-
Vol. 124, No. 10 HIGH ION
sphere, it may be avai lable for practical solid electro- lytes
under oxidizing atmosphere because of its high conductivity and
easy preparation.
Manuscript submitted Apri l 26, 1977; revised manu- script
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 tech- nique
is appl ied to the determinat ion of the kinetic parameters of the
com- pound "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 crys- tal
structures and in which the equil ibration of local differences in
composition occurs rapidly.
In general, in the presence of a compositional (stoi-
chiometric) 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 fac- tor , 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, electro- chemical 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 mea- sure 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 con- ductors, 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
stoichio- metric 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 param- eters of the material. The
particle fluxes wil l be ex- pressed in terms of the individual
kinetic and thermo- dynamic 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 move- ments 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 ex- pression 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 diffu- sion 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 coeffi- cient 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 ex- per imental
ly determined, may be el iminated from Eq. [4] by the condition
that, except for transient condi- tions 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 quan- tities 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 chem- ical
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 ref- erence to the
crystal lattice, which in the cases dis- cussed 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 co- efficient"
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 dif- ferent 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 en- hancement 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 mo- bi 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 galvano-
static intermittent titration technique (GITT). AEt is the total
transient voltage change of the galvanic cell for an applied gal-
vanostatic 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 diffu- sion 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 cur- rent, 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 con- ditions
[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 galvan- ostatic 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 concen- tration 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 coulo- metric 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 predomi- nant 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 de- scribed 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 prod- uct
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 con- stituent.
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 sys- tem 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 tech- nique it should be
pointed out that Li~Sb actual ly offers less than opt imum
conditions, for it has a com- parat 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 coulo- metric 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 com- positional 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 coulo- metric t itration curve and
plotted in Fig. 7 as a func- tion of the stoichiometry. As can be
seen, the chem- ical diffusion of the l i thium is enhanced by up
to a factor of near ly 70,000 relat ive to the component dif-
fusion coefficient (or the tracer diffusion coefficient divided by
the correlation factor).
In order to further test this technique, several ad- ditional
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 repre- sent 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 titra-
tion 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 calcu- lated 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 calcu- lated 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 re- lated 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 correla- tion 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 com- bines
transient and steady-state electrochemical mea- surements. 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 composi- tional
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 tran- sient methods. However, experimental problems in
other techniques related to resistance polarization, which are
typically only reduced by the use of refer- ence ~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 enhance- ment 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 thermody- namic 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 com- position dependence
of the chemical diffusion coeffi- cient 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 com-
position-dependent enhancement factor is to deter- mine the
stoichiometry 5 from the measured cell volt- age 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 num- ber 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 manu- script
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)
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