Weppner, Determination of the Kinetic Parameters of Mixed-Conducting Electrodes and Application, 1977

Vol. 124, No. 10 HIGH ION

sphere, it may be avai lable for practical solid electrolytes under oxidizing atmosphere because of its high conductivity and easy preparation.

Manuscript submitted Apri l 26, 1977; revised manuscript received June 10, 1977.

Any discussion of this paper wil l appear in a Discus- sion ~ection to be published in the June 1978 JOURNAL. Al l discussions for the June 1978 Discussion Section should be submitted by Feb. 1, 1978.

Publication costs o] this article were assisted by the authors.

REFERENCES 1. G. Gattow and H. Schrhder, Z. Anorg. Allgem.

Chem., 318, 176 (1962). 2. G. Gattow and D. Schutze, ibid., 328, 44 (1964). 3. M. G. Hapase and V. B. Tare, Indian J. Pure Appl.

Phys., 5, 401 (1967). 4. R. S. Sethi and H. G. Gauer, Indian J. Chem., 3, 177

(1955).

OXIDE CONDUCTION 1369

5. T. Takahashi, H. Iwahara, and Y. Nagai, J. Appl. Electrochem., 2, 97 (1972).

6. T. Takahashi and H. Iwahara, ibid., 3, 65 (1973). 7. C. N. R. Rao, G. V. Subba Rao, and S. Ramdas,

J. Phys. Chem., 73, 672 (1969). 8. C. A. Johnson, R. C. Bradt, and J. H. Hoke, J. Am.

Ceram. Soc., 58, 37 (1975). 9. T. Takahashi, H. Iwahara, and T. Arao, J. Appl.

Electrochem., 5, 187 (1975). 10. T. Takahashi, T. Esaka, and H. Iwahara, ibid., 5,

197 (1975). 11. R. S. Roth and T. L. Waring, J. Res. Nat. Bur.

Stand., Sect. A, 66, 451 (1962). 12. E. M. Levin and R. S. Roth, ibid., {}8, 202 (1962). 13. C. Wagner, Z. Phys. Chem., 21, 25 (1925). 14. T. Takahashi and H. Iwahara, Energy Convers., 11,

105 (1971). 15. T. Takahashi, T. Esaka, and H. Iwahara, in J. AppL

Electrochem., 7, 303 (1977). 16. H. Yanagida, R. J. Brook, and F. A. Krhger, This

Journal, 117, 593 (1970). 17. H. Schmalzried, Z. Phys. Chem. N.F., 38, 87 (1963).

Determination of the Kinetic Parameters of Mixed-Conducting Electrodes and Application to the System LLSb

W. Weppner and R. A. Huggins* Department of Materials Science and Engineering, Stanford University, Stanford, California 94305

ABSTRACT

An electrochemical galvanostatic intermittent t i trat ion technique (GITT) is described which combines both transient and steady-state measurements to obtain kinetic propert ies of solid mixed-conduct ing electrodes, as well as thermodynamic data. The derivation of quantities such as the chemical and component diffusion coefficients, the part ia l conductivity, the mobility, the thermodynamic enhancement factor, and the parabol ic rate constant as a function of stoichiometry is .presented. A description of the factors governing the equil ibration of composition gradients in such phases is included. The technique is appl ied to the determinat ion of the kinetic parameters of the compound "Li~Sb," which has a narrow composition range. For Lie.999~Sb the chemical diffusion coefficient is 2 10 -5 cm ~ sec -1 at 360~ This value is quite high, due to a large thermodynamic enhancement factor of 1.3 104. The l i thium component diffusion coefficient is comparat ively small at this composition, 1.5 1O-~ cm 2 sec -1. The part ia l conductivity and electrical mobi l i ty of l i thium are 1.5 10 -4 1% -1 cm -1 and 3 10 -s cm 2 V -1 sec - I , respectively, at the same stoichiometry and temperature. Because of the very large values of the chemical diffusion coefficient and the fact that 3 moles of l i thium can react per mole of antimony, this system may be of interest for use in new types of secondary batteries.

Quantitat ive knowledge of transport kinetics in solids is of importance both for increased understand- ing of transport mechanisms and the disorder in solid materials and for the technological application of many sol id-state reactions. Especial ly in the current search for new types of high power density batteries, success in several approaches strongly depends on the avai labi l i ty of solid electrodes which incorporate elec- troactive species from the electrolyte into their crystal structures and in which the equil ibration of local differences in composition occurs rapidly.

In general, in the presence of a compositional (stoichiometric) gradient the transport parameters of the different mobile species are interrelated. The quan- t ity usual ly used to describe the complex process of compositional equi l ibration (relaxation) is the chem-

ical diffusion coefficient, I~. Besides the thermo-

* E lec t rochemica l Soc iety Act ive Member , Key words: chemica l diffusion, ga lvanostat ic in termi t tent t i t ra-

t ion, par t ia l ionic conduct iv i ty , thermodynamic enhancement factor , parabo l ic ta rn ish ing ra te constant , ionic mobility.

gravimetric, conductometric, and radiotracer methods which have been used for a long t ime for the observa- tion of the exchange of solid compounds with adjacent l iquid or gaseous species, electrochemical techniques have recently come into use (1-4). The important ad- vantage of this latter approach is that the voltage of a suitable galvanic cell is directly related to appropr i - ate thermodynamic quantities, and the current can provide easily measurable kinetic information at the same time. For the case of ideal solutions and con- centrat ion- independent diffusion coefficients, electrochemical methods have also been used in several earl ier investigations [e.g., (5-7)].

To study chemical diffusion in mixed-conduct ing solids one may employ transient methods similar to those used for the electrochemical investigation of the kinetics in the vicinity of chemical ly inert solid electrodes in l iquid electrolytes (chronopotentiometry, chronoamperometry, voltammetry) [e.g., (8)]. In the case discussed here, however, the transport in the bulk of the electrode acts as the rate-determining step.

1570 J. Electrochem. Soc.: SOL ID-STATE SCIENCE AND TECHNOLOGY October 1977

Steady-state measurements are often used to measure the conductivity. However, they give information concerning the product of the concentration and the diffusion coefficient, rather than either one separately, without additional information. On the other hand, transient methods may sometimes be used to separate these quantities. An example is the so-cal led "Rosen- burg method" for the evaluation of defect concentra- tions and chemical diffusion in oxide layers (9-11).

In this paper, the application of an electrochemical cell method is introduced for the acquisition of both kinetic and thermodynamic information in mixed conductors, such as insertion compound electrodes. This method allows separate determinat ion of the chemical diffusion coefficient, as well as other useful quantities, in one set of experiments. This approach employs an exper imental ly tractable galvanostatic intermittent t i - ' t rat ion technique (GITT), which combines transient and steady-state measurements. It el iminates the cus- tomary problem of interference due to resistance polarization in the use of potentiostatic techniques. The desired data are conveniently accessible, and a precise coulometer or independent determinat ion of coulometric t itration curves are not necessary.

The investigation of tr i l i thium antimonide, "Li3Sb," by the use of this new technique is described here. Li~Sb is a mixed-conduct ing compound whose features make it potential ly interesting for use as an electrode constituent. Although the range of stoichiometry of this phase is rather narrow, so that only small stoichiometric variations may occur, this method is shown to be extremely sensitive and quite precise.

Basic Relations Chemical diffusion is a process in which the trans-

port of all species within the solid is involved. Gen- eral equations wil l be derived for its microscopic description and its relation to other kinetic parameters of the material. The particle fluxes wil l be ex- pressed in terms of the individual kinetic and thermodynamic propert ies of the different ionic and electronic species. This approach will show which factors control chemical diffusion under various circumstances. Effects result ing from volume changes and macroscopic movements wil l not be considered. For this purpose, one may often use the sublattice of one component as the frame of reference for the motion of all the other species (12). If necessary, other reference systems should be considered (13), of course. Also, it wil l he assumed that the diffusion length is small compared to the dimensions of the whole system.

In general, in an isothermal and adiabatic system and if Onsager's cross-coefficients (12) are negligible, the flux density of species i (in particles per square centimeter second) under the influence of a gradient of the electrochemical potential ~]i (related to one part ic le) is given in the one-dimensional case by

j~ = [1] z~Yq 2 Ox

r zi, and q are the part ia l electrical conductivity due to the transport of species i, the charge number (val- ence), and the e lementary charge, respectively. The electrical conductivity r may be replaced by an expression including the product of the concentration ci and the electrical mobi l i ty ui or, alternatively, the general mobi l i ty bi (the mean particle velocity per unit general acting force), where bi : ui/lzilq

~xi : Izilqciul : ziYqYcibi [2] Also, the electrochemical potential ~i may be divided into two terms, one containing the chemical potential gi (per part icle) or the activity ai, and the other the local electrostatic potential r

~li : m + ziq~ : ~i ~ + kT In a i + ziq~ [3]

~i ~ k, and T are the chemical potential of species i in the standard state (ai = 1), Boltzmann's constant, and the absolute temperature, respectively. Inserting Eq. [2] and [3] into Eq. [1] yields for the flux density of the component i

kTui[Olna, Oci ZiqCi ~b ] [4]

The factor in front of the bracket has the dimensions of a diffusion coefficient and is the component diffusion coefficient DKi, or the "diffusivity of the species i as a component" (12)

kTui DK i "- -- bikT [5] [z lq

This quantity is thus s imply proport ional to the mobi l - i ty of the species in question and reflects the same microscopic kinetic phenomena. This diffusion coefficient obeys the Nernst-Einstein equation regardless of whether the solution is ideal or nonideal, and is the diffusion coefficient which is direct ly measured in the case of ideal di lute solutions of neutral species. DK/ is a measure of the random motion of the particles of species i in the crystal, and is related to the (radio) tracer diffusion coefficient DT~ by DTI --- ]iDKi, where ]i is the correlation factor.

The inner electric field Oq~/ax, which cannot be exper imental ly determined, may be el iminated from Eq. [4] by the condition that, except for transient conditions involving the accumulation of a significant space charge, charge flux balance must be maintained. That is, for all species, if there is no external ly appl ied electric potential difference

z~ji = 0 [6] i

Inserting Eq. [4] into Eq. [6], and solving for Or and then using this expression in Eq. [4] yields the flux density for species i in terms of transport quantities and activity gradients related to it and to all other species j

[01na i ~ tj zi 01na i l 0ci [7a] 3i-- -- DKI 0 ]nc i j z--l-- ~ J o~z'

[ zi Olnaj] ,Sc, O ln a~ ~ tj : -- DKt ( I -- it) ~ ~ Zj 0 In cl Ox

[7b]

The summations include all other ionic and electronic

species. The symbol ti -- ~i / ~ ~j is the transfer- ] J

ence number for species i. As a result of the ionization equil ibrium, in which

the activities of neutral species are related to those of their ionic and electronic constituents within solids, we have

d In ai + zid In ae = d In ai -- zid In ah : d In al* [8]

where e, h, and i* represent electrons, holes, and neutral i species, respectively. Equation [7] may now be transformed into an expression containing only the (physical ly more relevant) activities and concen- trations of neutral atomic species

zt 01naj* ] ~ci 0 In ai* ~ tj 3i = -- DKi O In el* j~e.h Zj O In ci* ~X

[9a]

[ 01nj__~* E z, 01naj* ] ~c, =--DK i ( l - - t i ) _ _ ~j

0 In ci* j=2~i.e.h Zj 0 In ci* ~x [gb]

Vol. 124, No. I0 KINET IC PARAMETERS 157t

These equations give the flux density for any chemical component and hold both for the ionic species within the solid or for the effective flux density of such species in their neutral form. These equations are distinct from those which appeared in the earl ier discussion [7], however, in that the activity and con- centrat ion terms only involve neutral chemical species, and neither electrons nor holes.

These general equations have a form similar to the famil iar Fick's first law

8Cl Jl = - -D I [ i0]

ax

where /~ is the chemical diffusion coefficient (the effective diffusion coefficient) for species i (with reference to the crystal lattice, which in the cases discussed below, is identical to the laboratory frame) and

~)i = DKi W [11]

where the factor W is an enhancement factor, defined by the quantity in the square brackets

zi O In a j* ] 8 in ai.___~* E w= (1- t , ) 8 In ci* j:~i,e,h Zj ,O In c]*

[12]

The symbol W was selected for this purpose in honor of Carl Wagner, who first showed (14) the influence of the transport of charged species upon each other.

Di is sometimes called the "intrinsic diffusion coefficient" of species i, when referred to the laboratory frame. In the case that the crystal and laboratory frames are identical, these quantities are the same. In cases in which two atomic species interdiffuse, the term "interdiffusion coefficient" is sometimes used with reference to the crystal frame, which may move with respect to the laboratory frame.

In many systems of practical importance, including tarnishing processes (14), the situation is simplified by the fact that two types of species (either two different ionic species, or one ionic and one electronic species) dominate the transport phenomena. Under this situation, the chemical diffusion coefficients of

the two species are equal, and we can use the symbol for both (15). For the case in which only one ionic and one elect}onic species have to be considered, so that te ---- 1 -- ti, the enhancement factor becomes

and

O in ai* W = r e - - [13]

O In ci*

_- DKit e 0 In ai_____~* [14] 0 In ci*

If the sample is predominant ly an electronic conductor, so that te --> 1, we have s imply

O In ai* W -- - - [15]

0 In ci* and

5 - - - - - DK, O In ai__.....~* [16] 0 In ci*

This is the case previously discussed by Wagner (16). Using the definition of the activity coefficient "n = ai/ci, the enhancement factor can be writ ten in the form

W- - J1 81n~t O lnv i ] [17] -t-a--~-~nc~]=[1 +c~ Oci

derived for metals by Darken (17).

On the other hand, if the transference number of the ionic species is much larger than that for the electronic species, Eq. [13] can be rewrit ten as

(+:) 0 oo," W = [18]

d In ci*

and by using Eq. [2] and [5], we get

ceD~e 8 in ai* W -- [19]

Zi2CiDKi 0 In ct* and then

- - CeDKe 8 In ai* [20]

Zi2Ci 0 In ci*

Thus we see that, in this case, the chemical diffusion coefficient for the ionic species is dependent on the component diffusion coefficient of the electrons, not the ions.

Let us now further examine the value of the enhancement factor for several different situations for the common and important case in which only one ionic species and one electronic species (assumed to be excess or mobile electrons) have to be considered, the transference numbers of all other species being negligible. From Eq. [8] and [13], the enhancement factor for the ions becomes s imply

[ O In ai 0 In ae ] w = + j [21] For the special case in which we can assume that

both the ionic species and the electrons obey either Henry's or Raoult's law, i.e., "n and me are constant, and considering the electroneutral i ty condition

dCe -" zidci [22]

we derive from Eq. [21]

W : te 1 -~ Zi 2 ' [23] Ce

We can thus conclude that if te ~ 1, i.e., in the case of a solid electrolyte with dilute mobile ionic species, W wil l tend toward zero, and the chemical diffusion of ions in response to a composition gradient will be very sluggish.

On the other hand, if electronic conduction pre- dominates, te --> 1, the value of W will depend sensi- t ively upon the value of ci/ce. If ci Ce. This is possible if the electrons have a much greater mobi l i ty than the ionic species and can lead to unusual ly

large values of W, and thus of ~}i, as pointed out by Wagner (18), who cited a number of materials in which this has been found. It wil l be shown later in this paper that this is also true for Li~Sb.

Very large enhancement factors can also be present for materials in which one can assume that only the electronic species exhibit ideal solution behavior. In that case Eq. [21] becomes

F O In ai ci -1 w = te l + Zi 2 - J [24]

Ce

The large enhancement (or reduction) factor W is sometimes explained in terms of a microscopic model in which the more mobile species tend to move ahead of the others. If they have different charges (or species of the same sign move at different rates in opposite directions), this creates an internal electric field in which the slower species are accelerated, and the

1572 J. E tect rochem. Soc.: SOLID-STATE SCIENCE AND TECHNOLOGY October 1977

Thermodynamic assumptions

Table I. Value of enhancement factor under various conditions

(Cases with only one ionic species and electrons mobile) Concentration Transference assumptions no. assumption

Enhancement factor (W)

a. General General General

b. Genera l General t~

VoL 124, No. 10

Z td n, n.

0

I.D

tD

0 > .._1 J LI2 (D

I E0 - -

to to+V = T ime, t T

I~ ~ } I R OROP

T \ L ._A to to+T

~Time, t

IE 1

Fig. 2. Schematic illustration of a single step of the galvanostatic intermittent titration technique (GITT). AEt is the total transient voltage change of the galvanic cell for an applied galvanostatic current la for the time "c, AEs is the change of the steady-state voltage of the cell for this step.

voltage by a constant value but does not change the geometrical shape of the voltage-t ime curve.

After a t ime interval T, the current flux is inter- rupted, whereupon the composition within the Au+~B sample tends to again become homogeneous by diffusion of the mobile species. During this equi l ibration process the surface composition of the sample, and therefore the cell voltage, drifts back toward a new steady-state value El, corresponding to a new activity of A in the sample as a result of the change of the stoichiometry ~5 caused by the coulometric t i trat ion of A ions into it by the current

I o~MB a8 : - - [27]

ZAmBF

MB, roB, Io, and F are the atomic weight of B, the mass of the component B in the sample, the constant current, and Faraday's constant, respectively. After the electrode is again in equil ibrium, the procedure may be repeated, starting now with the new cell voltage El. This process may be continued unti l a phase change occurs in the electrode, or the electrolyte decomposes or becomes electronical ly conductive.

In order to calculate the voltage E as a function of t ime t during which the current is applied, the time dependence of the concentration ci at the interface x = 0 has to be determined by solving Fick's second law

~ci(x,t) : ~ O~ci(x,t) [28] Ot Ox 2

with the init ial and boundary conditions

c i (x , t=O) :Co (O_-0)

K INET IC PARAMETERS

[29]

t3D]

[31]

The last condition is due to the fact that the r ight- hand phase boundary of the sample is assumed to be impermeable,

1573

The solution of the differential Eq. [28] under conditions [29]-[31] is known (19, 20) and can be cast in the fol lowing form for x -- 0

2Ion/t" == c,(x=0, t)=Co+ v - ziq - -

----'O

( ie r fc [_~t ]_}_ ier fc[ ' (nW1)L

with ierfc(~) = [~-l /e exp(_~2) ] _ t + [~. erf(~.)], the first integral of the complement of the error func-

tion. At times t

1574 J. Electrochem. Soc.: SOL ID-STATE SCIENCE AND TECHNOLOGY October 1977

4 (mBVM~2(AEs~2

hEr is the total change of the cell voltage E during the current pulse, neglecting the IR drop.

Determination of Other Kinetic and Thermodynamic Quantities

Besides the determinat ion of the chemical diffusion

coefficient D, it is possible to evaluate other kinetic and thermodynamic parameters by use of the galvanostatic intermittent t i trat ion technique. The methods for obtaining some of the more important quantities are as follows.

The enhancement ]actor.~As discussed earl ier the chemical diffusion of species A is enhanced by the thermodynamic factor 0 In aA*/O In CA* relat ive to the component diffusivity. This enhancement factor can be determined from the local slope of the coulometric t i trat ion curve, dE/dS, at any composition. The change of In aA* is given by the change of the voltage E of the cell times zAq/kT, and the change in the concentration CA is NAdS/VM. Therefore, we have

0 In aA, ZAqCAVM dE ZAq(y -t- 8) dE 0 In cA. kTNA d8 kT d5

[39]

By determining the slope of the coulometric t itration curve at different stoichiometries the thermodynamic factor may be obtained as a function of the composition.

The partial conductivity.--For the case in which the chemical diffusion is predominant ly determined by one kind of ionic species, A z~ and either electrons or holes, Eq. [14] yields, with the substitution of DK by ~ according to Eq. [2] and [5]

tAre'--(ZAq)2CAD ( O ln aA* ) -1 kT 0 In CA* [40]

By using Eq. [39] the thermodynamic factor can be replaced by a term containing the slope of the coulometric t i trat ion curve, dE/dS, which is determined from the steady-state values of the cell voltages in between titrations and the corresponding change of the composition. This yields

9 ZAFD ( dE ~-I vM \--~" / [41]

from which ~rAt e may be calculated if the chemical

diffusion coefficient/~ has been determined as described above. It is obvious that if the sample is predominant ly electronical ly conducting, the product aAte is the part ia l ionic conductivity of the A species, CA. For a predominant ly ionic conductor ~Ate is the part ia l electronic conductivity, ~e, since t e - - ae / (~ 'e -~- aA) ~- 0"e/a" A.

If the coulometric t i trat ion curve can be considered l inear over the range of a single galvanostatic t i t ra- tion step, replacement of dE and d8 by their finite

values and using Eq. [37] for D yields

4 VMmBIoAEs ~At~ = -- (t

Vol. 124, No. 10 KINET IC PARAMETERS 1575

by plott ing DKA, which is determined according to Eq. [44] or [45], as a function of the cell voltage E.

This follows from Eq. [48] by using Eq. [16] for and Eq. [39] for the enhancement factor

kt -" ~ ZA DKAdE [49]

The integration extends from the voltage at the two- phase equi l ibr ium to the voltage corresponding to the activity aA* or the stoichiometric parameter 5 for which kt is evaluated.

In contrast to the usual tarnishing experiments in which the mater ia l is typical ly maintained in contact with a constant activity of A*, e.g., that corresponding to equi l ibr ium with another phase, the method described here permits the tarnishing rate constant to be evaluated as a function of the activity of A* over the whole stoichiometric range of the tarnish product phase.

Appl icat ion to Tr i l i th ium Ant imonide , Li~Sb The galvanostatic intermittent t i trat ion technique

has been appl ied to Li~Sb as a model electrode constituent. Its kinetic parameters determine the rate of al loying ant imony with l i thium at high l i thium ac- tivi.ties. Due to the Iarge amount (3 moles) of l i thium that can react per mole of antimony, this b inary system may be of practical interest for application in new types of high power density batteries, either at room temperature (22, 23) or at elevated temperatures. From the point of view of testing the general appl ic- abi l i ty of the galvanostatic intermittent t itration technique it should be pointed out that Li~Sb actual ly offers less than opt imum conditions, for it has a comparat ive ly narrow range of stoichiometry and a very composit ion-dependent enhancement factor.

A eutectic mixture of molten LiC1-KC1 salt (spec- troscopical ly pure, Anderson Physics Laboratories, Ur- bana, I l l inois) has been used as an electrolyte for l i thium ions in a set of exper iments on the Li3Sb phase. Both reference and counterelectrodes were com- posed of the sol id-state two-phase system of l i th ium- saturated A1 and the adjacent phase "LiAI," which has a composition at the phase boundary of about 47 atomic percent Li. This system has a potential about 300 mV less negative than pure l iquid l i thium at 400~ The LisSb electrode was prepared by coulometric t itration of l i thium into an antimony disk. The electric leads were made of molybdenum, which ex- hibits a negligible solubil ity for l ithium. The experi - ments were conducted within a Vacuum/Atmospheres Company recycl ing hel ium dry box. No impur i ty effects were observed.

The voltage range of the existence of the Li3Sb phase is between about +0.88 and 0V with reference to pure l i thium at 400~ This corresponds to a range of l i thium activity from about 2.6 X 10 -7 to 1. F igure 3 shows a typical voltage-t ime dependence for the case of galvanostatical ly t i trat ing l i thium into the Lis+aSb sample with a current density of 0.25 mA/cm 2, appl ied for 300 sec at 360~ The corresponding change of the stoichiometric parameter ~ of the sample is about 2.5 X 10 -4. The voltage is plotted as a function of the square root of t ime for the same measurement in Fig. 4. In agreement with theoretical considerations (Eq. [35]), a straight l ine is observed for times up to 200

sec. The deviation from l inearity at longer times is discussed later.

A sequence of selected galvanostatic t itrations of l i thium out of Li~Sb, distr ibuted over the entire range of the thermodynamic existence of the compound, is plotted in Fig. 5. The polarization of the voltage is observed to be much greater in the center of the stoichiometric range of existence than near both ends.

E vs.L[

[r.Vl

l 800

700

I I I I I

" L~ 3 Sb"

T = 360 *C I = 2.5x IO -4 A/cm 2

[ I ] I I 0 I00 20O 300

Fig. 3. Typical transient voltage change of the galvanic cell with a Li3Sb sample after a constant current of 2.5 X 10 -4 A cm 2 was applied. LiCI-KCI(e) was used as a molten salt electrolyte, and two-phase AI, LiAI mixtures used as counter and reference electrodes.

0 I0 30 60

E vsLL

t [sec]

120 ISO 240 300

"L~,3 Sb" T=360~C

I:2.5x10 -4A/cm 2 700

l I I t L__~L i I 0 4 8 12 16

Fig. 4. Representation of the transient voltage of the galvanic cell as a function of the square root of the time. The experimental conditions were the same as those of Fig. 3.

This is closely related to the shape of the coulometric t itration curve, determined from the steady-state cell voltages observed between titrations as a function of composition. The data for several runs (in both compositional directions) are shown in Fig. 6. Near the middle of the composition range the slope dE/d8 has a marked maximum, producing the greatest changes of the transient voltage with time, in accord- ance with Eq. [35].

The thermodynamic enhancement factor, a In au/O In cu, was calculated from the slope of the coulometric t itration curve and plotted in Fig. 7 as a function of the stoichiometry. As can be seen, the chemical diffusion of the l i thium is enhanced by up to a factor of near ly 70,000 relat ive to the component diffusion coefficient (or the tracer diffusion coefficient divided by the correlation factor).

In order to further test this technique, several additional parameters have been varied, and the results compared with the theoretical considerations. At the same voltage, or stoichiometric composition, the values of the kinetic parameters should be independent of the direction of the current flux. The slopes of the

i@76 J. EIectrochem. Sot.: SOL ID-STATE SCIENCE AND TECHNOLOGY October 1977

E vB. L i Imv]

T 800

60C

40C

20C

~ [ I 71 I

" Li3 S'b" r !

[ I r r I io00 io00

- - I

/

T = 360~ I = 5xlO-4A S = 1.5cm a

I ~ I r tq I0o o ICE) 0 I00 o

- - t rsecl L J

II qO0 0 ~30

Fig. 5. The transient voltages of the galvanic cell containing Li3Sb as a function of the time t for several different initial steady- state voltages plotted vs. pure lithium using the same galvanostatic current density of 3.3 X 10 -4 A cm -~ contact area between the electrolyte and the sample. The different starting voltages represent different stoichiometries of Lia+~Sb as determined from Fig. 6.

I000

E vs. Li.

[mY]

~ o t . % Li, 74,98 75 75 .02

I I I I I

T - 360 ~ 6OO

E vs. 800 - A~ ~'Li. Af'

[mv] 4OO

6OO

200

4O0

0

2OO

-200

0 I L i I I L I I -4 -2 0 2 4

- 3x103 in LL3+ ~ Sb

Fig. 6. The steady-state voltages of the galvanic cell with a LisSb sample vs. pure lithium and an AI, LiAI electrode as a function of the stoichiometry q3, i.e., the amount of lithium which is electrochemically pumped into the sample (coulometric titration curve). The horizontal part of the curve represents the two- phase region of Li~Sb in equilibrium with Li~Sb.

E vs. v'-t curves should be the same but of opposite sign for the same current density. This was found to be true, as demonstrated in Fig. 8. Also, the calculated kinetic quantities should be independent of the actual value of the current. According to Eq. [35] the slope of the plots of the voltage vs. the square root of t ime should be proportional to the current Io applied. The results of experiments carried out with current densities differing by a factor of 12 are shown in Fig. 9. Good agreement between experiments and theory is seen.

The chemical diffusion coefficient that was calculated from the slope of the straight l ine in Fig. 4 and

d (n qLi

d ~13 CLL

X 10 -4

74 .98 8 j k

L i3+~Sb

T = 560~

6

4

2 / f t

i i / i -4 -2

- - at. % LL 75 7 ,5 .02

t L i Iq

-1 dE IV ] Eg x 10 -2

- I0

8

- -6 i

i

4

x 2

"X~x~ I { I r x x~x~

0 2 4

8X103 in L i3+a Sb

Fig. 7. The composition dependence of the enhancement factor in Li3+aSb, which causes the chemical diffusion coefficient to be much greater than either the component diffusion coefficient or the tracer diffusion coefficient of lithium. The enhancement factor is determined from the slope of the coulometric titration curve (Fig. 6), which is plotted at the right-hand ordinate.

0 iO 20 40 60 80 I00 800 F- I I , I

E vs. L~

7OO x

LL3Sb

T = 360 ~

I = 3.3X10 -4 A/cm 2

600 I I I ' 0 2 4 6

I I _ _

8 I0

,/T [~ec ~]

Fig. 8. Two transient voltage changes of the galvanic cell with a li~Sb electrode with reference to pure lithium as a function of the square root of time for adding lithium to the sample and re- moving it with the same current density at about the same mean composition, i.e., cell voltage. The absolute value of the slopes is seen to be the same for both procedures.

the corresponding change of the steady-state cell volt-

age according to Fig. 6 is ~ ~ 2 X l0 -s cm 2 sec -1 at the stoichiometric deviation 6 ~ -- 6 X 10 -4 and 360~ This value is strongly enhanced by the enhancement factor of a In aL|/O In CLi = 1.3 I04, thus being related to a much lower component diffusion coefficient, DKLi ~ 1.5 10 -9 cm2 sec-Z at the same stoichiometry and temperature. The tracer diffusion coefficient has also the same order of magnitude, since the correlation factor is of the order of 1. The corresponding partial l i th ium ionic conductivity was calculated to have a value of ~Lt : 1.5 X 10 -4 f1-1 cm-Z for that composition, according to the general equations of the previous section. Using the Nernst-Einstein equa-

Vol. I24, No. 10 KINETIC PARAMETERS 1577

,o 2o 4o 6o 8o ,oo

E-E ( t=O) i T=560~ / /4mA

[mY] EI,:O.~ / / / -

.oo / / /

~OO

0 0 2 4 6 8 I0

Fig. 9. The transient voltage of the galvanic cell with a Li3Sb sample referenced to an initial ceil voltage of approximately 550 mV vs. pure lithium as a function of the square root of time. The results are shown for several current densities. The slopes increase almost linearly with the current density.

tion the general l ithium mobility was found to be bLi = 2 101' cm 2 V -1 A -1 sec -2, and the electrical l i thium mobil ity is ULi -- 3 10 -s cm 2 V -1 sec -1 at this composition. The parabolic rate constant kt for the formation of a Li~Sb tarnishing layer on a Li2Sb (24, 25) substrate is of the order of 10 -s cm 2 sec -1 at a l i thium activity aLi "-- 4 X 10 -7 and of the order of 10 -7 cm 2 sec -1 at a l ithium activity aLl .---- 1.

By use of this same procedure, values of the chem-

ical diffusion coefficient D have been determined over the whole range of stability of the Li3Sb phase, as il lustrated in Fig. 10.

Extensive experiments utilizing this galvanostatic intermittent titration technique to study the systems Li-Sb and Li-Bi will be reported elsewhere (25, 26).

Discuss ion

The galvanostatic intermittent titration method combines transient and steady-state electrochemical measurements. In one set of experiments a number of kinetic and thermodynamic quantities may be readily determined with considerable precision as a function of composition within a single phase. The compositional resolution in this technique is quite high, so that phases with only small ranges of stoichiometry can be investigated. The experimental equipment re- quirements are quite modest, consisting primari ly of

Jog 5

-4

-4'.4

J

LL 3 Sb

T = 360~C

/ o c{

/

i ~. . . . . ~ _ -o I

o o -48

-52 I~ -l -0,5

I I

o o ~

/ o /

/ /

I _ _ I 0 0.5

~x lO ~ in Li .3

Fig. 10. Variation of the chemical diffusion coefficient with composition over the whole range of stability of Li3Sb at 360~

a constant current source and an instrument that can measure voltage vs. time.

The information that is obtained from the observa- tion of the time dependence of the cell voltage during a galvanostatic current pulse is in principle no greater than that which can be obtained by using other transient methods. However, experimental problems in other techniques related to resistance polarization, which are typically only reduced by the use of reference ~lectrodes, are here totally eliminated.

In the common potentiostatic voltage step method, the current should theoretically be infinite in the first moment. This is not experimental ly realizable, and the imposed voltage difference creates large transient polarization effects. Thus the initial and boundary conditions usually applied to the assumed solution of Fick's second taw are not exactIy fulfilled. The result is a deviation from the theoretically expected l inear relation between the current and 1/x/ t at short times. Additionally, because of the inverse time scale, this part of the plot is expanded whereas the part with the more useful information is compressed. In the case of the galvanostatic titration method, however, the initial period contains the most valuable information because any effects due to a change in the enhancement factor or to the finite thickness of the sample do not become important unti l later. This early part of the experiment is fortuitously expanded by the use of the square root of time scale in the voltage plot.

If a deviation from the l inear dependence of the voltage on the square root of time is found at longer times in galvanostatic experiments it may have one or more of several explanations. Either the solution to the assumed boundary value problem is not valid at long times because of finite length effects due to a high diffusion coefficient or a thin sample, or there may be a significant change of either the thermodynamic enhancement factor or the chemical diffusion coefficient over the pertinent stoichiometric range. These different possibilities may be evaluated from the experimental results on the magnitude and composition dependence of the chemical diffusion coefficient and the enhancement factor. If necessary, a current pulse of shorter duration or a smaller value of current might be used.

Another way to eliminate the influence of a composition-dependent enhancement factor is to determine the stoichiometry 5 from the measured cell voltage E by using data from the coulometric titration curve and to plot 5 as a function of k/t. According to Eq. [35] the following relation should be found

d6 2VMIo (t

1578 J. Electrochem. Soc.: SOL ID-STATE SCIENCE AND TECHNOLOGY October I977

erful tool which can be employed to determine a number of important kinetic and thermodynamic quantities in mixed conductors. Of special interest is its ready appl ication to compounds with small stoichiometric ranges and appreciable variation of the thermodynamic enhancement factor with composition,

The values of chemical diffusion coefficient observed in the case of LiaSb are several orders of magnitude greater than those found necessary to produce neg- l igible diffusional polarization in fine part icle battery electrodes.

Acknowledgments This work was funded by a grant from the Institute

for Energy Studies at Stanford University, which has supported one of the authors (W. W.) on a Standard Oil of California visit ing professorship. The earl ier grant of a NATO Scholarship through the German Academic Exchange Service (DAAD) is also grate- ful ly acknowledged.

Manuscript submitted March 14, 1977; revised manuscript received May 23, 1977.

9 Any discussion of this paper wil l appear in a Discus- stun Section to be published in the June 1918 JOURNAL. All discussions for the June 1978 Discussion Section should be submitted by Feb. 1, 1978.

Publication costs of this article were assisted by Stanford University.

LIST OF SYMBOLS ai activity of species i bi general mool l i ty Of species i (am 2 sac -2 V-1 A -z) ct concentration of part icle i (cm-~)

~)i chemical diffusion coefficient of species i (am 2 sea- l )

DKi component diffusion coefficient of species i (am 2 sac -1)

DTi tracer diffusion coefficient of species i (am 2 sea- l )

E galvanic cell voltage (V) aE~ steady-state voltage change (V) AE~ transient voltage change iV) ] correlation factor F Faraday's constant, (A sac va l - : ) I electric cunent (A) Io applied constant electric current (A) Ji flux density of particles i (am-2 sea- l ) k Boltzmann's constant (V A sac ~ -1) kt parabol ic tarnishing rate constant (cm 2 sec-D L sample length (cm) mi mass of species i (g) Mi atomic weight of species i (g mo le - l ) NA Avogadro's number (mole -1) q e lementary charge (A sac) S contact area between electrolyte and sample

(am 2) t t ime (sec) t~ transference number of species i T absolute temperature (~ u~ electrical mobi l i ty of species i (am z V -1 sea- l ) V~ molar volume (cm a mole - I ) W enhancement factor x distance coordinate (cm)

y stoichiometric number zi vMence of species i "n activity coefficient of species i (era 3) 5 deviation from the (ideal) stoichiometry Tn electrochemical potential of species i (per par-

ticle) (V A sac) m chemical potential of species i (per part icle)

(V A sec) m ~ chemical potential of species i in the standard

state (V A sac) r part ia l electrical conductivity of species i (~-1

CITI- 1) pulse duration (sec)

r electrostatic potential (V)

REFERENCES 1. W. F. Chu, H. Rickert, and W. Weppner, in "Fast

Ion Transport in Solids, Solid State Batteries and Devices," W. van Gool, Editor, p. I8I, North Hol- land Publishing Co., Amsterdam-London (1973).

2. H. Rickert and W. Weppner, Z. Natur]orsch., Tell A, 29, 1849 (1974).

3. B. C. H. Steele, in "Mass Transport Phenomena in Ce " " ramlcs, A. R. Cooper and A. H. Heuer, Edi- tors, p. 269, P lenum Press, New York (1975).

4. J. B. Wagner, Jr., in "Electrode Processes in Solid State Ionics," M. Kleitz and J. Dupuy, Editors, p. 185, D. Reidel, Dordrecht (1976).

5. H..Rickert, in "Electromotive Force Measurements m High-Temperature Systems," C. B. Alcock, Editor, p. 59, The Institute of Mining and Metal- lurgy, London (1968).

6. R. A. Rapp and D. A. Shores, in "Techniques of Metal Research," Vol. 4, Part 2, R. A. Rapp, Edi- tor, p. 123, Interscience, New York (1970).

7. D. O. Raleigh, Electroanal. Chem., 6, 87 (1973). 8. R. N. Adams, "Electrochemistry at Solid Elec-

trodes," M. Dekker inc., New York (1969). A. J. Rosenburg, This Journal, 107, 795 (1960). E. Fryt, S. Mrowec, and T. Walec, Oxid. Met., 7, 117

(1973). S. Mrowec, J. Mater. Sci., 9, 1961 (1974). C. Wagner, in "Progress in Solid State Chemistry,"

Vol. 10, Part 1, J. McCaldin and G. Somorjai, Edi- tors, p. 3, Pergamon Press, Oxford (1975).

R. Haase, "Transportvorg~nge," Steinkopff Verlag, Darmstadt (1973).

C. Wagner, Z. Phys. Chem., Abt. B, 21, 25 (1933). C. Wagner, Acta Metall., 17, 99 (1969). C. Wagner, in "Atom Movements," p. 153, American

Society for Metals, Cleveland, Ohio (1951). L. S. Darken, Trans. AIME, I75, 184 (1948). C. Wagner, J. Chem. Phys., 21, 1819 (1953). H. S. Carslaw and J. C. Jaeger, "Conduction of Heat

in Solids," p. 112, Clarendon Press, Oxford (1967).

J. Crank, "The Mathematics of Diffusion," Oxford University Press, London (1967).

J. Bardeen and C. Herring, in "Atom Movements," p. 87, American Society for Metals, Cleveland, Ohio (1951).

J. O. Besenhard and H. P. Fritz, Electrochim. Acta, 20, 513 (1975).

Rheinisch-Westf~lisches Elektr iz it~tswerk AG, Fr. Demande 2,206,593 (1974); Get. AppL P. 2,254,- 870.5 (1972).

R. G~rardin and J. Aubry, C. R. Acad. Sci., Set C, 278, 1097 (t974).

W. Weppner and R. A. Huggins, To be published. W. Weppner and R. A. Huggins, Submitted to

J. SoLid State Chem.

9. 10.

11. 12.

13.

14. 15. 16.

17. 18. 19.

20.

21.

22.

23.

24.

25. 26.

Documents

Weppner, Determination of the Kinetic Parameters of Mixed-Conducting Electrodes and Application, 1977