Modelingof GalvanostaticChargeand Discharge of the Lithium:Polymer:InsertionCell

8/16/2019 Modelingof GalvanostaticChargeand Discharge of the Lithium:Polymer:InsertionCell

1/8

1526

J. Electrochem. Soc. Vol. 140 No. 6 Jun e 1 993 9 The Electrochemical Society Inc.

also may be suitable for the removal of oxide impurities

from other alkali chloride melts such as LiC1-KC1 and

KC1-NaC1.

Acknowledgment

This work was supported by the A ir Force Office of Sci-

entific Research, Grant No. 88-0307.

Manus cript subm itte d Nov. 30, 1992; revised man uscr ipt

received Feb. 26, 1993.

The University of Tennessee assisted in meeting the pub-

lication costs of this artic le.

REFERENCES

1. T. M Laher, L. E. McCurry, and G. Mamantov,

Anal.

Chem.

57, 500 (1985).

2. B. Gilbert and R. A. Osteryoung, J. Am. Chem. Soc.

100, 2725 (1978).

3. K. Zachariassen, R. W. Berg, and N. S. Bjerrum,

This

Journal 134, 1153 (1987).

4. I-W. Sun, E. H. Ward, and C. L. Hussey,

Inorg. Chem.

26, 4309 (1987).

5. I-W. Sun, K. D. Sienert h, an d G. Mamantov, This Jour-

nal

138, 2850 (1991).

6. I-W. Sun, K. D. Sienerth, and G. Mamantov, Unpub-

lished results.

7. B. Gilbert, S. D. Williams, and G. Mamantov,

Inorg.

Chem.

27, 2359 (1988).

8. P.A. Flower s an d G. Mamantov,

This Journal

136, 2944

(1989).

9. P. A. Flowers and G. Mamantov,

Anal. Chem. 61

190

(1989).

10. P A. Flowers and G. Mamantov,

ibid.

59, 1062 (1987).

11. J. H. Taylor, W. S. Benedict, and J. Strong, J. Chem.

Phys.

20, 1884 (1952).

12. A. H. Nielsen and R. J. Lagem an,

ibid.

22, 36 (1954).

13. N.B. Colthu p, L. H. Daly, an d S. E. Wiberley, Introduc-

tion to Infrared and Raman Spectroscopy

pp. 43-45,

Academic Press, New York (1975).

14. J.-P. Schoebrechts, P. A. Flowers, G. W. Hance, and G.

Mamantov,

This Journal

135, 3057 (1988).

15. G.-S. Chen, A. G. Edwards, and G. Mamantov,

ibid.

Submitted.

16. R. E W. Bader and A. D. Westland, Can. J. Chem. 39,

2306 (1961).

17. B.J. Brisdon , G. W. A. Fowles, D. J. Tidmarsh, a ndR . A.

Walton, Spectrochim. Acta 25A, 999 (1969).

18. M. Volloton and A. E. Merbach,

Helv. Chim. Acta

57,

2345 (1974).

19. I-W. Sun and C. L. Hussey, Inorg. Chem. 28, 2731

(1989).

20. E. Hond rogia nnis and G. Mamantov, Unpubl ishe d re-

sults.

21. C. K. Jorge sen,

Mol. Phys.

2, 309 (1959).

22. E. Tho rn-Cs anyi and H. Timm, J.

Mol. Cat.

28, 37

(1985).

Modeling of Galvanostatic Charge and Discharge

of the L ithium Polymer Insertion Cell

Marc Doyle, Thomas F. Fuller, and John New ma n

Department of Chemical Engineering University of California and Materials Sciences Division

Lawrence Berkeley Laboratory University of California Berkeley California 94720

ABSTRACT

The galvanostatic charge and discharge of a lithium anode/solid polymer separator/insertion cathode cell is modeled

using conce ntrat ed solution theory. The model is general enough to includ e a wide range of polymeric separat or materia ls,

lithium salts, and composite insertion cathodes. Inserti on of lithium into the active cathode material is simulated using

superposition, thus greatly simplifyi ng he numer ical calculations. Variable physical properties are permit ted in the model.

The result s of a simu lat ion of the c harge /disc harge b eha vio r of the LiIPEOs-LiCF~SO31TiS2 system are pre sente d. Criter ia

are established to assess the impor tanc e of diffusion in the solid mat rix a nd tra nspo rt in the electrolyte. Considerat ion is

also given to various procedures for optimization of the utilization of active cathode material.

There has been interest recently in the use of thin-film

rechargeable batteries for electric-vehicle applications.

Several groups have developed and tested rechargeable

cells incorporating a lithium anode, solid-polymer-elec-

trolyte separator, and a composite cathode consistin g of an

insertion material mixed with the solid polymer elec-

trolyte.

T

Generally, large energy densities are pre dicted for these

cells from theoretical calculations. In addition, the re-

versibil ity and large selection of materia ls makes inse rtion

compounds attrac tive for the cathodic process. 5 Anoth er

advant age of this system is the relative safety and dur abi l-

ity afforded by the solid separator i n comparison to a liquid

electrolyte. 6 The high reactiv ity of the lith ium a node m ay

be a significant problem; however, there is much evidence

that a protective film is formed on the electrode similar to

that in nonaqueous liquid electrolytes/ To date, experi-

men tal cells reported in the literat ure are small. The devel-

opment of a detailed mathematic al model is important to

the design and optimization of lithium/po lymer cells and

critical to their scale-up.

There have been few previous modeling efforts of thin-

film solid-state b atte ry systems using insert ion electrodes.

* Electrochemical Society Student Member.

** Electrochemical Society Active Member.

West et al. 8 treated insertion into the composite cathode

with porous electrode theory, modeling the electrolyte and

active cathode material as superimposed continua without

regard to microscopic structure (the separator was not in-

cluded). Transport in the liquid electrolyte phase was de-

scribed with dilute solution theory includin g diffusion and

migratio n. The insertio n process was assumed to be diffu-

sion-limited, and hence charge-transfer resistance at the

interface between electrolyte and active material was ne-

glected in West s model. Data for the o pen-cir cuit potential

vs. the am ount of lithium inserted were used to relate the

surface c oncentration of lithium in the solid matrix to the

electrolyte concent ratio n in the so lution phase.

While dilute solution theory has many useful applica-

tions, an incorrect number of transport properties is

defined because only interactions between the solute and

the solvent are considered. Investi gation s of the mech anism

of conduction in these electrolytes have concluded that ion

pair ing a nd io n association are importan t. 9 Thus, the more

complete concentrat ed solution theory is appropriate.

Furthermo re, the more rigorous theoretical frame work of

concentrated solution theory provides greater flexibility

over dilute solution theory in accounting for volume

changes and polymer flow. One also may wish to include

additional species in the polym er phase such as a low-

) unless CC License in place (see abstract).ecsdl.org/site/terms_useaddress. Redistribution subject to ECS terms of use (see 140.112.71.49ownloaded on 2015-08-16 to IP

http://ecsdl.org/site/terms_usehttp://ecsdl.org/site/terms_usehttp://ecsdl.org/site/terms_usehttp://ecsdl.org/site/terms_use


2/8

Z Electmchem. Soc.,Vol. 140,

No. 6 June 1993 9 The ElectrochemicalSociety lnc.

c o m p o s i t e

c a t h o d e

lithium foil

anode

polymer

electrolyte

8s

x O

c

Fig. 1. Lithium /polymer cell sandwich consisting of lithium-fo il

anode solid polymer electrolyte and composite cathode.

molecular-weight polymer phase or a second lithium salt.

Treating these complexities is straightforward with con-

centr ated soluti on theory.

There are limited data available on the kinetics of the

charge-transfer reaction at the surface of insertion com-

pounds. Pollard and NewmanTM have shown that the as-

sump tion of infinitely fast kinet ics for a porous electrode

leads to a spike in the local current density at the separa-

tor/cathode interface at short times. Assuming infinitely

fast kinetics also changes the natur e of the governing equa-

tions. We wish to keep the mode l general so that the kine t-

ics of the cathode can be include d when d ata are avail able.

Consequently, a charge-transfer resistance is assumed in

the present model.

An important objective of this model is to be general

enough to include the full range of materials currently used

in lithium/p olymer/ins ertion systems. This generality

should allow us to assess the performance of this class of

battery systems in general and to establish guidelines for

their optimization. Also, when data on a particu lar system

are available, we can provide specific guidelines on cell

configuration, assess the effects of kinetic and transport

limitations, a nd evaluate the per form anc e of the system.

Model Development

We have chosen to mode] the galvanostatic charge/dis-

charge beha vior of the cell sand wich show n in Fig. 1. We

consider one-dimensi onal transport from the lithium an-

ode through the polymer separator into the composite

cathode. It is desired that the most important pheno mena

be treated witho ut introducing und ue complexity. Conse-

quently, film formati on at the lithium/ polymer interface

and volu me change s dur ing operation are ignored.

The separator consists of an inert pol yme r material that

acts as the solvent for a lithium salt. Several po ly me r an d

salt combi nation s with widely varying properties have

bee n consi dered in the literature, u Transpo rt in the separa-

tor is mod ele d with co ncentrated solution theory, assu min g

a binary electrolyte and a single-phase pol yme r solvent.

Th us the electrical conductivity, the transferen ce nu mb er

of the lithium ion, an d the diffusion coefficient of the

lithium salt characterize transport in the polyme r. Since

each of these properties is concentration dependent, vari-

able physical properties are treated in the mode l. This

macro scopi c approach, u sing concentrated solution theory

an d variable physical properties, al lows one to deal rigor-

ously with the transport p henome na.

In concentrated solution theory New man , Ref. 12), the

driving force for ma ss transfer is the gradient in electro-

chemical potential

C,V~lb, = E g, j( vj -- Vi) [1]

1527

where the K, (Kii = Kii) are f rict ion al coefficients desc ribi ng

interactions between species i and j. For a solution of a

binary salt (e.g., LiX) plus solvent (polymer), because of the

Gibbs-Duhem equation, we have two independent trans-

port equations of the form given in Eq. 1. If we use the

polyme r as the reference species and take its velocity to be

zero, we can invert these equations to obtain

it+~ [2]

N = -v DVc z F

and

t o

N_ = - v_ DV c + z_ F [3]

c is the concentration of the lithium salt electrolyte (c =

cJv~). The Kij's can be related directly to the three me asur-

able tra nspor t properties D, t ~ and K.13

A material balance on the salt in the separator is then

given by

( ( d (l nc 0) \ \ i2.Vt~

Oe-vat D (c ) 1 ~ ) V c ) - z.v +F [4]

At this point we assume that the solvent concentration is

not a function of the electrolyte concentration, which im-

plies that the partial molar volume of the electrolyte is

zero. 12 The var iati on in pot enti al in th e separat or is calcu-

la ted from 12

(

Olnf• s

+ t ~ c [5]

= --K(c)Vqb~

K(c)RT 1 +

F Olnc] \nv . z .v§

where qb2 is measu red wi th a lit hiu m refere nce electrode. If

data are available, the variation of the activity coefficient

of the salt is included in this equation.

At the lithium anode (x = 0), a charge-transfer reaction

following Butler-Volmer kinetics occurs. Following Se-

queira

et al., 14

the reaction at the anode is assumed to take

the form

Li + | ~__ Li § | + e-

where Op represents a site in the polymer lattice. This cor-

responds-to an equilibriu m between occupied and unoccu-

pied lit hium sites in the solid -polymer lattice. Then we can

use the exchange current density data obtained by the

above authors for this reaction.

The general for m of the kinetic expression is taken to be

I is the supe rficial cur ren t den sit y of the cell, and ~lsl is the

local value of the surface overpotential

~sl = (I)l -- (I)2 -- U1 [7]

U1, the theoretica l ope n-circ uit cell potenti al, is zero. The

exchange current density takes the form

~cl aal

io~ = F(ka l) (kc ~) (Cmax--C) I(C)al [8]

The parameters k~l and k~ are reaction rate constants for

the anodic and cathodic reactions, respectively. The total

numb er of sties available in the polymer lattice is taken as

the solubility limit of the lithiu m salt, den ote d by c .... This

value is given in App end ix A for one particular lithium

salt/polymer combination. The current mode l is easily

modified to account for a simple charge-transfer process, as

wou ld be expected with a liquid electrolyte, for exampl e.

Howe ver, the experim ental evidence currently available

supports the abov e reaction stoichiometry for pol yme r sys-

tems.

Th e potential of the solid lithiu m pha se is arbitrarily set

to zero at this bou nda ry x = 0). The other bound ary condi-

tions include the flux of lithium ions equalin g the net trans-

fer of current at the interface

I

N. = ~ at x= 0 [9]




3/8

1528

J. Electrochem. Soc.

Vol. 140 No. 6 Jun e 199 3 9 The Electrochemical Society Inc.

The flux and con centration of each species and the poten-

tial in the solution phase are taken to be continuous be-

tween the separator and the composite cathode material

(x =

~ .

The composite cathode can consist of an inert c onducting

material, the polymer/sal t electro]fie, and the solid active

insertion particles, each of whose volume fractions should

be given. These phases are treated as superimposed con-

tinua,

so that a material balance on the lithium in the poly

mer/salt phase gives

a ( c) i~ . v t ~ ( c ) - t O)

9~ - = V . ( 9

+ a3~(1 [10]

z+ v+F v+

where 9 is the volume fraction of the polymer in the

cathode. The extra term here, j~, compared to Eq. 4, is the

pore wall flux of lithium ions across the interface, which is

averaged over the interfacia l area between the solid matri x

and the electrolyte. The pore wall flux is related to the

divergence of the current flow in the electrolyte phase

through

g in s i

~ V. i2 [11]

The current flowing in the electrolyte phase is given by

Eq. 5. Here, the diffusivity and conductivity are effective

values ac count ing for the actua l pat h lengt h of the species ~5

Keff - - -- K 9 5

and

D~ = De ~

As before, these quantities and the transference number

are treated as known functions of concentration.

The boundary conditions in the solution phase are that

the flux of each species is equal to zero at the cathode/c ur-

rent collector boundary

N~ = 0 at x = ~ + ~ [12 ]

The active cathode material is assumed to be made up of

spherical particles of radius R, with diffusion being the

mechanism of transport of the lithium. We take the direc-

tion normal to the surface of the particles to be the r-direc -

tion. Thus

oC, _ D r o + l

Ot L or 2 r or J

[13]

where c~ represents the concentrat ion of lithium in the solid

particle phase. The diffusion coefficient of lithium ions in

the solid phase has been assumed constant in this expres-

sion. From symmetry

0c~ _ 0

at r = 0 [14]

Or

The second boundary condition is provided by a relation-

ship betwe en the pore wall flux across the interface and the

rate of diffusion of lithium ions into the surface of the in-

sertion material

j~ -- -D ~ OC~orat r = R~ [15]

As the diffusion coefficient of the inserted lithium ions is

taken to be constant, this is a linear problem and can be

solved by the method of superposition (see Appendix B).

This is in contrast to the a pproach of West

et al . 8

This model is intended to be general enough to include a

wide range of insertion compounds as the active cathode

material. The open-circuit potential of insertion materials

varies with the amoun t of lithium inserted and is expressed

a s

This is similar to the expression proposed by West

et al . 8

The only difference is the deletion of the dependence on

electrolyte concentration, whic h is not included in this ex-

pression where the potential is defined using a reference

electrode in solution at the local concentration. The

parameters ~ and ~ can be thought oi as expressing activity

corrections and are taken to be constants that can be fit

from experimental data on open-circuit potential vs . state

of charge.

The insertion process at the cathode is represented by the

reaction

LF- -@p + | + e- ~ Li--@~ + @p

This leads to a kinetic expression of the form

/ ~ a F

i = Fk2(emax - c)~~ ~R ~ 01 - U ) )

(_ ~o F

- cT-c /exp [17]

where U' is given by

= u~ - u~of + ~ (~cs + ~) [18]

'

The overpotent ial appearing in this expression is defined as

~1 = r - r [19]

Using the para meters given by West

et al . ~

based on exper-

iment al data for the TiS2 system

U' = 2.17 + ~ (-0 .000558c~ + 8.10) [20]

The para met er k2 in Eq. 17 is a combinat ion of forward a nd

backward rate constants for the charge-transfer reaction,

similar to an exchange current density. Because the ex-

change current density of the charge-transfer process at

the TiS2 interface has not been reported, we set the para me-

ter k2 in Eq. 17 equal to a value corresponding to a nearly

reversible situation. An addit ional condi tion on the poten-

tial in the insertion phase is

V~ = 0 at x = ~, [21]

The current flowing in the ma trix is governed by

il =

-r [22]

The current in the two phases is conserved through

V 9 il + is) = 0 [23]

leading to the integrated form

I = il + i2 [24]

Thus, the current flows through either the polymer/salt

phase or the insertion phase.

The problem now is specified completely, and the equa-

tions above are solved simultaneously using the subroutine

BANDY The final equations and boundary conditions are

listed in Table I. The Crank-Nico]son implicit metho d was

used to evaluate the time derivatives. About one minute of

CPU time (VAX 6510) was nee ded to simu late a s ingle-d is-

charge cycle.

Results and iscussion

Appendix A gives transport properties for the polymer

electrolyte. Additional parameters used in this model are

listed in Table II. Quantities on the left are inherent proper-

ties of a specific system and are determined from experi-

mental measurements. Quantities on the right may be

varied to optimize a particula r batte ry design. The porosity

of the composite cathode is assumed to be constant during

operation. The max imu m conc entr atio n in the solid, cT, was

estimated assuming one lithium a tom per molecule of tita-

nium disulfide and using the density of TiS2.

Figure 2 shows the cell potential as a function of utiliza-

tion of cathode material for galvanostatic charge and dis-

The temperature was not reported in Ref. 8. We assumed that

the standard cell potential was independent of temperature.




4/8

J. Electrochem. Soc. V o l . 1 4 0 , N o . 6 , J u n e 1 9 9 3 9 T h e Electrochemical Society, Inc.

Table I. Equations used in the simulation.

1 5 2 9

Variable Equation Boundary condition

Separator

0c V.DVc i2.Vt~

Ot z+ v +F

n

V n _ I

~ ( s +

t~ c

-- \n~

Z+I +/

Composite cathode

~c i2 .Vt

~

a j ~ ( 1 - t o

c 9149 -++

-

Z y F v

~] V~] -( I- i2 )+ i2 +R T[

s+ t o

t - - jV ln c

K F \nv z+v§

i2 aj~ = -

n F V . i2

t OCs

c, Jn = -D , Jo ~ (R,, ~) ~ r~ (R, t - ~)d~

j. Eq. 17

N = I / F a t

x = 0

V c = O a t x = 8 ~ + ~ o

V~] = - I /a at x = , + ~

i2 = I at x = ~,

charge The util izati on s

u = C~,av~ [25]

aT

The dashed l ine i s the open-c i rcu i t po ten t ia l ca lcu la ted

from Eq. 16, and the current density is a parameter . I t is

apparent tha t the ma te r ia l u t i l iza t ion i s l imi ted a t h ighe r

discharge ra tes. For instance, a t a ra te of 20 A/m 2 the cell

poten t ia l d rops sha rp ly when about 30% of the ca thode

mater ia l is uti l ized. Similar results have been observed in

exp eri men tal dis charg e curves. 3'17 A ty pical cutoff vol tage

is about 1.7 V; bey ond thi s value t he cell is severely polar-

ized.

The con centr a tio n of the e lectrol yte over the t ime scale of

a full discharge cycle is depicted in Fig. 3. For a posit ive

current density the concent ration at the ano de increases

wit h time as lithium is discha rged into the electrolyte. The

concentr ation cha nges rapidly at first nd then the conc en-

tration profiles are nearly constan t over mo st of the dis-

charg e cycle. At long times the concent ration at the bac k of

the cathod e is low. At the front of the cathode x = 0.33 the

conc entr atio n dips for short times. This effect is easier to

see in Fig. 4 whe re concentr ation profiles at short times are

displayed.

One impr ovem ent that should suggest itself immed iately

wh en exa min ing Fig. 3 is a cha nge in the initial concentr a-

tion of electrolyte in the cell. Th e use of a one mo lar solu-

tion is pres umab ly due to the conductivity max imum that

occurs at approx imate ly this concentration. Howeve r

Fig. 3 sh ows that the bulk of the compo site cath ode is at a

significantly lowe r concentration whe re the conductivity

Table II. Pa rame ters used in the TiS2 simulation.

System specific Adjustable

Parame ter Value Ref. Param eter Value

D~ 5.0 • 10 -13 m2/s 16 T 100~

cr 1.0 • 104 S/m - - 52 50 I~m

io,1 12.6 A/mz 14 b 5c 100 ~m

c,a, c,r 0.5 a R~ 1.0 p.m

v+, v 1 - - c o 1000 mol /m3

cT 29,000 mol /m3 - - e 0.3

k2 10 -1~ m4/mol 9 ~ -- - -

n 1 - - - - - -

Data are not available for these parameters.

b Value given is at initial conditions.

is a lso much lower as a result . This leads to severe transpo rt

l imita tions in the depth of the e lectrode. A higher init ia l

concent ra t ion leads to a somewha t lower conduc t iv i ty in

the sepa ra tor but a much la rge r conduc t iv i ty in the com-

pos i te ca thode , where th is i s o f pr ime impor tance .

An impor tan t f ac tor in opt imiz ing the pe r formance of the

cell is good uti l ization of the active cat hode mater ia l . For a

specif ied batt ery perfor mance, the cell potent ia l sho uld fa ll

below its cutoff value only af ter near l y a l l the active mate-

r ia l is consumed. This requires an understanding of the

t ranspor t l imi ta t ions in each phase of the composi te ca th-

ode, as these lead to nonuniform reaction distr ibutions.

The importance of diffusion in the solid can be assessed

f rom the d imensionless pa rame te r Sc

R : I

[26]

S ~ - D ~ F ( 1 - e)CwSc

and is the rati o of diffus ion time to disc harg e time. For Sc


5/8

1 5 3 0

J. Electrochem. Sac.

Vol. 140 No. 6 Ju ne 1 993 9 The Electrochemical Society Inc.

150 0~, , I I I I

composite

cathode

m S= 0 3

1000

~ ~

2 0 0

\ ~ s e p a r a t 0 r ~

500 ~ .~ - - - - -4 oo ~ - -

400~

12,600

0 I I I I

0,0 0.2 0.4 0.6 0.8 1.0

Fig. 3. Concentration profiles at long times; I = | 0 A /m 2 discha rge.

Dashed line divides the separator and composite cathode. Initial

concentration is 1000 mo l/m 3.

0.0001. Therefore, the concen tration at the surface an d the

avera ge concen tratio n in the solid are nearly identical, and

we do not present concen trati on profiles in the solid. No te

that the radius of the particles would have to be on the

sam e order as the thickness of the cathode for diffusion

limitations to exist in the solid pha se in this system. Alter-

natively, if the diffu sion coefficient in the solid wer e de-

creased, diffusion limitations could be co me importa nt.

An analogo us par amet er can be calculated relating the

time consta nt for transport of the electrolyte to the time of

the discharge

I

S ~ = ~ + h a ) 2 D F I - e ) c ~

[27]

For ] = I0 A/ m 2, we find that S~ = 0.15. At high current

densities, the low rate of transport in the electrolyte pha se

is the main factor causing the sharp drop in cell voltage at

less tha n co mpl ete utilization of the cathode. Th e solution

is depleted of electrolyte whi ch cannot be replenished be-

cause of transport limitations. Therefo re, for I = 20 A/ m 2,

the cell potential drop s off at a low value of utilization of

active material. This par am ete r also affects the concen tra-

tion dip in Fig. 4 me nt io ne d above. If transport in the elec-

1200

I

,-I

0

C

1150

1100

1050

separator

composite

cathode

s

1

\ h \ 5

95O

10~

2

900 I , I

0 .0 0.2 0.4 0.6 0,8 .0

E:

Fig. 4. Concentration profiles at short times. I 10 A /m 2. Dash ed

line divides the separator and composite cathode. Initial concentra

tion is 100 0 mol /m ~.

0 0 j I

I I I ~

iOOO

~ -0.5

?

1 0 I ~

12,600 2800 7000

M

~ -I .5

-2.0

~ 2 . s

-3.0

0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fig. 5. Pore wall flux of lithium as a function of dimensionless

distance from the anode. I = 10 A/ m 2. Negative values of i . are for

insertion.

trolyte phase were the dominant limiting factor, that is

Ss >> 1, the dip in co ncen trat ion wo uld be more pron ounce d

and would propagate through the cathode.

Figure 5 shows the local pore wall flux of lithium ions

across the composite cathode at various times during dis-

charge. This figure is analogous to a graph of the current

distr ibuti on f or the system. New man t2 gives four di men-

sionless parameters that characterize the current distribu-

tion in a porous electrode. These parameters describe the

balance between ohmic and kinetic limitations, but not

concen tratio n effects. At short times, the conce ntrati on of

electrolyte is nearly constant, and these paramet ers can be

used to descr ibe the current distribution.

The dimensionless current density and excha nge current

densities are

~

1+ 1 / [28]

Fa i f l ~

If either of these parameters is significantly larger than

unity, then we expect that the ohmie drop dominates the

current distr ibution in the porous electrode. The exchange

current density in the cathode can be determine d from the

reaction rate pa ramete r k2 through

io2 = F k2) Cmax - CY~ -- esya~ cs)~~ [30]

For these calculations, the eoneentration s are taken to be at

the ir i nit ial value s. In our ease, we find tha t g = 1.95 an d 1, =

68, and we expect the ohmic drop to dominate at short

times. This is understandable when considering the re-

versibility

usually ascribed to the charge-transfer process

for insertion materials. Wh en oh mi c effects domi nate , the

reaction distribution can be characterized by the ratio of

the electronic conduct ivity in the insertion material to the

ionic conductiv ity in the po ly me r electrolyte. This ratio is

O 105 ) for this system, caus ing the reaction to occu r prefer-

entially at the front of the electrode. This analysis is sup-

ported by the short-time current distributions sho wn in

Fig. 5.

As the discharge p roceeds and the active material in the

front of the catho de begins to fill up, the reacti on shifts

tow ard s the center of the electrode. T his is an effect of

concen tratio n polarizations in bot h the solid an d elec-

trolyte phases. In this case the pre dom ina nt cause is the

solid pha se concentration, as a relatively small increase in

the concent ration in the solid has a large effect on the po-

tential at the start of the dischar ge Eq. 16). Th e reaction

rate initially increases at the ba ck face of the electrode, but

bec aus e of transport limitations in the electrolyte phase it




6/8

J. Electrochem. Soc. Vol. 140 No. 6 Ju ne 1993 9 The Electrochemical Society Inc. 1531

tapers off at long times. Figure 3 shows that the c once ntra -

tion in the electrolyte phase rapidly decreases at the back

face. The degree to which the conce ntrat ion is depleted at

the back of the cathode depends on the transference num-

ber of lithium, the diffusion coefficient, and the current

density.

Predicting the curre nt distr ibution at long times is a

more difficult problem because of the ubiquitous natu re of

the effect of concentrati on. Not only does the deple tion of

the electrolyte cause conce ntrat ion polariz ations to occur,

but it also affects the kinetic expression and the tran sport

properties. For example, in this system the transference

number rapidly decreases with concentration (Ap-

pend ix A), approac hing zero in the depleted region near the

back face of the electrode. This contributes to the poor

utilization of material that is seen in this region.

For a given rate of discharge, one should be able to opti-

mize the performa nce of a system by exa mini ng the reac-

tion di strib utio n in the electrode, Fig. 5, along with a graph

of the local uti liza tion of active material. Fi gure 6 shows

the local utilization, which is proportional to the average

concentratio n in the intercalation material in the r-direc-

tion. This figure allows one to examine the relationship

between electrode thickness and active material utiliza-

tion.

One optimization sch eme is to vary the thickness an d

porosity of the cathode while holding its theoretical capac-

ity constant. This coul d lead to a ma xi mu m in utilization

wh en the transport limitations in the electrolyte pha se are

minimize d. The use of this met hod for the current syst em

led to the conclus ion that, for a current density of I0 A/m 2

an d a separa tor thickness of 50 l~m, there is a ma xi mu m in

utilization at a porosity of 0.60. This value resulted in a

utilization of 97 of the active material before the cutoff

potential wa s reached, significantly higher than the 84

that was obtained previously with a porosity of 0.30.

In general, thinner electrodes m ak e better use of active

materi al wh en transpor t limitations in the electrolyte exist.

This does not consider the disadvantages that may be asso-

ciated with processing ultrathin compos ite electrodes.

There also wou ld be a weigh t increase with ma ny thin cells

in com pari son to fewer thicker cells. This is an opt imiza-

tion pro ble m that requires detailed information on the bat-

tery configuration, ene rgy an d powe r density require ments

of the system, and cost of compo nents ; these issues are not

discussed here.

A general a ssessment of the perf orma nce of this system

can be made by calculating the average and peak po wer for

1 . 0

I I I I I I

0 . 8 - 14,400 S

0 . 6

-.4

0 2 ~ ' - ' - - - ' - 1 ~ 4 0 s

0 0 .... I 40 s I I I I

0 . 4 0 , 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0

Fig. 6. Local uti l ization of intercalation material in the cathode as

a function of dimensionless distance from the anode. Current density

is 10 A /m ~.

N~

~

2 0 0

i i '] i

1 5 0

1 0 0

5 0

80

/ / /

/ / /

. - 2

9',',//f'

2 0

3 . 0

2 . 5

2.0

1,5

O

1.0 .2

0.5

0 I I I 0 . 0

0 4 0 6 0 8 0 1

O0

Curre nt (A/m )

Fig. 7. Power at 1, 50, and 80 of d ischarge at the 3-h rate

(12.1 A/m 2) as a function of current density is shown by the dashed

lines. Corresponding cell potential curves are depicted by the solid

lines.

a given discharge rate predic ted by the model. Using a 3 h

discharge rate (12.1 A/m2), we determine the power avail-

able for a 30 s pulse of current. The power and cell pote ntia l

are plotted in Fig. 7 at different depths of discharge. The

values in Fig. 7 could be converted to W/kg from an esti-

mate of the mass of materia l and size of the system. Basing

calculations only on the mass of active cathode material

used, the present s imula tion predicts a verage specific

power to be 90.8 W/kg and pe ak power to be a bout 450 W/

kg at 1 depth of discharge dropping to 105 W/kg at 80

dept h of discharge. This represe nts a m ax im um of 1.67 h -1

for the ratio of peak power to average specific energy,

which is lower than that desired for electric-vehicle appli-

cations. Generally, one would want this ratio to be appr ox-

imately 2 to 4, depending on the desired performance and

range of the vehicle. The predicted specific energy for this

system is 272.4 Wh/kg.

Several improvements have been made to the model of

West et al. of the insertion cathode, the most important

being the consideration of the full cell sandwich. We also

chose to mode l a solid-p olymer-elec trolyte system, while

West's simulati on was for a liqu id electrolyte. One can first

analyze the validity of their assumption that the concen-

tration of electrolyte at the separator/cathode interface is

constant. From Fig. 3, it is apparent that this concentratio n

varies by about 15 from its initial value for the present

system. Fixing the concentra tion at this boundary is a non-

physical condition, as it violates the principl e of conserva-

tion of mass for the electrolyte. To compare discharge

curves directly, we have run a simulation using the same

parame ters as in the West model,8 including co nstant phys-

ical properties for a lithium perchlorate/propylene carbon-

ate electrolyte. The comparison of discharge curves can be

seen in Fig. 8, for a separator thickness of 100 ~m.

Although the separator is an a dditional ohmic resistance,

correcting the nonphysical bounda ry conditions of West's

model causes a significant mprovement in the performance

of the system. The conce ntration gradients that develop in

the separator provide an extra driving force for transport

of the electrolyte. Whereas the earlier model predicted

severe electrolyte depletion in the interior region of the

porous electrode, this does not occur in the cu rrent si mula-

tion. The final mater ial util izat ion predicte d for the system

has been increased from 80 to near ly 100 by using the

correct bound ary condition at the separator/cathode inter-

face.

The curren t model gives a theoretical sim ulat ion of the

charge or discharge behavior of a given lithium/polyme r/

insert ion system for a single cycle. The program could be




7/8

1532 J. Electrochem. Soc. Vol. 140 No. 6 Ju ne 1993 9

The Electrochemical

Society inc.

3 . 0

2.5

>

2.0

r~

~ 1 .5

0

,.-I

, - I 1.0

0

0.5

2

I=5

A m

0.0 I I I I

0.0 0.2 0.4 0.6 0.8 1.0

u utilization

Fig. 8. Compar ison of our model resul ts wi th those of West

et ai.

Dashed l ine represents the open circuit potential . The circles are the

simulation results of West et aL and the solid l ine depicts our results.

used to predic t multi ple discharge and charge cycles; how-

ever, the only differences between successive cycles would

be the result of conce ntration gradients in the cell and the

differing local states of charge in the solid particles. This

could be used to predict the effect of relaxation time be-

tween charge and discharge, for ex ampl e) 8

Long- term degr adati on of the cell due to irreversible re-

actions or loss of interfacial contact is not predictable u n-

der the cu rrent model. Losses of contact betwee n the vari-

ous phases of the composite cathode wou ld be expected to

occur during extended cycling. This represents a major

problem in the fabrication and operation of these systems,

but is beyond the scope of the present model.

Summary

A full cell model is presente d for a li thiu m anode, solid

polyme r electrolyte, and ins ertion composite cathode. The

model treats the electrolyte as a binary salt plus polymer

solvent using con centra ted soluti on theory, The velocity of

the polymer is assumed to be zero and volume changes in

the system are neglected. Film formation at the lithium/

polymer inte rface also is neglected. The composite cathode

is modeled using porous electrode theory with bo th kin etic

and diffusional resistances to solid pha se transport. This

mode l is intended to give a general compr ehens ive treat-

ment to the majo r phe nome na involved in this class of sys-

tems whil e providin g the flexibility for seco ndar y ph eno m-

ena to be co nsider ed in the future.

Simu lati on results are presen ted for one particular poly-

mer/salt combina tion using data taken f rom the literature.

Concentration profiles and discharge curves are given and

anal yzed to dete rmin e the beha vior of the cell duri ng dis-

charge. Various methods of optimizingthe system param e-

ters are discussed with several conclusions mad e involving

the limitations of the system.

cknowledgment

This work was supported by the Assistant Secretary for

Conservati on and Renew able Energy, Office of Transp orta-

tion Technologies, Electric an d Hybr id Propul sion Division

of the U.S. Depart ment of Energy unde r Contract No. DE-

AC03-76SF00098.

Manus cript subm itted Oct. 30, 1992; revised manu scri pt

received Jan. 12, 1993.

The U n iv e rs i t y o f Ca l i f o rnia a ss i s t e d in m e e t in g the pub -

l i c a t ion c os t s o f t h i s -a r t i c l e .

APPENDIX A

Transport Properties

We chose to model the polymer-electrolyte system con-

sisting of polyethylene oxide-lit hium trifiuoromethane sul-

fona te (PEO-LiCF3SO3). This part icu la r salt was chosen

because of the relative abunda nce of data available on this

system in comparison to other possible electrolytes. The

concentration dependence of the conductivity and the

transference numb er were ob tained from data available in

the literature. The diffusion coefficient of the salt was

take n to be constant, since reproducible data were difficult

to obtain. Act ivity coefficient data have not be en reported.

The c ond uct ivi ty of PEO-LiCF3SO319 was fit to a th ird -

order polynomial. The transference numbe r2~was fit to the

equation

t+ = 0.0107907 + 1.48837 • 10 -~ c

The dif fusi on coeff icien t was t ak en to be 7.5 • 10 -~2 m2/s. 2~

The solubility limit of lith ium trifluoromethane sulfonate

in PEa was assumed to occur at the transiti on from amor-

phous behavior to mixed-phase behavior on the phase dia-

gram for this system, lea din g to ,x = 3920 mol/m3.~

APPENDIX B

Superpositian

Since the equations describing transport in the active

cathode ma teria l are linear, contr ibuti ons to the flux from

a series of step changes in surface con cent ratio n can be

superposed. This is an example of Duhamel's superpo sition

integral

Oc~ = t Oc~ R~, ~) OCs R~, t -

~)d~ [B-l]

~ r (R~ t ) J0 a t ~ -

where c represents the solution to Eq. 13 for a unit step

change in concentration at the surface. The above integral

is calculated numerica lly using the method suggested by

Wagner21 an d by Aerivos an d Chambr& 22 Where

0Cs (Rs, t) = ~2 (Cs,k+1 _ Cs,k An-k + (Cs, -- Cs,n-1)dl

[B-2]

a~- z~ At At

k=

where

A~_k = a[ n - k)At ] - a[ (n - k - 1)At] [B-3]

and

a t ) = ; tw ~ R~, ~)d~ [B-4]

s Or

Two expressions for a t ) were developed with Laplace

transforms: at long times

~.~1 1 [1 - - n 2 ~ 2 7 ) ] [B-5]

a (7) = ~ n-

exp

=

and for short times

a , ) : - ~ + 2 ~ / / 2 [ l + 2 ~ e x p

- n (~)u2 erfc (~ )]

n l

[B-6]

is dimensi onless time; T = tDJR~. These expressions are

each uni form ly valid; however, the lat ter ex pression con-

verges more quickly with fewer terms at very short times.

The values of a (~) and A,-k can be calcula ted separ ately a nd

used whenev er Eq. B-2 must be evaluated. This procedure,

applicable to linear diffusion into the cathode matrix, is

consequ ently more efficient than solving for the tw o-di-

mensional transport directly.

LIST OF SYMBOLS

a specific int err aci al area, m2/m3

c conc entra tion of electrolyte, mol/m 3

Co conc entra tion of polymer solvent, mol/m 3

ci conc entra tion of species i, mol/m 3

c~ maxim um concentration in solid, mol/m~

c~ conc entra tion in solid, mol/m 3

c~,~ maxim um concentration in polymer, mol/m3

c ~ initial concentration in polymer, mol/m3

D, D~

diffusion coefficient of electrolyte in the polymer

and of lith ium in the solid matrix, m2/s

f activity coefficient

F Fara day 's cons tant , 96,487 C/eq

i curre nt density, A/m2




8/8

J. Electrochem. Soc.

Vol. 140 No. 6 Jun e 1993 9 The Electrochemical Society Inc. 1533

o

I

o

ks

kal

kcl

~j

n

N i

r

R~

Si

S

t

tg

T

U

U

Vi

V

Z

Z

Z i

O~a, O~

Op

O~

K

V

1J , 1,

f f

T

q~

exchange curre nt density, A/m 2

superficial current density, A/m2

pore wall flux of lithium ions, mol/m2 9

reaction rate co nstant at cathode/polymer interface,

m4/mol, s

anodic reaction rate constant, mJ/s

cathodic reaction rate constant, mJ/s

fricti onal coefficient, J 9 /m 5

numb er of electrons transferred in electrode

reaction

mola r flux in x di rection of species i, mol/m 2 9

distance normal to surface of cathode material, m

univ ersal gas constant, 8.3143 J/mol 9K

radius of cathode material, m

stoichiometric coefficient of species i in electrode

reaction

dimensio nless ratios defined in Eq. 26 and 27

time, s

transference number of species i

temperature, K

utilization of intercalation material

open-circuit potential, V

velocity of species i, m/s

cell potential, V

distance from the anode, m

dimension less distance from the an ode

charge nu mbe r of species i

transfer coefficients

activ ity coefficient correction

dimensionless current density

thickness of separator, m

thickness of composite cathode, m

porosity of electrode

activity coefficient correction, or dummy variable

of integratio n, s

surface overpot ential, V

site concentrat ion in polymer

site concentration in solid matrix

conductivity of electrolyte, S/m

dimensionless exchange current density

numb er of cations and anions into which a mole of

electrolyte dissociates

conductivity of solid matrix, S/m

dimensionless time

electroc hemical potentia l of species i J/tool

electrical potential V

Subscripts

c cathode

r reference state

s solid phase, or separa tor

T maxim um concentration in intercalatio n material

1 solid matrix

2 solution phase

Superscripts

0 solvent, or init ial condit ion

@ standar d cell potential

REFERENCES

1. A. Hooper and J. M. North, Solid State lonics 9 10,

1161 (1983).

2. F. Bonino, M. Ottaviani, B. Scrosati, and G. Pistoia,

This Journal

135, 12 (1988).

3. M. Gauthier, D. Fauteux, G. Vassort, A. Belanger, M.

Duval, R Ricoux, J.-M. Chabagno, D. Muller, R

Rigaud, M. B. Arma nd, a nd D. Deroo, ibid. 132, 1333

(1985).

4. M. Z. A. Munshi and B. B. Owens, Solid State Ionics 26,

41 (1988).

5. K. M. Abraham,

J. Power Sources

7, 1 (1981/1982).

6. F. M. Gray, Solid Pol ymer Electrolytes VCH, New York

(1991).

7. D. Fauteux,

Solid State Ionics

17, 133 (1985).

8. K. West, T. Jacob sen, a nd S. Atlu ng, This Journal 129,

1480 (1982).

9. M. A. Ratner, in

Polymer Electrolyte Reviews-I J. R.

MacCatlum a nd C. A. Vincent, Editors, Elsevi er Ap-

plied Science, London (1987).

10. R. Pollard a nd J. Newman,

Electrochim. Acta

25, 315

(1980).

11. M. B. Armand, in Polymer Electrolyte Reviews-I J. R.

MacCal]um and C. A. Vincent, Editors, Elsevi er Ap-

plied Science, London (1987).

12. J. Newman, Electrochemical Systems Prentice-Hall,

Englewood Cliffs, NJ (1991).

13. T. F. Fuller and J. Newman, This Journal 139, 1332

(1992).

14. C. A. C. Sequeira and A. Hoope~,

Solid State Ionics

9 10, 1131 (1983).

15. J. Newm an and W. Tiedema nn, AIChE J. 21, 25 (1975).

16. S. Basu and W. L. Worrell, in

Fast Ion Transport in

Solids P. Vashista, J. N. Mundy, and G. K. Shenoy,

Editors, p. 149, North -Hol land Publi shing Co., Am-

sterdam (1979).

17. B. C. H. Steele, G. E. Lagos, R C. Spur den s, C. Forsy th,

and A. D. Foord,

Solid S tate Ionics

9 10, 391 (1983).

18. R. Pollard and J. Newman,

This Journal

128, 503

(1981).

19. D. Fauteux, ibid. 135, 2231 (1988).

20. A. Bouri dah, F. Dalar d, D. Deroo, an d M. B. Arm and , J.

Appl. EIectrochem. 17, 625 (1987).

21. C. Wagner,

J. Mathematics and Physics

34, 289 (1954).

22. A. Acrivos and P. L. Chambr~,

Ind. Eng. Chem.

49,

1025 (1957).

) l CC Li i l ( b t t)dl / it /tdd R di t ib ti bj t t ECS t f (140 112 71 49l d d 2015 08 16 t IP

Documents

Modelingof GalvanostaticChargeand Discharge of the Lithium:Polymer:InsertionCell