A818.full

Journal of The Electrochemical Society,152 ~4! A818-A829 ~2005!A818

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Cyclable Lithium and Capacity Loss in Li-Ion CellsJohn Christensen* ,z and John Newman**

Department of Chemical Engineering, University of California, Berkeley, California 94720, USA

In lithium-ion cells, there are several different classes of capacity loss, both reversible and irreversible, that limit the cell’sexploitable specific capacity and can lead to eventual cell failure. We attempt to clarify what is meant by capacity loss and cyclablelithium loss by defining these terms in the context of electrode state-of-charge restrictions. We define irreversible capacity loss asthat associated with active material loss and define two types of reversible capacity loss associated with balanced and unbalancedside reactions. We also examine several methods of compensating for cyclable lithium loss associated with passive-film formationand calculate the effect each has on a cell’s specific energy. Preforming the negative electrode, adding cyclable lithium to thepositive electrode, and introducing lithium powder into the negative electrode appear to be the most attractive methods in termsof specific energy, but practical constraints such as fabrication cost must be evaluated to determine which is superior.© 2005 The Electrochemical Society.@DOI: 10.1149/1.1870752# All rights reserved.

Manuscript submitted July 13, 2004; revised manuscript received October 19, 2004. Available electronically March 14, 2005.

0013-4651/2005/152~4!/A818/12/$7.00 © The Electrochemical Society, Inc.

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Cyclable lithium is consumed to form passivating films on etrode materials during the formation cycles of the lithium-ion cwhich consist of one to several charge-discharge cycles. The sof cyclable lithium is that which is initially contained in the positand negative electrode insertion materials, as well as that whadded to either electrode via side reactions from the lithium sthe electrolytic solution. The latter source typically accounts fsmall but significant fraction~less than 15%!of the total amount olithium in the cell.1

Because some lithium is consumed irreversibly~a process thatoften misleadingly referred to as “irreversible capacity loss”! duringthe first few cycles, it is common practice to include excess capin the positive electrode when manufacturing low-power cthereby allowing full utilization of the negative electrodcapacity.2 Such “mismatched” cells have a lower overall specenergy than an idealized cell would have, in which there is noof cyclable lithium.

One means of avoiding this ostensibly necessary evil is to iduce cyclable lithium by some means other than those alreadytioned. FMC Corporation has demonstrated the successful intrtion of passivated lithium powder into the composite negaelectrode of a lithium-ion cell.3 This additional cyclable lithiumcompensates for that which is lost in passive-film formation, wminimizing the addition of mass to the cell. Furthermore, the omally balanced cell that results can be operated over a higherage cell-voltage range.

In the present work, we attempt to clarify what is meantcapacity loss and loss of cyclable lithium, the two being equivaonly when a loss of lithium effectively limits the cycling rangethe cell. Perhaps counter to intuition, increasing the amounlithium contained in the positive and negative electrodes canlead to capacity loss. To elucidate these concepts, we evaluateral methods for balancing the loss of lithium during passiveformation. We also examine capacity loss due to loss of activeterial and distinguish between “reversible” and “irreversible” caity loss.

Conceptual Framework

An earlier paper introduced the state-of-charge~SOC!operatingwindow used to examine capacity fade and capacity baphenomena.1 We summarize the pertinent equations here. Letx bethe negative electrode SOC, or amount of lithium in LixN, where Nrepresents the electrochemically active negative electrode ma~e.g., C6!; let y be the positive electrode SOC, or amount of lithiin Li yP, where P represents the positive electrode material~e.g.,Mn2O4!. The capacity ratio is defined as

* Electrochemical Society Student Member.** Electrochemical Society Fellow.

z E-mail: [email protected]

address. Redistribution subject to ECS terms140.112.4.206aded on 2014-03-18 to IP

e

--

-

v-

l

z =Dx

Dy=

C+L+«+r+

C−L−«−r−

f1g

whereCi is the specific capacity of the active material in the etrode i, «i is the volume fraction of active material,Li is the electrode thickness, andri is the active-material density. The optimcapacity ratio, in which the entire capacity range of both electris realized, is

zopt =Dxmax

Dymaxf2g

where the maximum cycling range is set by physical considera~e.g., Li deposition at highx and Jahn-Teller distortion aty . 1, forsome materials!.2 Figure 1 is a schematic SOC operating windwith cycle paths drawn for a perfectly balanced cell and amatched cell that has excess positive electrode capacity.Dxmis is thecycling range of negative electrode SOC for the mismatchedand in this case it is equal toDxmax. Likewise,Dymis is the range opositive electrode SOC, which is less thanDymax when there iexcess positive electrode capacity. In the absence of side reaand capacity loss, the precise limits of the positive electrode cyrange depend entirely on the initial SOCs of both electrodesinstance, if we were to build the same cell with additional lithiumthe positive electrode material, the mismatched cell’s cyclewould be shifted upward in the operating window. However,slope of the path would remain the same, because it is dictatthe relative capacity of the two electrodes.

Figure 2 is an alternative representation of capacity balancing open-circuit potential~OCP! curves for positive and negatielectrodes. Figure 2a illustrates the case in which the two eleccapacities are perfectly balanced. In Fig. 2b, the additional poelectrode capacity, which may be introduced by increasing the tness of the positive electrode or the volume fraction of activeterial, is represented by stretching the positive electrode curvetive to that of the negative electrode. By shifting the two curelative to one another along the capacity axis, we explore diffportions of the positive electrode curve, and therefore differenttive electrode SOC’s and cell potentials, while maintaining thepacity ratio.

To examine capacity-fade and side-reaction phenomena, weintroduce equations for the amount of lithium contained in diffeparts of the cell. At any given time, the amount of lithium contain the negative electrode active material, per unit separator ar

nLi,− =C−L−«−r−x

Ff3g

whereF is Faraday’s constant. Similarly, the amount containethe positive electrode material is

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A819Journal of The Electrochemical Society, 152 ~4! A818-A829 ~2005! A819

Downlo

nLi,+ =C+L+«+r+y

Ff4g

We refer to the sum of these two amounts as the total cyclithium. By our definition, lithium corresponding toy , ymin andx , xmin is called cyclable, even though it is never accessed dcell operation. There is also “noncyclable” lithium containedother parts of the cell, most notably the electrolyte, which cancome cyclable lithium through the occurrence of a side reactionarea-specific quantity of lithium in the electrolyte is given by

nLi,E = cLiX sL−«E,− + L+«E,+ + LS«E,Sd f5g

wherecLiX is the concentration of lithium salt in the electrolyte,LSis the separator thickness, and«E,−, «E,+, and «E,S are the volumfractions of electrolyte in the negative electrode, positive electand separator, respectively. This typically accounts for less thanof the total lithium in the cell. Lithium can also be present inform of reduction or oxidation products that make up passivafilms on the electrodes or are otherwise insoluble. Passivated lipowder is another source of lithium, which becomes cyclable wit is introduced into the active electrode material.

Cyclable lithium can be added or destroyed via side reactioeither the positive or negative electrode. The main reaction anegative electrode is

xLi+ + xe− + N Li xN f6g

and a generic side reaction is

S + e− S− f7g

Similarly, the main reaction at the positive electrode is

Li yP yLi+ + ye− + P f8g

and a generic side reaction is

S S+ + e− f9g

Here S is any species~e.g., solvent, anion, additive, or binder! thatcan be oxidized or reduced.

The total superficial current density at the negative electrod

Figure 1. The SOC operating window, with limits onx andy correspondingto a typical LixC6/Li yMn2O4 cell. The optimum cycle path, a, has a slope−1/zopt = −Dymax/Dxmax, while the cycle path for a mismatched cell, b, haslope of −1/zmis = −Dymis/Dxmis. The shaded areas represent unutilized ptive electrode capacity for the mismatched cell.


t

i = −C−L−«−r−dx

dt+ isn f10g

where isn is the combined current density of side reactions anegative electrode and is negative for the forward direction ofaction 7.

At the positive electrode, the total superficial current densit

i = C+L+«+r+dy

dt− isp f11g

where isp is the combined current density of side reactions apositive electrode and is positive for the forward direction of Rtion 9. From Eq. 3, 4, 10, and 11, the rate of change in cyclithium is

dnLi

dt=

d

dtsnLi,+ + nLi,−d =

isn + isp

Ff12g

We have here neglected the introduction or loss of cyclable litby diffusion between solids. As we demonstrate, such nonelechemical changes can be accounted for by modification of10-12.

Before proceeding with specific examples, we define cell caity, reversible capacity loss, and irreversible capacity loss as the

Figure 2. OCP functions of positive and negative electrodes for~a! a per-fectly balanced cell and~b! a mismatched cell with excess positive electrcapacity. The empirical fits shown here are for Lonza KS6 graphite~fromBelcore!and spinel Mn2O4 ~upper plateau!, taken from the program Dualfoavailable on our website at http://www.cchem.berkeley.edu/;jsngrp/




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A820 Journal of The Electrochemical Society, 152 ~4! A818-A829 ~2005!A820

Downlo

used throughout the paper. We define the “full cell capacity” a

minimum of the negative electrode capacitysC−L−«−r−Dxmaxd and

positive electrode capacitysC+L+«+r+Dymaxd, because the two anot necessarily equal. The “effective cell capacity,” defined asactual discharge capacity of the cell, is lower than the full celpacity if the SOC cycling ranges are limited by running the cella corner of the operating window. Capacity loss~or gain! occursonly if there is a change in capacity from one discharge to the

Figure 3 depicts three classes of capacity loss. The first, rsented by cycle path a, is reversible capacity loss due to imbalside reactionss isn Þ −ispd, which lead to a change in cyclablithium. This class could also be called a “restricted cycling loand is manifested in the SOC operating window as translation ocycle path toward the upper right or lower left corner. This capaloss is a change in the effective cell capacity, but not the fullcapacity. An example is the formation of a passivating film, cathe solid electrolyte interphase~SEI!, at the negative electrode,which cyclable lithium is consumed, and the cycle path movesthe lower left corner of the SOC window~see Fig. 3, path a!. Thslope of the cycle path at the end of this process~provided that thside reaction ceases! is the same as the initial cycle path slope. Sreactions could also add cyclable lithium to the cell, movingcycle path into the upper right corner and resulting in restricycling loss.

Whether this type of capacity loss is truly reversible dependthe nature of the side reactions that cause it~are they reversible?and how the cell is put together~are there lithium reservoirs that creplenish the cyclable lithium that is lost?!. We call this capacity losreversible in the sense that a hypothetical external lithium sourc~orsink! could be used to bring the cell back into balance, althougdo not have the luxury of opening the cell in most practical apcations. We stress that loss or addition of cyclable lithium neeresult in capacity loss at all. Mismatched cells may shift posiwithin limits, without restricting the cycling range of the cell~seethe section on excess positive electrode capacity under CaBalancing!.

The second type of reversible capacity loss~shown in Fig. 3 aarrow b! is reversible capacity loss due to balanced side reacs isn = −ispd, which do not change the amount of cyclable lithiuThis is manifested in the operating window as movement aloninitial cycle path, in which useful energy is not obtained from

Figure 3. Different types of capacity loss encountered in lithium-ion c~a! cycle path after “restricted cycling loss” corresponding to a loss oclable lithium,~b! reversible capacity loss due to balanced side reactiothe two electrodes, and~c! cycle path after irreversible capacity loss assated with the loss of positive electrode active material. All losses are reto an ideally balanced cell, represented by the dashed cycle path.


-d

y

cell ~e.g., during self-discharge of the cell!. This would be encountered when a redox couple in the system removes some lithiumthe negative electrode and inserts a commensurate amoulithium into the positive electrode as it is shuttled across the srator. This type of capacity loss is reversible in a classical sensea temporary drop in the effective cell capacity and should be reered on a subsequent discharge. If the cell is charged, thecharged immediately at a rate high enough to make the fracticurrent due to side reactions insignificant, the full capacity shourealized.~A caveat to this statement is that, at high discharge rhigh cell impedance can limit the extent of discharge.1 Thus, wemay not be able to observe the full capacity of the cell.!

In terms of the variables defined previously, reversible caploss is purely restricted cycling loss when eitherisn or isp is nonzeroand the other is zero. From Eq. 12, it is clear that reversible caploss is entirely of the second variety ifisn = −isp Þ 0. In generaisn Þ −isp, and both side-reaction currents are nonzero, in wcase the cell may undergo a combination of both types of revecapacity loss.

Finally, our definition of irreversible capacity loss applies onlloss of active material, manifested as a change in the cycleslope ~see Fig. 3, path c!. In this case, the full cell capacitdiminished only if the active material is lost from the limiting eltrode. Active material loss has a number of possible causes, ining dissolution, exfoliation, or particle isolation.

This process can be shown more clearly on an extensive-caoperating window. Let

C− = FnLi,− = C−L−«−r− x f13gand

C+ = FnLi,+ = C+L+«+r+y f14g

Then, for a perfectly balanced cell,DC+ = DC− and dC+/dC− =−1. If active material is lost, the size of the window changes, buslope of the cycle path does not change. In Fig. 4, we shoexample of this, in which positive electrode material is lost~DC+decreases!. This effectively limits the negative electrode SOC rwhile maintaining the positive electrode SOC range, and is eqlent to the cycle path change, from the optimum path to cycle pashown in Fig. 3.

In summary, irreversible capacity loss results from a changthe size of the~extensive-capacity! box, while reversible capaciloss comes from a change in cell balance, so that cycling reachconstraint~of y or x! on one electrode before the other, or frmovement along the cycle path due to balanced side reactionscan be compared and contrasted with the traditional view of reible and irreversible capacity loss. If a charged battery is lefopen circuit, it may lose capacity in the sense that a discharge

Figure 4. Extensive-capacity operating window for a perfectly balancedbefore and after 25% of the positive electrode active material is lost. Irepresentation, active material loss is represented by shrinkage of thdow, and, in the absence of side reactions, the cycle path does not mchange slope. The dotted area represents negative electrode capacitstill available but not utilized. This figure corresponds to the change fromoptimum cycle path to cycle path c represented in Fig. 3.




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stand, shows less capacity. If the battery is recharged, some ocapacity may be recovered. The capacity not recovered woucalled irreversible capacity loss.

Irreversible capacity loss~or simply “irreversible capacity”! hasalso been used extensively to mean the difference between thcharge capacity and first discharge capacity, often associatedSEI formation. Our definition of effective cell capacity rejectsuse of the term irreversible capacity in this sense. Dependinhow the cell is designed~i.e., whether it is purposefully mimatched!, SEI formation may or may not result in capacity lossit does not result in irreversible capacity loss unless it actuallylates or destroys active electrode material. We prefer to classifformation as consumption of cyclable lithium.

Capacity Balancing

An optimum balance between negative and positive eleccapacities can be achieved after formation-cycle lithium loss byeral means. We discuss here~i! the building of excess capacity in tpositive electrode,~ii! the manufacture of positive electrodes whigh lithium contentsy . 1d, ~iii! the use of pretreated carbon innegative electrode,~iv! the inclusion of passivated lithium powderthe negative electrode, and~v! the use of a high salt concentrationthe electrolyte. We compare the influence of each method ocell’s specific energy and examine practical difficulties thatpreclude the use of some methods in manufacturing commcells.

Some clarification should be given about capacity balancingthe intended application. Because of transport limitations in caaceous materials, the concentration of lithium may be much hat the particle surface than the average concentration in the paThus, lithium tends to plate at the surface at a much lower aveSOC in high-power applications than in low-power applicationsavoid this plating, which leads to further SEI formation and loscyclable lithium, one should actually design the cell with excnegative capacity. An alternative way of stating this is thatxmaxshould be lowered as the peak power requirements are incrFrom this perspective, all of the following discussion of “capabalancing” is valid for both low-power and high-power cells.

Excess positive electrode capacity.—Lithium-ion cells are typically manufactured with enough excess capacity in the positivetrode to compensate for the amount of lithium that is consumformation of the SEI on the graphitic negative electrode. Thequired increase in mass of positive electrode material, beyondof a perfectly balanced cell, is proportional to the fraction ofclable lithium that is consumed in SEI formation. This increasloading, in units of mass per unit area, can be achieved by increeither the thickness of the composite electrode or the volumetion of active material. Thus, we have

DsL+«+d =C−L−«−r−

C+r+

DxSEI

Dymaxf15g

whereDxSEI is the amount of cyclable lithium consumed duringformation, expressed in terms ofx. This amount depends on actmaterial particle size, carbon type,4,5 chemical and thermtreatment,4,6 electrode porosity,7 and the choice of electrolyte,8 but istypically around 8-15% ofDxmax for graphitic negative electrodes.4-8

Because we can vary thickness and volume fraction independthe required increase in cell mass is not uniquely determined bpredicted SEI-related cyclable lithium loss, but depends on hoadd active material to the cell. We would select the value of ttwo parameters to optimize the cell for a particular application~e.g.,electric vs. hybrid electric vehicles!. Assuming that only the thickness is changed, the mass increase per unit separator area is g


s

t

I

l

.

d.

-

t

g

,

by

Dm = DL+f«E,+rE + «+r+ + s1 − «E,+ − «+drbg f16g

where rE is the electrolyte density andrb is the volume-averagdensity of any polymeric binder or conductive filler contained inpositive electrode.

The specific energy of an electrochemical cell is directly protional to the average cell voltage and inversely proportional tomass of the cell. Thus, the decrease in specific energy of a pro“mismatched” cell below that of a “balanced” cell is

DE = E0

m0DVav

Vav− Dm

m0 + Dmf17g

whereE0 is the specific energy of an idealized balanced cell, witside reactions,m0 is its mass,Vav is the average voltage that tidealized cell supplies between its cycle limits, andDVav is thechange in the average cell voltage between the mismatcheidealized cells. The average cell voltage can be roughly estimfrom the difference in OCPs of the positive and negative electrComparison of Fig. 2a and b reveals that there is a slight differin Vav between the two cells because the cycle limits of the poselectrode have changed.

Table I contains parameters and properties of a typical lithion cell. From the preceding equations, we calculate that themass must be increased by 15.8 g/m2 ~3.50%! to account for filmformation, and that the average cell voltage would increase bymV ~0.759%!, resulting in a specific energy loss of 4.70 Wh~2.65%!. Although the mass of the positive electrode increasesin this case, the cell’s mass increase is much smaller due tadditional weight of the negative electrode, separator, and cucollectors. Furthermore, even though the difference in averagvoltage is small, it can have a big impact on the specific cap

change~i.e., if we neglectedDVav in Eq. 17, DE would be 28%larger!. The mass increase and specific energy decrease fomethod discussed here are summarized in Table II.DVav is zero foreach of the other methods because the ultimate cycle pathsame as that of the idealized balanced cell.

We can quantitatively determine the path traced out in theoperating window for a mismatched cell that undergoes passiv

Table I. Typical cell parameters.

Negative electrode Separator Positive elect

Li smmd 70 25 61.24«i 0.4 0.4«E 0.596 0.4 0.596ri sg/cm3d 2 sN = C6d 4.5 sP = Mn2O4drE sg/cm3d 1.3 1.3 1.3rb sg/cm3d 1.8 1.8 1.8

CismAh/gd 372 sN = C6d 148 sP = Mn2O4dxmax ~or ymax! 0.65 1.0xmin ~or ymin! 0 0.17r i smmd 5 3Lcc smmd 12.5 12.5rcc sg/cm3d 8.95 2.71

Other parameters

DxSEI 0.065rLi smmd 0.5rSEI sg/cm3d 2rLi2CO3

sg/cm3d 2.1rLi sg/cm3d 0.534cLiX smol/Ld 1.0MwLi sg/mold 6.941MwX sg/mold 144.9642sX = PF6d




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with the use of Eq. 1, 10, and 11. In this case we assume thisp= 0 and that the only side reactions at the negative electrodthose that contribute to SEI formation. Combining the equationshave

dy = −1

z

i

i − isndx f18g

in which isn is a function of the negative electrode potential. Tdependence can be garnered from cyclic voltammograms ocharge curves.

The side reaction of interest is reduction of the solvent afilm-solution interface, which we assume to be the result of electransfer through the film. To calculate the side reaction currentsity, we must examine charge transfer at the electrode-film andsolution interfaces, as well as the transport of electrons througfilm. For the sake of this example, let us assume that charge traat the electrode-film interface is fast, so that the equilibrium elecconcentration at that interface is

ce,ef = K expS−F

RTsfe − ffdD f19g

whereK is the ratio of anodic and cathodic rate constants,R is thegas constant,T is the temperature,fe is the potential of the eletrode,ff is the potential of the film, and the subscript ef refers toelectrode-film interface. The current through the film is given a

i film = −FDe

LSEIsce,ef − ce,fsd f20g

whereDe is the diffusion coefficient of electrons in the film,LSEI isthe film thickness, and the subscript fs refers to the film-soluinterface. Note that the current density is inversely proportionthe film thickness, in accordance with previous SEI models.9-11 Wehave assumed here that the potential drop through the filmtherefore migration of electrons, is negligible.

Finally, we assume that solvent reduction at the film-soluinterface is irreversible and follows a Tafel dependence

i fs = − Fkcce,fs expS−acF

RTsff − fsdD f21g

where kc is the cathodic rate constant,ac is the cathodic transfecoefficient, andfs is the potential in the solution.

The superficial side-reaction current density is equal to thetronic current density through the film and at each interface,rected by a factor which relates the electrochemical surface athe negative electrode to the superficial area of the cell. Thus,bining Eq. 19-21, we have

Table II. Changes in cell mass and specific energy.

Idealized cell Method~i! Method ~i

m0 = 474.8 g/m2 Dm sg/m2d 15.8 0.351~%! 3.50 0.0776

Vav0 = 3.86 V DVav smVd 29.5 0~%! 0.759 0

E0 = 177.8 Wh/kg DE0 sWh/kgd −4.70 −0.138

~%! −2.65 −0.077

The methods of capacity balancing entail~i! excess capacity in the posinegative electrode,~iv! passivated Li powder in the negative electrode, a~

the five methods of capacity balancing. Change in specific energy wh


-

r

f-

isn = −

a−L−K expS−F

RTsfe − ffdD

1

FkcexpS−F

RTacsff − fsdD +

LSEI

FDe

f22g

where the dimensionless factora−L− is the electrochemical surfaarea per unit separator area. We assume further that the podrop across the electrode-film interface is independent of thetrode SOC, and is equal to the constantF. If the main reaction ifast and the rate of the side reaction is small compared to ththodic contribution to the main reaction, then we can assume thfilm-solution potential drop roughly follows the OCP of the mreaction, minus the constantF

ff − fs < Usxd − F f23g

whereUsxd is an experimentally determined function of the SONote that the constant,F, in Eq. 23 can be absorbed into the eqlibrium constantK and the cathodic rate constantkc in Eq. 22.

For graphite~see Fig. 2 caption!, we use

Usxd = 0.7222 + 0.13868x+ 0.028952x1/2 − 0.017189x−1

+ 0.001914x−3/2 + 0.28082 expf15s0.06 −xdg

− 0.79844 expf0.44649sx − 0.92dg f24gThe rate of change in film thickness is proportional to the

reaction current density

dLSEI

dt= gisn f25g

whereg depends on the composition and density of the film. Cbining Eq. 10 and 25, we have

dLSEI = − gC−L−«−r−isn

i − isndx f26g

Combining Eq. 22, 23, and 26, we have the first-order diffetial equation

dLSEI =gC−L−«−r−

1 + IFexpSacF

RTUsxdD +

LSEI

bG

dx f27g

where

Method ~iii! Method ~iv! Method ~v! No compensatio

0.336 0.355 7.67 00.0743 0.0785 1.70 0

0 0 0 7.140 0 0 0.185

−0.132 −0.139 −2.97 −17.5

−0.0743 −0.0784 −1.67 −9.83

lectrode,~ii! excess Li in the positive electrode,~iii! pretreated carbon in thh salt concentration. Changes in specific energy are calculated by Eq.

cell is not compensated is given byDE0 = E0s0.9Vav0+DVav

Vav0− 1d.

i!

5

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I =i

a−L−FKkc expSFF

RTsac − 1dD f28g

is the dimensionless total current density, and we define the peter

b =De

kcexpS− ac

FF

RTD f29g

Depending on the exact mechanism for film formation, one mayup with a result somewhat different from Eq. 27.

These equations are readily solved on a spreadsheet by stthrough values ofx. Figure 5 is a representation of passive-fiformation in the SOC operating window, in which we have usedvalues z = 1.1Dxmax/Dymax = 0.861,b = 0.05 nm,ac = 1

2, and

gC−L−«−r− = −155 nm. We assume that the film stops growonce it has reached a thickness of 10 nm. This is an idealizednario; cells may be cycled several times before growth taperand some small degree of side reaction, possibly involving conous film growth, can occur throughout the life of the cell.

The three solid curves in the figure correspond to three diffedimensionless current densities, −0.01, − 0.025, and −0.07.higher current density, the current efficiencys=1 − isn/id is higher,which is why the film does not reach a thickness of 10 nm untinegative electrode has attained a higher SOC.

Regardless of the particular path taken, once the film has stogrowing, the cell cycles along the lower curve in Fig. 5. Noticethis curve has the same slope as the dashed curve, which wotraced out if there were no film formation. The slope is dictatethe invariant capacity ratio,z.

The mismatched cell does not lose any capacity after film fotion; it cycles between the same limits ofx and with the sameDy asin a hypothetical cell that does not undergo passivation. Incases, the negative electrode capacity is the limiting capacity ocell, and this capacity has not been diminished in forming aneven though some cyclable lithium has been consumed. Onebe careful when using the term “irreversible capacity loss” injunction with SEI formation, because in this case, the utilizedpacity of both positive and negative electrode, proportional toDyandDx, respectively, remains constant.

Alternatively, if a cell were manufactured without excess poselectrode capacity, SEI formation would result in capacity loss,

Figure 5. Cycle paths for a mismatched cell with a cyclable lithiuconsuming side reaction at the negative electrode and an initial capacitof 0.861. Intermediate curves a, b, and c are first-charge paths with dsionless superficial current densities of −0.01, −0.025, and −0.07, retively.


-

g

-

d

e

t

cifically restricted cycling loss, as shown in Fig. 6. In this casecycling ranges of both the positive and negative electrode are dished as we run into a corner of the cycling window. Note thacycling window losses,DxL andDyL, are not necessarily equal, bthe capacity losses in the two electrodes are. Even in this cascapacity loss cannot be termed “irreversible,” as it is conceivthat cyclable lithium can be added to either electrode from slithium reservoir within the cell, as described in later subsectThis could negate a portion or all of the capacity loss exhibiteFig. 6.

Because of the cyclable lithium loss, the uncompensated ce10% less capacity than an idealized cell, and 9.83% less spenergy. The specific energy does not scale precisely with thcapacity because there is a slight increase in the average cell v~7.14 mV!. In Table II, we clearly see that any compensateddesign is far superior to the uncompensated cell design.

Excess lithium in the positive electrode.—Rather than increathe entire positive electrode mass, it has been demonstrateenough additional lithium can be inserted into the active mabefore cell assembly to account for the lithium consumed duringformation at the negative electrode.12 For instance, if it is believethat 10% of the cyclable lithium would otherwise be consumedpositive electrode material could be fabricated with a stoichiomof y = 1.083. The operating window and cycle path for this cellshown in Fig. 7.

The benefit of this technique is that much less mass muadded to the cell to achieve post-formation balance. The addilithium mass is

DmLi =C−L−«−r−DxSEIMwLi

Ff30g

whereMwLi is the molar mass of lithium. From the parameters liin Table I, the additional lithium mass is 0.351 g/m2 ~0.0776%!Because only lithium is added, this is equal to the additionalmass. Thus, according to Eq. 17~with DVav = 0!, the specific energdecrease is only 0.138 Wh/kg~0.0775%!, or 2.93% of the decreafor the method involving excess positive electrode active mate

One drawback of using this method is that some positivetrode materials undergo phase changes when additional lithiinserted in the host matrix. For instance, LiyMn2O4 undergoes JahTeller distortion wheny increases above 1,2,13-15 although this dis

--

Figure 6. Cycle paths for an initially balanced cellsz = 0.65/0.83dthat goesout of balance and loses capacity due to a cyclable lithium-consuminreaction at the negative electrode. Intermediate paths a, b, and c redimensionless superficial current densities of −0.01, −0.025, and −0.0spectively.DxL andDyL represent the losses in negative electrode andtive electrode cycling range, respectively.




has

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tortion is diminished when the original starting materialy . 1.05.2,12,16 Strain between the surface tetragonal Li2Mn2O4phase and bulk cubic LiMn2O4 phase can lead to particle fractuand loss of particle contact.15 Furthermore, some authors suggthat LiyMn2O4 may be more susceptible to manganese dissolutilow potentials when trace acid is present in the electrolyte17-20

There is evidence that the dissolution also occurs when the cfully charged.21,22

Pretreated carbon.—Another means of avoiding cyclablithium consumption in the cell is to preform the negative electrthat is, form an SEI on the active material before assemblingcell. This results in a cell of slightly higher mass than a balancedthat is not preformed. The additional mass depends on the dand thickness of the SEI and the electrochemical surface areanegative electrode

Dm = rSEIa−L−LSEI f31g

wherea− = 3«−/r− is the electrochemical surface area per unitume of the composite electrode andr− is the radius of active matrial particles. For the set of parameters we consider here, theincrease is 0.336 g/m2 ~0.0743%!, and the decrease in specificergy is 0.132 Wh/kg~0.0743%!. As there are no film-formation sreactions in the preformed cell, the operating window would beof Fig. 1, with cycle path a.

Although this appears to be an attractive solution to the loscyclable lithium, there are several practical considerations thaclude its widespread implementation. If the carbon particlessomehow preformed before a composite negative electrodepressed, the electronically insulating SEI surrounding them wprevent the establishment of a conducting matrix. Thereforemust first build a composite uncharged electrode, then chargetemporary cell, and finally remove the preformed electrode fromcell and use it as the negative electrode of a perfectly balacommercial cell. This additional processing step could prove coand the increase in specific energy over the method of excesstive electrode capacity may not offset the additional cost.

Lithium powder.—A fourth alternative was presented at a recsymposium in Berkeley. FMC Corporation has been able to proa passivated lithium powder by exposing grains of lithium meta carbon dioxide atmosphere.3 The powder does not corrode inand can be included during the fabrication of composite neg

Figure 7. Cycle paths for a perfectly balanced cell that is fabricated w10% excess of cyclable lithium in the positive electrode and loses cadue to a cyclable lithium-consuming side reaction at the negative elecIntermediate paths a, b, and c represent dimensionless superficialdensities of −0.01, −0.025, and −0.07, respectively.


ye

s

s

i-

electrodes. Figure 8 is a diagram of a portion of the compelectrode. Because the lithium particles are passivated, they ain electrical contact with the active electrode material. HoweThere is a pathway for lithium transport through the Li2CO3 SEI.The excess lithium contained in the powder can thus becomclable lithium if it diffuses into the graphitic host material@Fig. 8a~i!#.

The additional mass requirement should be much less thaof the method of excess positive electrode capacity and is depeupon the total surface area of the lithium particles and thicknethe Li2CO3 film surrounding them. The additional mass of the aclithium is given by Eq. 30. The ratio of Li2CO3 mass to lithium masis approximately

mLi2CO3

mLi=

3rLi2CO3LLi2CO3

rLirLif32g

where rLi is the radius of the lithium particles andLLi2CO3is the

thickness of the Li2CO3 film. Thus, the total additional mass is

Dm =C−L−«−r−DxSEIMwLi

FS1 +

3rLi2CO3LLi2CO3

rLirLiD f33g

Given the parameters listed in Table I, the mass of the cell wincrease by 0.355 g/m2 ~0.0785%!, and the specific energy wodecrease by 0.139 Wh/kg~0.0784%!.

There are several mechanisms for transport of lithium tonegative or positive electrode material. Depending on the timestants and driving forces that are involved, one mechanismdominate, or they could occur simultaneously. One such mechis diffusion of lithium from the powder, through the lithiumcarbonate coating, and into the graphitic host material. Presumthe lithium diffuses more rapidly into the graphite when it is vacIf this is the case, the transport could take place as soon acomposite electrode is manufactured, resulting in a preformedmuch like that examined in the method of pretreated carbonlithium powder method has an advantage over the method otreated carbon only if the additional step of manufacturing pvated lithium powder is more economical than the other meapreforming the negative electrode. The difference in specific enbetween the two cells is small.

Perhaps the diffusion process is slow enough that it doeproceed fully until the cell is already in use. In this case, we mmodify Eq. 10 to account for diffusion

.t

Figure 8. Schematic of lithium transport in a porous negative electrodecontains passivated lithium powder. Lithium can~a! enter the negative eletrode by ~i! diffusing through the lithium-carbonate film into an adjacgraphite particle or~ii! dissolving into solution and intercalating into tgraphite, with the negative electrode matrix providing an electronic pathor ~b! enter the positive electrode by dissolving and migrating acrosseparator. The former may occur under any condition, including open cand is driven by the difference in electrochemical potential betweelithium metal and LixC6. The latter can occur only during cell discharge, othere is a corroding side reaction at each electrode.




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i = − C−L−«−r−dx

dt+ isn − NLiFa−L− f34g

Here, NLi is the flux, per unit electrochemical surface arealithium from the powder to the active material. It depends, amother things, on the area of contact between the powder anactive material.

If the Li2CO3 coating has a high enough electrical conductithat the lithium metal can be electrochemically oxidized, the lithdissolves into solution. A shorted electrochemical cell can be sbetween the lithium metal and graphite, in which electrons areried through the conductive matrix, including lithium particgraphite particles, and current collector@Fig. 8a~ii!#. This may occuduring charge or discharge, or at open circuit.

Alternatively, the dissolved lithium could migrate to the posielectrode during discharge~Fig. 8b!. Then the addition of cyclablithium to the system may occur during the first discharge, rathan before the first charge. The dissolved lithium could be intelated into the positive electrode material, thereby rebalancingcell’s capacity after SEI formation during the charge step. Incase, the negative electrode side reaction is dissolution of LiEq. 18 becomes

dy = −1

z

i

i − iLidx f35g

whereiLi is the current density due to lithium dissolution. Negleing concentration gradients in the solution, the rate of dissolushould be relatively constant. Therefore, we get a straight line ioperating window during discharge, with a slope that differs f−1/z by i /s i − iLid. Once the lithium has completely dissolved,slope returns to −1/z. If the proper amount of lithium powder hbeen added to the negative electrode, the optimum cycle preached, and the capacity of the cell is perfectly balanced. Thepath traced out during cell operation in this case is depicted in9a.

Clearly, if we have too much lithium powder, the cycle povershoots the optimal path and runs into the upper right cornthe operating window, as shown in Fig. 9b. This effectively limthe cycling range in bothx and y, reducing the capacity of boelectrodes equally. Thus, the cell can experience capacity losswhen cyclable lithium is added to the electrode material. This cterintuitive example underscores the confusion that can resultpacity is not defined carefully.

This capacity loss is considered reversible, as there is thesible existence of a side reaction that consumes cyclable lithiuone does exist, it is likely for the cycle path to again overshoooptimal path and run into the lower left corner of the operawindow. This sort of “capacity fade” is usually exhibited in lithiuion cells. Even if no such side reaction exists, we still refer toprocess that limits capacity through a change in cyclable lithiu“reversible capacity loss.”

High lithium-salt concentration.—We have already suggestthat there is insufficient lithium in the electrolytic solution to replthe cyclable lithium that is lost during SEI formation.1 Dependingon the cell design and type of SEI that is formed, there maenough to balance the cell, but not without severely diminishingconductivity of the electrolyte. A technique that may be tried iincrease the concentration of lithium salt in the solution by enoto compensate for the loss of cyclable lithium that occurs duformation.

There are several immediate difficulties with this approach. Fincreasing the salt concentration may decrease the conductivthe short term. Lithium-salt concentrations between 1 and 2 M aretypically chosen because they result in a conductivity maximu23

Too high a concentration can increase the solution’s viscositypoint where the conductivity is adversely affected.

It would seem that if the lithium in solution is intercalatedcyclable lithium, this issue would be resolved. But for cycla


n

-

lithium to be created, a side reaction is required, as per EqOxidation of the solvent or some additive at the positive electwould yield a cation that could replace the lithium removed fsolution. This probably would not help decrease the solution’scosity, unless the new salt precipitated as a solid. Even then, itaffect transport by plugging pores or forming surface layers.

Another possibility is that the anions themselves are oxidizeeven intercalated irreversibly into a host material in one of thetrodes. A number of electrochemically active polymers can bedized with the concomitant insertion of anions,24 including poly~3-butyl thiophene! ~P3BT!, which has been used in lithium-ion cefor overcharge protection.25,26 Typical lithium-ion battery anionsuch as PF6

− can be inserted into P3BT, which is oxidized at potials above the normal operating potential of the positive electThus, additional cyclable lithium could be generated at the negelectrode while anions are inserted into P3BT in the positivetrode if the battery is overcharged. However, as anion insertionthe polymer is reversible, so is the addition of cyclable lithium anegative electrode.

In any case, the system becomes much more complicatedincreasing the salt concentration is chosen as a strategy to bthe cell. The components must be chosen such that the desire

Figure 9. Cycle path for a perfectly balanced cell, in which cyclable lithlost during film formation is replaced by passivated lithium metal thatsolves during discharge. Plot~a! shows the cell returning to the optimal pwhen the correct amount of lithium powder is contained in the negelectrode, while~b! shows the cycle path running into the upper right cowhen there is too much lithium powder. The dashed line in~b! is the opti-mum cycle path.




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reactions occur, and if they do occur, there must be some mechfor stopping them once the lithium salt falls to the desired contration. One can imagine that tailoring the negative electrode SEendeavor that has been the focus of intense research over thseveral decades, would be a simpler task. Furthermore, it is unexactly how much extra mass must be added to the cell to acthe appropriate side reactions.

In the ideal case, only the additional salt must be added. Fosalt Lin+Xn−, the additional mass would be

Dm =C−L−«−r−DxSEIMwLi

FS1 +

n−MwX

n+MwLiD f36g

wheren+ andn− are the numbers on cations and anions, respectinto which the salt dissociates. Even this minimum value is larg

a heavy anion like PF6−, for which Dm is 7.67 g/m2 ~1.70%!andDE

is −2.97 Wh/kgs−1.67%d. These values are smaller than, but ofsame order of magnitude as, those calculated for the methexcess positive electrode capacity.

A specific side reaction mechanism is required to determincycle path traced out in the operating window. Just imagining apossible paths makes evident the challenge of ending up odesired optimal path. Using lithium powder is much simpler becthe side reaction~or diffusion process! can be allowed to continuuntil the reactant~Li! is completely consumed. In the methodhigh salt concentration, we do not want to consume all of the lithsalt.

Loss of Active Material

So far we have examined processes that involve changesamount of cyclable lithium, which can lead to cell imbalancereversible capacity loss. We have already defined “irreversiblpacity loss” as capacity loss associated with the loss of activetrode material. In terms of an electrode’s capacity, this loss of amaterial manifests itself as a reduction in active material volfraction. The full capacity of each electrode can be obtainedEq. 13 and 14

DC− = C−L−«−r−Dx f37g

DC+ = C+L+«+r+Dy f38gThus, any decrease in the volume fraction of active material yieproportional decrease in capacity.

We can illustrate the effect this type of capacity loss has oncycle path through a few examples. We first examine positivetrode material loss while the cell is held in the discharged state~i.e.,y = 1!. Two possible electrochemical reactions are

2LiP + A+ → Li2P + P+ + A f39g

where A+ is a cation in solution that can be reduced~e.g., H+!, and,for P = Mn2O4

4H+ + 2LiMn2O4 → 2Li+ + Mn2+ + 3l-MnO2 + 2H2O f40gThe latter is known as Hunter’s reaction, and it is a well-stumechanism for manganese dissolution in the presence ofacid.2,18,22,27,28In Eq. 39, the amount of cyclable lithium in tpositive electrode remains constant~assuming the lithiumcontaining product can still deintercalate lithium!, but y increaseabove 1. The loss of active material results in an increase islope of the cycle path, according to Eq. 1. During charge, it wappear as though no capacity has been lost, but on the subsdischarge, the capacity is limited by the maximum value ofy in theoperating window. One could conceivably discharge the cell athis value and recover the full capacity, but this may have a derious impact on the intercalation host~e.g., via Jahn-Teller distotion!. The cycle path for this type of capacity loss is shown in10.

In Reaction 40, both the active material and cyclable lithiumconsumed. The value ofy does not change, but the cell still falls o


str

f

e

-

e

nt

of balance, and the slope of the cycle path becomes steeper thoptimal slope. Upon cycling, the cell follows the cycle path in11. In this case, there is no chance of going outside the opewindow without further side reactions, which replace the cycllithium that has been lost, resulting in the cycle path drifting toright. Notice that the value ofDx over which the negative electrois cycled may be the same regardless of whether cyclable lithilost along with the active material. However, the actual limitsxare different, meaning the cell potential that is achieved masomewhat different in the two cases.

A variation on Reaction 39 which has been proposed for lithdissolution is2,29

12LiMn2O4 → Li2MnO3 + 5Li2Mn4O9 + 3Mn2+ + 6e− f41g

Li2MnO3 is electrochemically inert and Li2Mn4O9 has very limitedcapacity at the potential over which the positive electrode is cytherefore, cyclable lithium is effectively lost from the positive etrode in the dissolution process. However, because electrongenerated at the positive electrode, there must be a commenintercalation of lithium into the negative electrode. Because 12

Figure 10. Cycle path of a cell that loses positive electrode active matbut maintains its cyclable lithium, while it is stored in the discharged sThe dashed line is the optimum cycle path.

Figure 11. Cycle path of a cell that loses positive electrode active matas well as cyclable lithium, while stored in the discharged state. The dline is the optimum cycle path.




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ums are consumed and only 6 electrons are generated in Re41, roughly half of the cyclable lithium lost at the positive electris recovered at the negative electrode. The Mn2+ ions replace the Li+

that is intercalated from the electrolyte into the negative electOverall this active material consumption leads to an increascycle path slope that limits that charge capacity of the negelectrode, accompanied by a shift inx that divides the loss into twnearly equal portions of the negative electrode capacity. Becavery small amount of Li in the Li2Mn4O9 product is still cyclablelithium, y actually increases slightly, and the cycle path is shiftethat it passes just to the right of the operating window’s center.scenario is shown in Fig. 12.

If we combine the preceding side reaction~Reaction 41!with thepositive electrode main reaction~Reaction 8!, we can eliminate telectrons to yield

6

yLi yMn2O4 + 6Li+ → Li2MnO3 + 5Li2Mn4O9 + 3Mn2+

+6s1 − 2yd

yMn2O4 f42g

At the negative electrode, manganese cations are reduced

Mn2+ + 2e− → Mn f43gCombining this with the negative electrode main reaction~Reaction6!, we have

2

xLi xC6 + Mn2+ → 2Li+ +

2

xC6 + Mn f44g

We can see that, at open circuit, Reactions 42 and 44 descself-discharge of the cell that involves the dissolution of activeterial in the positive electrode. Overall, there is one lithium alost ~if we assume Li2MnO3 and Li2Mn4O9 to be inert! for eachMn2O4 unit that is consumed in Reaction 42. It is interesting talthough this set of equations describes a self-discharge procesnecessary that some lithium be present in the positive electrode~i.e.,the positive electrode cannot be fully charged!. We can say tha

Figure 12. Cycle path~represented by the solid line! of a cell that losepositive electrode active material via an electrochemical side reaction bthe cell is charged, according to Reaction 41. The dashed line is the opcycle path; the dotted lines indicate the cycle paths of~a! a cell that losepositive electrode active material and cyclable lithium in the dischastate,~b! a cell in which the lithium-containing products in Reaction 41completely inert, and~c! a cell that loses the same amount of positive etrode active material in the charged state. These paths are chosenparticular capacity ratio to serve as a frame of reference for the solidpath, which would shift to the left or the right depending on the ratiLiMn2O4 that is lost to the electrons that are generated.


n

a

a

is

cyclable lithium is lost ify . ymin, whereas ify = ymin, the lithiumthat is lost is inert. The overall SOC of the positive electrodenot change during this process, whereas the SOC of the neelectrode decreases, driving the potential higher. Thus, thepath is shifted toward the left within the operating window to cpensate for the loss in positive electrode active material, as shoFig. 13.

Line a of Fig. 14 is the cycle path for a cell in which positelectrode active material is consumed when the cell is in the chstate. This could occur if oxidation of solvent at high potenforms a film that electronically isolates active material or if disstion proceeds in the charged cell. As with Reaction 39, no cyclithium is lost in this case, and although the slope of the cycleincreases, the cell could recover its full capacity if lithium winserted into the positive electrode abovey = 1.

Electronic isolation may also occur in the negative electrespecially in the charged state. In this case, both active matericyclable lithium contained therein are made unavailable. Theof the cycle path decreases from the optimal value, as shown i

a

Figure 13. Cycle path followed due to manganese dissolution frompositive electrode and deposition at the negative electrode, accordingactions 42 and 44. The process illustrated here occurs at open circuiy . ymin. The dashed line is the optimum cycle path.

Figure 14. Cycle paths for capacity loss due to electronic isolation of amaterial in~a! the positive electrode and~b! the negative electrode, while tcell is in the charged state. The cycle path directions are given as~i! firstcharge,~ii! first discharge, and~iii! second charge.




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14, line b. Because cyclable lithium is lost, there is no chancovershooting the maximum value ofx unless side reactions carry tcycle path upward~in the direction of positivey!. The danger incharging past the upper limit onx is that lithium metal can depositthe electrode surface, leading to rapid reduction of the solvenaltering the character of the SEI.

Loss of active material in an actual cell is probably even mcomplicated than in the examples discussed already. Active mamay be lost as part of a side reaction that occurs during chardischarge. Take, for instance, Reactions 41 and 43, in whichreactions decrease the amount of active material in the positivetrode. LetRdis be the net rate of Reaction 41. In terms ofRdis, therate of the side reaction is

isp = 6FRdis f45g

and the corresponding decrease in the volume fraction of amaterial is

d«+

dt= − 12

Mw+

L+r+Rdis f46g

whereMw+ is the molecular weight of the active material. Hen

d«+

dt=

d«+

dx

dx

dt= − 2

Mw+

FL+r+isp f47g

Combining this equation with Eq. 10, we have

d«+

dx= 2Isu

«+0

z0f48g

wherez0 is the initial capacity ratio,«+0 is the initial volume fractionof active material in the positive electrode,Is = isp/s i − isnd, andu

= Mw+C+/F. We could assume thatisn = −isp ~i.e., every Mn atomthat dissolves from the positive electrode is plated at the negelectrode!, although this is not necessarily true, because someganese cations can remain in solution. Equation 12 implies thaclable lithium is not consumed whenisn = −isp. However, Eq. 12does not account for changes in active material mass, and arigorous derivation shows that

dnLi

dt=

isn + isp

F+

C+L+r+y

F

d«+

dt+

C−L−r−x

F

d«−

dtf49g

To minimize the complexity of our example, let us assume thIsis constant. Then Eq. 48 can be integrated to yield

«+ = «+0F1 + 2Isu

z0sx − x0dG f50g

Substituting this expression into Eq. 11, we have

dy

dx= −

1 + Isz0 + 2Isusx − x0d

f51g

where we have again made use of Eq. 10 to eliminate thederivative. Integrating once more yields

y = y0 −1 + Is2Isu

lnF1 + 2Isu

z0sx − x0dG f52g

Figure 15 shows the cycle path that is followed according to Eqwith u = 0.96 ~using the molecular weight and upper-plateaucific capacity of Mn2O4! and Is = 1/3 ~i.e., isp = −isn = i /2!. As in-dicated in the figure, we assume that the side reaction sudceases, after which the cycle path obeys the relationship

dy

dx= −

1

zf53g

z being the new capacity ratio after Mn2O4 has stopped dissolvinThe point at which the side reaction stops has been chosentrarily. Because MnO dissolves more readily at higher tempe
2 4

lr

-

-

e

i-

tures, this might correspond to abruptly lowering the temperatuthe cell.

The elementary but important point brought forth by theseamples is that a change in the amount of active material leadpermanent change in the slope of the cycle path, whereas sidetions that do not consume active material change the slopewhile they occur. If such a side reaction stops, the slope returthat dictated by the capacity ratio of the cell. Side reactions thconsume active material create a combination of temporary anmanent slope changes.

Conclusions

Capacity loss occurs whenever the cycle path is moved fuaway from its optimum, either through changes in cyclable lithor loss of active material. The former is termed “reversible,”cause side reactions or appropriate lithium sources and sinkhelp rebalance the cell. The capacity loss in this case manifestsas the cycle path running into a corner of the SOC operatingdow, the lower left if cyclable lithium is consumed, and the upright if cyclable lithium is generated. Loss of active materiatermed “irreversible capacity loss”, although it is possible to mtain the initial capacity if the cell is cycled beyond its normal oating limits. This capacity loss manifests itself as a change inslope of the cycle path.

There are several techniques that may be used to replace cylithium that is lost when forming the SEI during the first sevcycles. Because the current collector’s mass is a significant pof the overall cell mass, compensating for a 10% cyclable lithloss increases the cell mass by at most a few percent. Howaddition of lithium by increasing the active mass of the poselectrode or the concentration of electrolyte has a much greatepact on the mass and specific energy than the other methodcussed. Preforming the surface of the negative electrode, alithium powder, and increasing the initial lithium content in the ptive electrode material all have a negligible influence~ca. 0.08%increase for a typical cell! on the cell mass. The main practiconsideration in choosing among these latter methods is the cadditional steps in the cell fabrication process.

Acknowledgments

The authors thank Venkat Srinivasan for helpful discussiongarding high-power applications and passivated lithium powThis work was supported by the Assistant Secretary for Energ

Figure 15. Cycle path that is followed according to Eq. 52, withu = 0.96and Isp = 0.5. As indicated in the figure, we assume that the side reasuddenly ceases, after which the cycle path obeys the relationshipdy/dx=−1/z. The dashed line is the optimum cycle path.




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ficiency and Renewable Energy, Office of FreedomCAR and VeTechnologies of the U.S. Department of Energy under contracDE-AC03-76SF00098.

The University of California at Berkeley assisted in meeting the pubtion costs of this article.

List of Symbols

a specific electrode area, cm2/cm3

ce concentration of electrons in the SEI, mol/m3

cLiX concentration of salt in the electrolytic solution, mol/m3

C separator area-specific capacity of electrode, mAh/cm2

C mass-specific capacity of active material, mAh/gDe diffusion coefficient for electrons in the SEI, m2/s

E specific energy of cell, Wh/kg

E0 specific energy of idealized balanced cell, Wh/kgF Faraday’s constant,F = 96,487 C/moli superficial current density, A/cm2

i film electronic current density through the SEI, A/cm2

isn superficial current density of all side reactions at the negative electroA/cm2

isn0 exchange current density of a side reaction at the negative electroA/cm2

isp superficial current density of all side reactions at the positive electrodA/cm2

I dimensionless total current densityIs dimensionless side-reaction current densitykc cathodic rate constant, m/sK equilibrium constant, mol/m3

L electrode thickness, cmLSEI SEI thickness, nm

m cell mass, g/cm2

m0 mass of idealized balanced cell, g/cm2

Mw molar mass, g/molnLi amount of cyclable lithium in cell~or electrode!, mol/cm2

N negative electrode chemical formulaNLi flux of Li from powder to active material, mol/cm2-s

P positive electrode chemical formular particle radius, cm

Rdis rate of positive electrode dissolution, mol/cm2-sS electrolyte speciest time, s

U OCP, VV potential, V

V0 equilibrium potential for SEI formation, VVav average cell potential, V

x stoichiometry of lithium in negative electrodey stoichiometry of lithium in positive electrodez negative-to-positive capacity ratio

z0 initial capacity ratiozopt optimum capacity ratio

Greek

ac cathodic transfer coefficient of side reactionb proportionality constant, nmg proportionality constant, cm3/C« volume fraction

«+0 initial volume fraction of active material in positive electrodeh surface overpotential, Vu dimensionless positive electrode capacityn electrolyte dissociation numberr density of active electrode material, g/cm3

F fe − ff, Vfe potential of the electrode, Vff potential of the film, Vfs potential of the solution, V

Subscripts

b polymeric binder/conductive filler property


ct charge transferE electrolyte propertyef electrode-film interfacefs film-solution interfacei species index

Li lithium propertyLi2CO3 lithium carbonate property

max maximum stoichiometry, dictated by electrode properties, or maximustoichiometric range, dictated by electrode properties

min minimum stoichiometry, dictated by electrode propertiesmis stoichiometric range for mismatched cell

S separator propertySEI SEI property

X anion property+ positive electrode property− negative electrode property

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