A128ndashA131 2010

[22] B S Haran B N Popov and R EWhite ldquoDetermination of thehydrogen diffusion coefficient in metal hydrides by impedancespectroscopyrdquo Journal of Power Sources vol 75 no 1 pp 56ndash631998

[23] E Tatsukawa and K Tamura ldquoActivity correction on elec-trochemical reaction and diffusion in lithium intercalationelectrodes for dischargecharge simulation by single particlemodelrdquo Electrochimica Acta vol 115 pp 75ndash85 2014

[24] W Fang O J Kwon andC-YWang ldquoElectrochemical-thermalmodeling of automotive Li-ion batteries and experimentalvalidation using a three-electrode cellrdquo International Journal ofEnergy Research vol 34 no 2 pp 107ndash115 2010

[25] A P Schmidt M Bitzer A W Imre and L GuzzellaldquoExperiment-driven electrochemical modeling and systematicparameterization for a lithium-ion battery cellrdquo Journal of PowerSources vol 195 no 15 pp 5071ndash5080 2010

[26] K A Smith C D Rahn and C-Y Wang ldquoModel-based elec-trochemical estimation and constraint management for pulseoperation of lithium ion batteriesrdquo IEEETransactions onControlSystems Technology vol 18 no 3 pp 654ndash663 2010

[27] Y Qi H Guo L G Hector Jr and A Timmons ldquoThreefoldincrease in the youngrsquos modulus of graphite negative electrodeduring lithium intercalationrdquo Journal of the ElectrochemicalSociety vol 157 no 5 pp A558ndashA566 2010

[28] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill New York NY USA 3rd edition 1970

[29] J CrankTheMathematics of Diffusion Oxford UK Clarendon2nd edition 1956

[30] A V Shobukhov and D S Maximov ldquoExact steady state solu-tions in symmetrical Nernst-Planck-Poisson electrodiffusivemodelsrdquo Journal of Mathematical Chemistry vol 52 no 5 pp1338ndash1349 2014

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ElectrochemistryInternational Journal of



CatalystsJournal of


account for mechanical failure of active materials within theelectrodes in batteries applied in EVs Cheng and Verbrugge[4] recently analyzed the evolution of stress and strain energydue to DIS in a spherical insertion electrode particle undereither galvanostatic or potentiostatic condition Deshpandeand coworkers [8 9] established relationships between sur-face stress surface energy and the magnitude of DISs innanowires and investigated the effects of surface elasticity andsurface energy on the stress evolution in spherical electrodesThey also correlated elastic strain energy DISs and stressdiscontinuities at the phase boundaries of the different phasesthat form during lithiationdelithiation using a core-shellstructural model [10]

The literature on modeling of stress and crack formationwithin active electrode particles is quite extensive Stressesand fracture in electrodes undergoing volume changes werepredicted in a one-dimensional model by Huggins and Nixin [11] Furthermore a terminal particle size below whichparticles are not expected to crack was also predicted in theirmodel Crack formation of electrode particles was modeledwith facture mechanics by Aifantis and Dempsey [12] Otherresearchers [4 13ndash21] noted that the maximum stress withinactive particles is in proportion to the product of particlesize and the concentration gradient The maximum stressincreases with current density radius of spherical particlesand lower Li-diffusivity in the electrode material all of whichlead to steeper concentration gradients between the surfaceand the core They were also verified in literatures [6 9ndash11]

Despite significant advances in the theoretical investiga-tion of stress and strain evolution within electrode activematerials of Li-ion batteries some innovative researches arestill needed to explore the mechanisms of mechanical failureIn this paper the SPM was established in Section 2 Thesolutions of the diffusion equations with initial and boundaryconditions and analytic expression of evolution of DISs forthe galvanostatic control within the SPM were presented inSection 3 In Section 4 a commercial Li-ion battery wascycled in a constant current (CC) chargedischarge processwith a battery cycler in the potential range of 30 Vndash42 V Forcomparison the other fresh battery belonging to the samebatch was disassembled in this paperThe anodes dismantledfrom fresh and degraded batteries were analyzed by XRDSEM and TEM The measurements from XRD reveal thelattice parameter changes and SEM and TEM were appliedto verify the microstructure differences of the graphiticnanoparticles between the fresh and degraded anodes Theseare the main contributions and are helpful to understandthe DISs in this manuscript Besides the complex interplaybetween the origins and evolution ofmechanical degradationand structure changes during electrochemical reaction inelectrode materials were also discussed

2 Single Particle Electrode Model

The single particle model (SPM) was first proposed byHaran et al [22] in determination of the hydrogen diffusioncoefficient within a metal hydride electrode and used tocalculate the Li-ion diffusion of the insertion electrode duringdischargingcharging process [23] A schematic illustration of

electrochemical-based Li-ion battery is depicted in Figure 1The Li-ion battery is considered as three parts negativeelectrode (graphite anode) separator and positive electrode(metallic oxide cathode) as shown in Figure 1 with singlespherical particle representing respective electrode of Li-ionbatteries in the dischargecharge process according to [24ndash26] During charge process Li-ions diffuse to the surface ofmetallic oxide particles in the cathode where they extractthe lithiated particles travel through the electrolyte separatorand are inserted and diffuse in solid phase of graphite par-ticles in the anode Simultaneously electrons emitted fromthe cathode transmit through the external circuit towards theanode During discharge positive Li-ions and electrons flowin the reverse direction see Figure 1

3 Evolution of Stresses within a SPM

For a spherical particle several groups provided evidence forlithium ion diffusion and phase transformations in mechan-ical properties Qi et al [27] showed that material propertieschange substantially upon lithium insertion Deshpande et al[9ndash11 15] considered the effects of elastic properties depen-dent on solute concentration on diffusion induced stressesin single phase systems Furthermore analytic solutions forstresses [3 4 8ndash11 14ndash17 19ndash23] can help elucidate the effectsof lithium ions diffusion on DISs We study DISs within aSPM caused by lithium insertion and extraction in Li-ionbattery electrodes under galvanostatic charging strategies

31 Solid Mechanics of DIS The stresses induced by Li-iondiffusion are considered in a SPM of radius 119877 The analogybetween thermal and DIS is based on the assumption thatthe bulk of the spherical SP is an isotropic linear elastic solidThe expressions of stress-strain relationships are establishedfor the radial and tangential components in the sphericalcoordinate system by employing the analogy [4 8ndash11 20 21]

120576119903=

1

119864(120590119903minus 2]120590

120579) +

1

3Ω119888

120576120579=

1

119864[(1 minus ]) 120590

120579minus ]120590119903] +

1

3Ω119888

(1)

where 119864 is Youngrsquos modulus ] is Poissonrsquos ratio Ω is thepartial molar volume of the solute and 119888 is the molarconcentration Besides the elastic properties are assumed tobe constant and have nothing to do with the concentration 119888

For infinitely small formulation of deformation the radialand tangential strains of the spherically symmetric particleare given by the following

120576119903=

119889120583

119889119903

120576120579=

120583

119903

(2)

where 120583 is the radial displacement As atomic diffusion ismuch slower than rates of elastic deformation in solidstransients are often neglected in solving the problem of solid


Charge

Discharge

r r

Separator



ACharge

Discharge

Negativeelectrode

Positiveelectrode

L

x

e

ee

e

electrolyte

Li+

Li+

Li+

minus

120575120588120575n 120575Sep

Li C6 Li MnO2

Cs

x y



119889120590119903

119889119903+

2

119903(120590119903minus 120590120579) = 0 (3)



120590119903(119903) =

2


120590120579(119903) =

119864Ω


(4)

where 119888av(119903) equiv (31199033

) int119903

0

1199032





120590119903(119903) minus 120590

120579(119903)

2=

119864Ω

6(1 minus ]) (119888 (119903) minus 119888av (119903)) (5)



120597119888

120597119905=

119863

1199032

120597

120597119903(1199032120597119888

120597119903) (6)



119883 =119903

119877

120591 =119863119905

1198772

119910 =119888 minus 1198880

119888119877minus 1198880

(7)


120597119910

120597120591=

1

1198832

120597

120597119883(1198832120597119910

120597119883) (8)




119910 (0119883) = 1199100 120591 = 0 0 le 119883 le 1

120597119910 (120591 119883)

120597119883

10038161003816100381610038161003816100381610038161003816119883=0

120591 ge 0

120597119910 (120591 119883)

120597119883

10038161003816100381610038161003816100381610038161003816119883=1

=119868

119865 120591 ge 0

(9)


119910 =119868119877

119865119863 (119888119877minus 1198880)(3120591 +

1

21198832

minus3

10

minus2

119883

infin

sum

119899=1

(120582minus2

119899csc (120582

119899) sin (120582

119899119883) exp (minus120582

2

119899120591)))

(10)


119899) =


be expressed as

119910av (120591 119883) =3119868119877

119865119863 (119888119877minus 1198880)[120591 +

1

10(1198832

minus 1) minus2

1198833

sdot

infin

sum

119899=1

(119899120587)minus4 csc (119899120587)


1205872

120591))]

119910av (120591 1) =3119868119877120591

119865119863 (119888119877minus 1198880)

(11)


3120590119903(1 minus ]) 119865119863119864Ω119868119877

=1

5(1 minus 119883

2

) +4

1198833

infin

sum

119899=1

(119899120587)minus4 csc (119899120587)


1205872

120591))

= 120585119868

119903(120591 119883)

3120590120579(1 minus ]) 119865119863119864Ω119868119877

=1

5(1 minus 2119883

2

)

+ 2

infin

sum

119899=1

exp (minus1198992

1205872

120591)

119899120587 sin (119899120587)(119899120587119883)

minus1

(sin (119899120587119883)


minus1 cos (119899120587119883))

= 120585119868

120579(120591 119883)

(12)

where 120585119868119903(120591 119883) and 120585

119868



3 (120590119903minus 120590120579) (1 minus ]) 119865119863

2119864Ω119868119877=

1198832

10+

3

1198833

infin

sum

119899=1

(119899120587)minus4 csc (119899120587)


1205872

120591))

minus1

119883

infin

sum

119899=1


2

1205872

120591)

(13)


minus1




2


2


4 Validation



02 03 04 05 06 07 08 09 101rR

1205917 = 04

1205916 = 03

1205915 = 02

1205914 = 01 1205913 = 00574

1205912 = 001

1205911 = 0001

0

02

04

06

08

1FD

1003816 1003816 1003816 1003816C(120591r)minusC

01003816 1003816 1003816 1003816IR


02 03 04 05 06 07 08 09 101rR

3(1


) 1205911 = 0001

1205912 = 001

1205913 = 00574

1205914 = 01

1205915 = 04

minus4

minus3

minus2

minus1

0

1

(b) Radial stress

02 03 04 05 06 07 08 09 101rR

1205915 = 04

1205913 = 00574

1205911 = 0001

1205912 = 001

1205914 = 01

minus02

minus01

0

01

02

3(1

minus120574)FD120590120601(EΩIR

)


02 03 04 05 06 07 08 09 101rR

1205912 = 01

1205911 = 00574

1205913 = 02

1205914 = 04minus5

0

5

10

15

20

(120590rminus120590120601)

[2EΩ(c

Rminusc 0)]

(3(1

minus))

(d) Shear stress




times104

2120579 (∘)

times103

FreshDegraded

Enlarged

(111

)

(311

)

(400

)

545250484644

20 30 40 5010

2120579 (∘)

0

1

2

3

4

5

6

7

Inte

nsity

(au

)

0

2

4

6

8

10

12

Inte

nsity

(au

)




TEM


TEM




5 Conclusions


Competing Interests


Acknowledgments


References












































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


Charge

Discharge

r r

Separator



ACharge

Discharge

Negativeelectrode

Positiveelectrode

L

x

e

ee

e

electrolyte

Li+

Li+

Li+

minus

120575120588120575n 120575Sep

Li C6 Li MnO2

Cs

x y



119889120590119903

119889119903+

2

119903(120590119903minus 120590120579) = 0 (3)



120590119903(119903) =

2


120590120579(119903) =

119864Ω


(4)

where 119888av(119903) equiv (31199033

) int119903

0

1199032





120590119903(119903) minus 120590

120579(119903)

2=

119864Ω

6(1 minus ]) (119888 (119903) minus 119888av (119903)) (5)



120597119888

120597119905=

119863

1199032

120597

120597119903(1199032120597119888

120597119903) (6)



119883 =119903

119877

120591 =119863119905

1198772

119910 =119888 minus 1198880

119888119877minus 1198880

(7)


120597119910

120597120591=

1

1198832

120597

120597119883(1198832120597119910

120597119883) (8)




119910 (0119883) = 1199100 120591 = 0 0 le 119883 le 1

120597119910 (120591 119883)

120597119883

10038161003816100381610038161003816100381610038161003816119883=0

120591 ge 0

120597119910 (120591 119883)

120597119883

10038161003816100381610038161003816100381610038161003816119883=1

=119868

119865 120591 ge 0

(9)


119910 =119868119877

119865119863 (119888119877minus 1198880)(3120591 +

1

21198832

minus3

10

minus2

119883

infin

sum

119899=1

(120582minus2

119899csc (120582

119899) sin (120582

119899119883) exp (minus120582

2

119899120591)))

(10)


119899) =


be expressed as

119910av (120591 119883) =3119868119877

119865119863 (119888119877minus 1198880)[120591 +

1

10(1198832

minus 1) minus2

1198833

sdot

infin

sum

119899=1

(119899120587)minus4 csc (119899120587)


1205872

120591))]

119910av (120591 1) =3119868119877120591

119865119863 (119888119877minus 1198880)

(11)


3120590119903(1 minus ]) 119865119863119864Ω119868119877

=1

5(1 minus 119883

2

) +4

1198833

infin

sum

119899=1

(119899120587)minus4 csc (119899120587)


1205872

120591))

= 120585119868

119903(120591 119883)

3120590120579(1 minus ]) 119865119863119864Ω119868119877

=1

5(1 minus 2119883

2

)

+ 2

infin

sum

119899=1

exp (minus1198992

1205872

120591)

119899120587 sin (119899120587)(119899120587119883)

minus1

(sin (119899120587119883)


minus1 cos (119899120587119883))

= 120585119868

120579(120591 119883)

(12)

where 120585119868119903(120591 119883) and 120585

119868



3 (120590119903minus 120590120579) (1 minus ]) 119865119863

2119864Ω119868119877=

1198832

10+

3

1198833

infin

sum

119899=1

(119899120587)minus4 csc (119899120587)


1205872

120591))

minus1

119883

infin

sum

119899=1


2

1205872

120591)

(13)


minus1




2


2


4 Validation



02 03 04 05 06 07 08 09 101rR

1205917 = 04

1205916 = 03

1205915 = 02

1205914 = 01 1205913 = 00574

1205912 = 001

1205911 = 0001

0

02

04

06

08

1FD

1003816 1003816 1003816 1003816C(120591r)minusC

01003816 1003816 1003816 1003816IR


02 03 04 05 06 07 08 09 101rR

3(1


) 1205911 = 0001

1205912 = 001

1205913 = 00574

1205914 = 01

1205915 = 04

minus4

minus3

minus2

minus1

0

1

(b) Radial stress

02 03 04 05 06 07 08 09 101rR

1205915 = 04

1205913 = 00574

1205911 = 0001

1205912 = 001

1205914 = 01

minus02

minus01

0

01

02

3(1

minus120574)FD120590120601(EΩIR

)


02 03 04 05 06 07 08 09 101rR

1205912 = 01

1205911 = 00574

1205913 = 02

1205914 = 04minus5

0

5

10

15

20

(120590rminus120590120601)

[2EΩ(c

Rminusc 0)]

(3(1

minus))

(d) Shear stress




times104

2120579 (∘)

times103

FreshDegraded

Enlarged

(111

)

(311

)

(400

)

545250484644

20 30 40 5010

2120579 (∘)

0

1

2

3

4

5

6

7

Inte

nsity

(au

)

0

2

4

6

8

10

12

Inte

nsity

(au

)




TEM


TEM




5 Conclusions


Competing Interests


Acknowledgments


References












































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



119910 (0119883) = 1199100 120591 = 0 0 le 119883 le 1

120597119910 (120591 119883)

120597119883

10038161003816100381610038161003816100381610038161003816119883=0

120591 ge 0

120597119910 (120591 119883)

120597119883

10038161003816100381610038161003816100381610038161003816119883=1

=119868

119865 120591 ge 0

(9)


119910 =119868119877

119865119863 (119888119877minus 1198880)(3120591 +

1

21198832

minus3

10

minus2

119883

infin

sum

119899=1

(120582minus2

119899csc (120582

119899) sin (120582

119899119883) exp (minus120582

2

119899120591)))

(10)


119899) =


be expressed as

119910av (120591 119883) =3119868119877

119865119863 (119888119877minus 1198880)[120591 +

1

10(1198832

minus 1) minus2

1198833

sdot

infin

sum

119899=1

(119899120587)minus4 csc (119899120587)


1205872

120591))]

119910av (120591 1) =3119868119877120591

119865119863 (119888119877minus 1198880)

(11)


3120590119903(1 minus ]) 119865119863119864Ω119868119877

=1

5(1 minus 119883

2

) +4

1198833

infin

sum

119899=1

(119899120587)minus4 csc (119899120587)


1205872

120591))

= 120585119868

119903(120591 119883)

3120590120579(1 minus ]) 119865119863119864Ω119868119877

=1

5(1 minus 2119883

2

)

+ 2

infin

sum

119899=1

exp (minus1198992

1205872

120591)

119899120587 sin (119899120587)(119899120587119883)

minus1

(sin (119899120587119883)


minus1 cos (119899120587119883))

= 120585119868

120579(120591 119883)

(12)

where 120585119868119903(120591 119883) and 120585

119868



3 (120590119903minus 120590120579) (1 minus ]) 119865119863

2119864Ω119868119877=

1198832

10+

3

1198833

infin

sum

119899=1

(119899120587)minus4 csc (119899120587)


1205872

120591))

minus1

119883

infin

sum

119899=1


2

1205872

120591)

(13)


minus1




2


2


4 Validation



02 03 04 05 06 07 08 09 101rR

1205917 = 04

1205916 = 03

1205915 = 02

1205914 = 01 1205913 = 00574

1205912 = 001

1205911 = 0001

0

02

04

06

08

1FD

1003816 1003816 1003816 1003816C(120591r)minusC

01003816 1003816 1003816 1003816IR


02 03 04 05 06 07 08 09 101rR

3(1


) 1205911 = 0001

1205912 = 001

1205913 = 00574

1205914 = 01

1205915 = 04

minus4

minus3

minus2

minus1

0

1

(b) Radial stress

02 03 04 05 06 07 08 09 101rR

1205915 = 04

1205913 = 00574

1205911 = 0001

1205912 = 001

1205914 = 01

minus02

minus01

0

01

02

3(1

minus120574)FD120590120601(EΩIR

)


02 03 04 05 06 07 08 09 101rR

1205912 = 01

1205911 = 00574

1205913 = 02

1205914 = 04minus5

0

5

10

15

20

(120590rminus120590120601)

[2EΩ(c

Rminusc 0)]

(3(1

minus))

(d) Shear stress




times104

2120579 (∘)

times103

FreshDegraded

Enlarged

(111

)

(311

)

(400

)

545250484644

20 30 40 5010

2120579 (∘)

0

1

2

3

4

5

6

7

Inte

nsity

(au

)

0

2

4

6

8

10

12

Inte

nsity

(au

)




TEM


TEM




5 Conclusions


Competing Interests


Acknowledgments


References












































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


02 03 04 05 06 07 08 09 101rR

1205917 = 04

1205916 = 03

1205915 = 02

1205914 = 01 1205913 = 00574

1205912 = 001

1205911 = 0001

0

02

04

06

08

1FD

1003816 1003816 1003816 1003816C(120591r)minusC

01003816 1003816 1003816 1003816IR


02 03 04 05 06 07 08 09 101rR

3(1


) 1205911 = 0001

1205912 = 001

1205913 = 00574

1205914 = 01

1205915 = 04

minus4

minus3

minus2

minus1

0

1

(b) Radial stress

02 03 04 05 06 07 08 09 101rR

1205915 = 04

1205913 = 00574

1205911 = 0001

1205912 = 001

1205914 = 01

minus02

minus01

0

01

02

3(1

minus120574)FD120590120601(EΩIR

)


02 03 04 05 06 07 08 09 101rR

1205912 = 01

1205911 = 00574

1205913 = 02

1205914 = 04minus5

0

5

10

15

20

(120590rminus120590120601)

[2EΩ(c

Rminusc 0)]

(3(1

minus))

(d) Shear stress




times104

2120579 (∘)

times103

FreshDegraded

Enlarged

(111

)

(311

)

(400

)

545250484644

20 30 40 5010

2120579 (∘)

0

1

2

3

4

5

6

7

Inte

nsity

(au

)

0

2

4

6

8

10

12

Inte

nsity

(au

)




TEM


TEM




5 Conclusions


Competing Interests


Acknowledgments


References












































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


TEM


TEM




5 Conclusions


Competing Interests


Acknowledgments


References












































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of
































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

research article electrochemical and mechanical failure of

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