Application of A-C Techniques to the Study of Lithium Diffusion in

Vol. 127, No. 2
SEMICONDUCTOR ELECTRODES
6. S. N. F r a n k and A. J. Bard, ibid., 97, 7427 (1975).
7. A. u
Gokhshtein, Sov. Electrochem. Engl. Trans.,
7, 13 (1971).
8. R. E. Malpas, R. A. Fredlein, and A. J. Bard, J.
Electroanal. Chem. Interracial Electrochem., 98,
171 (1979).
9. R. E. Malpas, R. A. Fredlein, a n d A. J. Bard, ibid.,
98, 339 (1979).
10. R. E. Malpas and A. J. Bard, Anat. Chem., I n press.
343
11. J. F. McCann and L. J. Handley, S u b m i t t e d to Faraday Society Meeting (1979).
12. Y. G. Berube and P. L. DeBruyn, J. Colloid. Inter5ace Sci., 27, 305 (1968).
13. W. H. Laflere, R. L. Van Meirhaeghe, F. Cardon,
and W. P. Gomes, Surf. Sci., 59, 401 (1976).
14. H. Yoneyama, H. Sakamoto, a n d H. Tamura, Electrochim. Acta, 20, 341 (1975).
Application of A-C Techniques to the Study
of Lithium Diffusion in Tungsten Trioxide Thin Films
C. He,* I. D. Raistrick,* and R. A. Huggins*
Btanford University, Stanford, Cali$ornia 94305
ABSTRACT
The small signal a-c impedance of the cell
Li]LiAsF6 (0.75M) in propylene carbonatel
Lip V~O3 thin film on tin oxide covered glass substrate
has been measured at room t e m p e r a t u r e as a function of frequency from
5 • 10 -4 Hz to 5 X l0 s Hz at various open-circuit voltages. The diffusion
equations have been solved for the appropriate finite b o u n d a r y conditions, and
analysis of the impedance data by the complex plane method yields values for
the chemical diffusion coefficient, the component diffusion coefficient, the p a r tial ionic conductivity of lithium, and the t h e r m o d y n a m i c e n h a n c e m e n t factor
for LiyWO3 as a function of y. The films of WO3 were prepared b y v a c u u m
evaporation and were largely amorphous to x-rays. The chemical diffusion
coefficient has a value of 2.4 • 10 -12 cm2/sec at y _-- 0.1, increasing to 2.8 X
10 -11 cm2/sec at y ---- 0.26. At short times (t ~ 0.5 sec) the interracial charge
transfer reaction is important, but at longer times the rate of l i t h i u m injection
is d e t e r m i n e d by the diffusion kinetics.
Considerable interest has developed recently i n the
possibility of fabricating display devices based on the
electrochemical i n j e c t i o n of m e t a l or h y d r o g e n atoms
into t r a n s i t i o n m e t a l oxides, u s u a l l y WO3. The reaction m a y be w r i t t e n
WO8 (colorless) + yM + + y e - ~ MuWO3 (colored)
For each atom injected a n electron enters the conduction b a n d of the host oxide and a deep blue coloration develops. At the same time, the electronic conductivity of the oxide rapidly increases. The range of
solid solubility of M i n WO3 m a y be quite extensive;
typically a y value of about 0.1-0.2 is necessary for
good optical contrast. The activity of M in M~WO3,
a n d hence the electrode potential of the electrochromic oxide will v a r y continuously w i t h y as long
as a single phase solid solution is formed. If the reaction proceeds r a p i d l y and reversibly, t h e n the metal
atoms m a y be repeatedly injected and removed by
controlling the electrode potential of the oxide electrode with respect to a counterelectrode which acts
as a source a n d sink of M.
Although reactions of this type have been k n o w n
for several years, few data are available on basic
electrochemical parameters, such as diffusion coefficients of M i n the oxide or exchange c u r r e n t densities,
even though the e v e n t u a l response time of the device
will be largely d e t e r m i n e d by these parameters. It is
not even clear w h e t h e r the rate of injection is limited
b y diffusion or by charge transfer at times of interest
for display applications.
* E l e c t r o c h e m i c a l Society A c t i v e M e m b e r .
K e y w o r d s : diffusion, WOn, lithium injection, e l e c t r o c h r o m i c
films, a-c technique.
This paper describes a steady-state a-c technique
which m a y be used to d e t e r m i n e the r a t e - l i m i t i n g
process and the i m p o r t a n t kinetic p a r a m e t e r s of a n
electrochromic system. It is illustrated b y results obtained for the l i t h i u m - t u n g s t e n trioxide system.
Although a n y of several t r a n s i e n t techniques could,
in principle, be used to study this system, a n a-c
method was chosen since it seemed likely that, using
complex plane methods, data analysis could be m u c h
simplified, especially since semi-infinite diffusion conditions cannot be assumed for the case of thin film
electrodes.
F e w studies of the electrochromic effect of alkali
metals in W Q have been reported (1-10). G r e e n et al.
have studied sodium and l i t h i u m injection i n WO3
using both liquid and solid electrolytes (2-5, 10).
The diffusion coefficient of l i t h i u m was estimated to
be 2 • 10 -12 cm2/sec (3). Mohapatra (8), however,
has recently reported a m u c h higher value of 5 •
10-9 cm2/sec. T h e r m o d y n a m i c data for l i t h i u m in
amorphous W Q films have recently also appeared
(9, 10).
Theory of the A-C Method
A l t e r n a t i n g - c u r r e n t and voltage methods have, of
course, been extensively used i n m a n y kinds of electrochemical systems, and several review articles are
available (11-13).
In general, either c u r r e n t or voltage m a y be the
controlled variable, and the phase and m a g n i t u d e of
the dependent variable (voltage or current, respectively) are d e t e r m i n e d with respect to this. If measurements are made over a w i d e - e n o u g h frequency
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344
J. Electrochem. Soc.: E L E C T R O C H E M I C A L S C I E N C E A N D T E C H N O L O G Y
Experimentally, the electrode is held at a constant
potential E, with respect to a reference electrode of
pure A, u n t i l e q u i l i b r i u m is reached. The composition
of the sample at e q u i l i b r i u m is u n i f o r m a n d equal to
A~B. The chemical potential of A i n A~B is
CDL
# A -----~ A ~ q- R T
Fig. I. The Randles equivMent circuit for the o-c response of o
system with chorge tronsfer ond diffusion of the electrouctive
species.
range, then different physical processes m a y be separated through their different time constants.
The p r o b l e m of the a-c response of a simple electrochemical system with either charge transfer or diff u s i o n - l i m i t e d kinetics has been considered by several
authors. The e q u i v a l e n t circuit for this situation is
due to Randles (14) and is shown i n Fig. 1. This circuit will also be valid for the diffusion of a metal into
an oxide electrode, if it can be assumed that diffusion
is d r i v e n b y a gradient i n composition a n d not b y an
electric field.
I n Fig. 1, R1 is the u n c o m p e n s a t e d ohmic resistance
of the electrolyte and electrode, CDL is the double
layer capacitance of the electrode-electrolyte i n t e r face, 0 is the charge transfer resistance, a n d Zw* is a
complex impedance arising from the diffusion of the
electroactive species.
For the case of semi-infinite diffusion
--
jAw-W
[1]
where ~ is the radial frequency, j ~_ X/ -- 1~ and A
is a constant which contains a concentration i n d e p e n d e n t diffusion coefficient. Zw* is often k n o w n as
the W a r b u r g impedance (15). The explicit form of
A, and a more general expression for Zw*, which is
also valid for a finite diffusion situation, are derived
below.
The charge transfer resistance 0, is related to the
exchange c u r r e n t density (Io), through a linearization
of the B u t l e r - V o l m e r equation for small overpotentials (SE)
bE
RT
RT
e
-
i
-
- -
nloF
5E
In
TAX A
[~]
Here XA is the mole fraction of A, and VA is its activity
coefficient. If we choose VA ~ 1 as XA ~ 1 then ~Ao is
the chemical potential of pure A at the same temperature. Thus
@
Zw* = A~-'/g
February 1980
<
<
--
F
[2]
It is assumed in the derivation of this e q u i v a l e n t circuit that small signal conditions are met, since CDL,
and A are i n general voltage or concentration d e p e n d ent and lead to n o n l i n e a r behavior.
The frequency response of this circuit will be
governed by the relative importance of charge t r a n s fer and diffusion in d e t e r m i n i n g the current: for very
slow diffusion Zw* ~ 0, and for very slow charge
transfer kinetics Zw* < 0. Since Zw* is a function of
frequency and e is not, then an electrode reaction rate
m a y be controlled by diffusion at low frequencies
(long times) and by charge transfer at high frequencies (short times).
Derivation of Zw* for a thin film electrode.--The
magnitude of the W a r b u r g impedance Zw* is determined by the chemical diffusion coefficient of the
n e u t r a l electroactive species (A) i n the electronically
conducting oxide electrode (B), and the explicit form
of Zw* can be derived by solving Fick's laws with
suitable initial and b o u n d a r y conditions. The situation
is fully analogous to the d e t e r m i n a t i o n of chemical
diffusion coefficients in solid solution electrodes by a
t r a n s i e n t galvanostatic technique discussed previously
by Weppner and Huggins (16, 17).
E
RT
=
-- ~
[4]
In vAXA
zF
z is the charge carried by A cations, F and R are the
gas constant and Faraday, respectively.
A small a l t e r n a t i n g voltage
6E -- vo sin ~t
[~]
is now applied b e t w e e n the AyB working electrode and
the reference electrode, which drives the surface of
the working electrode to a n e w time dependent concentration XA + 8XA (which corresponds to a stoicniometry of Ay+6yB). Eventually, a steady state is
reached and we require an expression for the a m plitude a n d phase of t h e c u r r e n t flowing t h r o u g h the
cell. Since the c u r r e n t density depends on the gradient in concentration at the interface, through Fick's
law, and the voltage depends on the chemical potential at the interface, it is necessary to either make dilute solution assumptions or else to have knowledge
of the activity coefficient of A i n A~B. Since the solid
solutions u n d e r consideration here are quite concentrated (XA m a y be as high as 0.3) dilute solution behavior cannot be assumed and a n a l t e r n a t i v e approach is adopted here which is e q u i v a l e n t to a k n o w l edge of 7A as a function of y. If we can assume that
for very small changes i n voltage
6E
8XA
dE
----
[6]
dXA
We can substitute Eq. [6] into Eq. [5] and obtain
5XA (t) -~ (dE/dXA)xA - I . v o s i n w t
[7]
This equation describes the change i n mole fraction of A at the electrode-electrolyte interface for a
very small change in applied a l t e r n a t i n g voltage.
(dE/dXA)XA m a y be obtained b y differentiation of
the coulometric titration curve at the appropriate
composition (XA), or as will be seen later, may, in
the present case, be obtained directly from very low
f r e q u e n c y measurements. E q u a t i o n [7] represents a
linearization of the dependence of concentration on
voltage. The mole fraction of A at the interface is
therefore a sinusoidal function of time as long as
(dE/dXA)xA is constant over the range of composition
5XA.
For a diffusion e x p e r i m e n t it is c o n v e n i e n t to write
Eq. [7] in terms of the n u m b e r of atoms per u n i t volume (CA), and the stoichiometry (y)
~C.~(t) = ~
9
y
9 Vo sin ~t
[8]
Here N is Avogadro's n u m b e r and VM is the m o l a r
volume of AyB, assumed constant over the range 5y.
We therefore seek solutions of the equation
a [~cA (x,t) ] _ ~ 0 2 [,~cA]
Ot
[o]
ax 2
with the initial condition of u n i f o r m concentration
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Vol. 127, No. 2
APPLICATION
6CA(x,O)
--
]lax
: 0
6CA(X,t) = aB sin (at + @)
ax
[12]
x=L
zY
1"~"
[24]
has an a m p l i t u d e
io--
[11]
corresponding to a n interface i m p e r m e a b l e to A.
The diffusion coefficient used h e r e is the chemical
diffusion coenicient, w h i c h is a p p r o p r i a t e for the diffusion of a species in an a c t i v i t y g r a d i e n t (16). W i t h
c e r t a i n assumptions, it can be r e l a t e d to the particle
diffusion coefficient of the components (A ~+ ions and
electrons).
The solution of Eq. [9] for these conditions is given
by C a r s l a w and J a e g e r (18)
345
id._.. ~ / a[~CA] )
[i0]
0
and the x = l b o u n d a r y condition given b y Eq. [8].
Here 1 is the thickness of the oxide electrode; the
coordinate system used is shown in Fig. 2. The second
b o u n d a r y condition is
O[hCA (0,t)
OF A-C TECHNIQUES
zF (-~y
dE ~-'
/ " vo(~3)1/" ( h2 +
2VM
s~)Va
d2
[25]
and a phase of # with respect to the a p p l i e d voltage.
Note that both the phase and the a m p l i t u d e a r e f r e quency dependent.
Two excreme cases are of interest. First, consider the
value of io for high frequencies, small D, or thick s a m ples, i.e., kl > > 1. U n d e r these conditions (h 2 + s 'z) l/2/d
= Z, and so
zFVo(dE)
io --
VM
-1
~Y
-
9 (~D) 1/,
[26]
and
# -- a r c t a n (1) ----~/4
with
(dE 5-~
: VM
\"dyy/
[13]
S = lil
[14]
= arg (])
[15]
and
1=
Vo
kx 9 (1 -l-J)
cosh
cosh k~ 9 (i + j )
k
=
where
(~/2D) ~/~
=
Thus, the phase difference b e t w e e n the c u r r e n t a n d the
voltage is i n d e p e n d e n t of f r e q u e n c y and is equal to
45 ~ Expressing this r e s u l t as a complex i m p e d a n c e
[16]
with
[17]
Note that this is the steady-state solution. There is
also a transient caused by starting the sinusoidal oscillation at t = 0. The form of this transient is given in
Ref. (18).
In order to determine the current density in the system, it is necessary to e v a l u a t e (o[bCA]/OX)z=b F r o m
Eq. [12] it m a y be shown t h a t
IZl=
Iz]
-7-~. (1
v-
I v~ I
ioa
=
-
j)
[26]
[ VM(dE/dy)
z~l/)~=~
"
~-'/,I
[29]
w h e r e a ----surface area.
Comparison of Eq. [28] w i t h Eq. [1] shows that for
these conditions the p r e e x p o n e n t i a l factor A in the
V/arburg i m p e d a n c e has the form
A =
( O[hCA] )x =~
[27]
VM(dE/dy) I
~ zF~a
[303
[18]
The other limiting case occurs at v e r y low frequencies,
where
x : (ak/~/2d) (h 2 +
an d
p=arctan
s 2) '/=
(h+s
\'-h-~--s )
[19]
[2O]
for v e r y thin samples or for large
If
kL << 1
(i.e., ~ << 2D/l 2)
zFvo (
io-
with
h : sinh (2kl)
s
sin (2kl)
:
[22]
p = arctan (~o) = -2
8CA
X = [Z[ s i n ~ = ]Z l =
x=O
R :
[VMCdE/dy)/zF~la[
IZl cos ;9 = [Z 1 2k212/3a
1
="~L
[34]
[35]
i, = - zFD (~[BCA]
\ ax
)x:~
N
-I
w h e r e CL and RL are the limiting low f r e q u e n c y capacitance a n d resistance. A n i n t e r e s t i n g consequence
of Eq. [34] and [36] is that for thin samples, it is not
ELECTROLYTE
x=+l
[33]
Thus, under these conditions the current is 90 ~ out of
phase with the voltage and is independent of the diffusion coefficient.
Expressed as the imaginary (X) and real (R) parts
of the complex impedance for these conditions
Thus, the c u r r e n t d e n s i t y
ELECTRONIC
CONDUCTOR
'~-'
[21]
d = cosh2(k/) " cos2(kl) + sinh2(k/) 9 sin2(k/) [23]
ELECTROCHROMI
CI
MATERIAL
dE
[31]
+X
Fig. 2. The coordinate system used far the solution of the diffusion equations.
necessary to obtain a coulometric t i t r a t i o n curve since
simultaneous solution of these equations yields values
for both the diffusion coefficient and (dE/dy)u. A l t e r natively, if this last q u a n t i t y is k n o w n i n d e p e n d e n t l y ,
then the sample thickness m a y be calculated. Of course,
no information about ( d E / d y ) ~ or I m a y be obtained
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346
J. Electrochem. Sot.: E L E C T R O C H E M I C A L S C I E N C E A N D T E C H N O L O G Y
February 1980
from the high frequency data since the diffusion is
then semi-infinite i n nature, and only D m a y be d e t e r mined.
At intermediate frequencies, when n e i t h e r of t h e
two limiting cases can be applied, the frequency r e s p o n s e of the electrode m u s t be analyzed using the full
impedance expressions
Zw* = Rw -- jXw
with
Io,
[37]
IZ] sin #
[38]
R w = IZI cos #
[39]
Xw =
-X
w h e r e p is given by Eq, [20], a n d
IZwl = Ivo/ioal
w h e r e io is given by Eq. [25].
Complex p~ane representation.--Since the form of
Zw* is now known, the impedance of the entire circuit of Fig. 1 can be computed as a function of frequency. The most useful way of representing this result is by plotting the real (R) and i m a g i n a r y (X)
components of the total circuit impedance vs. one
another in the complex plane as functions of frequency. Three such computer generated plots are
shown in Fig. 3.
I n Fig. 3 ( a ) , the diffusional impedance is large a n d
a straight line of 45 ~ slope is seen over most of the
frequency range. The kinetics of the system are limited almost e n t i r e l y by the rate of the diffusional
process. The high frequency real axis intercept gives
the value of R1 and D m a y be found b y analysis of the
straight line portion of the curve using Eq. [28] and
[29].
I n Fig. 3(b) a transition is obtained such that the
kinetics pass from diffusion control at low frequencies
to charge transfer control at high frequencies. The
semicircle at high frequencies is due to the parallel
combination of CDn and e, the values of which m a y i n
principle be obtained by analysis of the high frequency
data. The extrapolated real intercept of the straight
line region lies at [11]
Rs = Rz + e -- r
[41]
r = 2Cm,A2
[42]
R
RZ
[40]
-X
(b)
e
%
-x
R;'
thickness effect. D may be obtained from either the
45 ~ portion of the line or from RL, d e t e r m i n e d from the
limiting resistance, a n d Eq. [36].
E x p e r i m e n t a l Procedures
diffusion experiments
were carried out on thin films of t u n g s t e n trioxide
deposited on tin oxide (700A thick) covered glass substrates. Amorphous W Q films were prepared by t h e r mal evaporation of the oxide at a rate of 30 A/sec.
The substrate t e m p e r a t u r e during evaporation was
100~C, and the pressure was m a i n t a i n e d at 2 • 10 -4
Tort of air. The method of preparation and the characterization of these films was discussed in Ref. (19),
where it was also demonstrated that such films are
stable in the electrolyte/solvent system used i n the
present experiments.
Sample
preparation.--The
R
(c)
/%J
with
where A is given by Eq. [30].
At very low frequencies the phase angle begins to
increase due to the onset of finite length effects. This
process is complete in Fig. 3(c), where R has reached
its limiting value given by the sum of RL (Eq. [36])
and (RI + 6). If the diffusion kinetics are in the
proper range, there can be good separation b e t w e e n
the charge transfer semicircle at high frequencies, the
45 ~ line of Zw* at i n t e r m e d i a t e frequencies, and the
vertical line at very low frequencies, due to the finite
I
%+0
(RI+ 8+RL)/" R
RI
Fig. 3. Computer simulation of the complex impedance of the
circuit shown in Fig. 1 for various values of the diffusion coefficient.
The results reported below were obtained on 1500A
thick films which had been h e a t - t r e a t e d in pure oxygen for 6 hr at 350~ This procedure was adopted i n
order to dry the films and to ensure that the stoichiometry was in fact WO~. X - r a y diffraction experiments
indicated that the initially amorphous films, which
st.owed only a broad diffuse scattering centered on a d
value of 3.6A, developed some crystallinity on u n d e r going this h e a t - t r e a t m e n t . Several sharp reflections
developed, which could be indexed on the same u n i t
cell as the high t e m p e r a t u r e tetragonal modification
of W Q (in agreement with the observations of Green
[10], but with slightly reduced lattice p a r a m e t e r s ) .
The sharp reflections were superimposed on a diffuse b a c k g r o u n d similar to that of the unheated,
treated films.
A densitly of 5.2 g/cm 3 was used in the calculation
of the results below, based on a comparison of the
measured film volume (using i n t e r f e r o m e t r y ) and
chemical analysis (19).
Electrochemical measurements.--Measurements w e r e
made
using a
three
electrode
configuration. B o t h
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Vol. 127, No. 2
347
A P P L I C A T I O N OF A-C TECHNIQUES
counter and reference electrodes were pure l i t h i u m
metal, and the electrolyte was 0.75M LiAsF6 i n p r o p y l ene carbonate. The electrolyte was dried by prolonged
agitation with l i t h i u m chips. All materials a n d the
electrochemical cell were kept in a helium-filled dry
box. Measurements were made at 300~
The t h i n film working electrode was held at constant voltage (and hence composition) b y a P r i n c e t o n
Applied Research Model 173 potentiostat. A small
a l t e r n a t i n g signal from a Hewlett Packard HP3110B
oscillator was then superimposed on the constant voltage and the c u r r e n t flowing in the c o u n t e r - w o r k i n g
electrode circuit was monitored via a s t a n d a r d resistance. The circuit is shown i n Fig. 4. The a m p l i t u d e
of the signal applied to the sample was always less
t h a n 10 mV and the frequency range covered extended
from 5 • i0 -4 Hz up to 5 kHz.
Above 20 Hz, the amplitude and phase of the current with respect to the applied voltage were measured directly using a PAR Model 5204 two-phase
l o c k - i n analyzer. Above about 100 IIz, small corrections were made to compensate for the a t t e n u a t i o n
and phase shift introduced into the signal by the potentiostat. This was achieved by comparing the a-c
p e r t u r b a t i o n of the potential difference b e t w e e n the
reference and w o r k i n g electrodes directly with the
output of the oscillator.
Below 20 Hz, the signals corresponding to c u r r e n t
and voltage were amplified using a PAR Model 113
preamplifier and fed to the a n a l o g - t o - d i g i t a l converters of a Digital E q u i p m e n t Corporation Lab 8e
computer, where the two signals were averaged over
several cycles and compared.
I
3.5
I
--
.ore
o
tf)
--
3.C
This Work
......
Reference [9]
.......
Reference [lO]
E
:3
e4-
e-,
2.5
ql)
>
LL
t.IJ
2.0
--
0
I
I
0.1
0.2
COMPOSITION
(yinLivWO
0.3
3)
Fig. 5. Coulometric titration curve for LiyW08. The numbers indicate the order in which the data were taken. Results of Mohapatra
and Wagner (9) and Green (10) are shown for comparison.
Results
The change in open-circuit voltage (E) with composition (y) is plotted i n Fig. 5. The order in which
the points were t a k e n is indicated, showing that the
m e a s u r e m e n t s were in fact made at equilibrium. Results of the two other t h e r m o d y n a m i c studies are
also shown for comparison (9, 10).
The complex impedance data are s u m m a r i z e d i n
Fig. 6 and 7. In Fig. 6, the higher frequency data are
shown for four open-circuit voltages. At the highest
voltage (2.6V with respect to pure l i t h i u m ) , a straight
line of slope 45 ~ was obtained from about 0.Ol Hz up to
5 kHz, corresponding to complete diffusion control
of the electrochemical insertion reaction over the
whole of the available frequency range. At lower
open-circuit voltages (corresponding to higher lithi u m concentrations i n the film), the separation of a
semicircular region is evident at higher frequencies
and, at the same time at lower frequencies, the 45 ~
line begins to give way to a vertical line u n d e r the
influence of finite length effects. The 5 Hz points are
m a r k e d on the plots, and it can be seen that at this
frequency a gradual transition from diffusion-controlled to charge t r a n s f e r - c o n t r o l l e d kinetics takes
place as l i t h i u m is added to the film.
Complex impedance data for the lowest frequencies
are shown i n Fig. 7. In this f r e q u e n c y range, the rate
OSCILLATOR ~
NPOT
I REFERENCE
TWO'PHASE I
ANALYZER I
OR
I
I
I
I
I SIGNAL
E'AMPL' 'ERI I
PO
INPUTS
WORKIN
rh ..........
.
/
~
I
I I REslSIANGE I
T ........
~
DC SOURCE
_L
"
500
200
-X
(ohms)
I00
400
500
600
700
R (ohms)
Fig. 6. High frequency complex impedance data
of l i t h i u m injection is always diffusion-controlled.
The onset of finite length effects shifts to higher frequencies and the low frequency limiting resistance
decreases as more l i t h i u m is added to the film, i n d i cating that the diffusion coefficient is steadily increasing. Values of the chemical diffusion coefficient, I~,
were obtained from either the 45 ~ straight line region
or else from the low frequency limiting behavior. A t
high voltages, in the semi-infinite diffusion regime,
data analysis presented no problems. Both the real
and the i m a g i n a r y parts of the impedance are proportional to ~-1/2. This is illustrated i n Fig. 8 for the highest voltage case. D was calculated from such plots
I
HEMICAL CELL
Fig. 4. Experimental circuit usedfor the A-C impedance measurements.
using Eq. [28] and [29]. Alternatively, D can be calculated from the low f r e q u e n c y limiting resistance
(Eq. [36]). The low frequency l i m i t i n g capacitance
(obtained from Eq. [34]) yields values of (dE/dy)y
which were used i n the computation of D. R e l e v a n t
data are given i n Table I.
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J. Electrochem. Soc.: E L E C T R O C H E M I C A L S C I E N C E A N D T E C H N O L O G Y
348
I
T
February 1980
I
I
.00:5
I
2 . 6 volts
2000
R
1500 -
4000
I
-X
-
.003
~1000
x
005
3000
-X
Ohms)
5po
0
.005
0
I
2
3
I/V~- ( Hz -"z)
2000
Fig. 8. Real and imaginary parts of the complex impedance
plotted vs. co- 1 1 2 .
Frequencies in Hz
o 2.6V
9 2.5V
I000
o 2.4V
& 2.3V
v 2.2V
/
I
]
I
IO00
2000
3000
R (ohms)
Fig. 7, Low frequency complex impedance data
Use of RL for the calculation of D, however, has a
d i s a d v a n t a g e in the p a r t i c u l a r case of the d a t a u n d e r
consideration here. RL is m e a s u r e d from the point at
w h i c h the 45 ~ s t r a i g h t line w o u l d i n t e r c e p t the r e a l
axis (see Fig. 3b), and thus, if %his point is n o t w e l l defined, due to o v e r l a p of the diffusion- and charge
t r a n s f e r - c o n t r o l l e d regions, an i n a c c u r a c y is i n t r o duced into the calculation of/~. The o v e r l a p also p r e vents accurate evaluation of CDL and 0.
These difficulties could, in principle, be overcome
b y fitting the low f r e q u e n c y i m p e d a n c e data to the
full i m p e d a n c e expression (Eq. [38] a n d [39]). H o w ever, the complicated n a t u r e of these functions m a k e s
this a difficult t a s k which was not a t t e m p t e d here.
Values of the diffusion coefficients c a l c u l a t e d for
lower voltages (2.2 a n d 2.3V) are, therefore, of l o w e r
accuracy t h a n those for h i g h e r voltages.
Discussion
The a-c method.--The technique described in this
p a p e r shows itself to be a useful tool for the s t u d y
of diffusion in thin film solid electrodes, a l t h o u g h it is
b y no m e a n s l i m i t e d in a p p l i c a t i o n to such systems.
The method appears to have a n u m b e r of advantages
over alternative, transient techniques (such as voltage
or c u r r e n t step e x p e r i m e n t s ) , the most important
being that m o r e i n f o r m a t i o n is o b t a i n e d t h r o u g h r e l a t i v e l y simple d a t a analysis procedures using the c o m p l e x i m p e d a n c e plane method. Even in situations
w h e r e t h e r e is extensive o v e r l a p of the charge t r a n s f e r - and diffusion-controlled regions, or w h e r e the
diffusion coefficient is sufficiently high to introduce
finite length effects, much i m p o r t a n t information can
be obtained b y inspection of the complex i m p e d a n c e
plot. The same i n f o r m a t i o n is, of course, also contained i n the t r a n s i e n t response of linear systems, b u t
d a t a acquisition ( p a r t i c u l a r l y at short times) and
i n t e g r a l t r a n s f o r m analysis so complicate the p r o cedure that often only d a t a o b t a i n e d over a l i m i t e d
time period (for example, w h e n the change in d e p e n d ent v a r i a b l e is p r o p o r t i o n a l to ~/t) are a c t u a l l y useful.
As m a y be seen from the p r e s e n t results, the region in
which the i m p e d a n c e is p r o p o r t i o n a l to ~/~ (or in
which E w o u l d be p r o p o r t i o n a l to tY~ in a c u r r e n t
step e x p e r i m e n t ) m a y be v e r y l i m i t e d or nonexistent.
U n d e r these circumstances, the complex i m p e d a n c e
m e t h o d at the v e r y least gives an indication of the e x p e r i m e n t a l l y i m p o r t a n t p h e n o m e n a contributing to
the time response of the system. Often a ' u s e f u l l i m i t ing region can be identified (for example, the v e r y
low f r e q u e n c y response for thin films) which can be
used to e x t r a c t n u m e r i c a l values for kinetic p a r a m eters.
A f u r t h e r a d v a n t a g e of using a s t e a d y - s t a t e m e t h o d
is that the response signal can be critically e v a l u a t e d
at each frequency, and f u r t h e r a v e r a g i n g (at low
frequencies) or filtering (at high frequencies) can be
employed.
L i k e the galvanostatic i n t e r m i t t e n t t i t r a t i o n t e c h nique described b y W e p p n e r and Huggins (16), the
a - c m e t h o d is capable of v e r y high resolution w i t h
respect to composition and is therefore applicable to
phases with r a t h e r n a r r o w ranges of stoichiometry.
The a p p l i c a t i o n of this m e t h o d is, of course, d e p e n d e n t on the v a l i d i t y of a n u m b e r of assumptions.
TaMe I. Thermodynamic and kinetic data
Composition*
(y in L~WOs)
O p e n cir~
cult voltage
(vs. L~)
0.097
0,138
0,170
0,201
0,260
2.60
2,59
2.40
2.30
2 20
~
( c m ~ sec -1)
2,4
4.9
1.5
2.6
2.8
•
x
•
x
x
10-~t
lO-~t
lO-n#t
lO-ntt
lO-ntl
(dE)**
-dy ~
5.0
3.6
4.0
3.8
3.6
d In aL,
d
In CLi
Dk
( c m ~ sec-D
18.9
19.4
26.5
29.8
36.5
1.3
2.5
5.7
8.7
7.7
•
x
x
x
x
10-73
i 0 -~
10-13
10-la
10-13
~m
(~-1 c m - D
1.04 x 10-~
3.0 • I0 ~9
8.1 x 10-9
1.5 x 10-~
1.7 x 10-8
" Calculated a s s u m i n g a density of 5.2 g c m -~.
** Calcuiated f r o m Eq. [34].
i ' D calculated f r o m Eq. [29].
t t ' D calculated f r o m Eq, [36].
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Vol. 127, No. 2
349
A P P L I C A T I O N OF A-C TECHNIQUES
The first of these is that th e b o u n d a r y conditions used
in the solution of Fick's diffusion equation are correct.
The greatest difficulty will u s u a l l y lie in the assumption that the potential of the cell is a m e a s u r e of the
l i t h i u m activity at the interface b e t w e e n the electrolyte a n d the electrode. This is only true if the electrode is a p r e d o m i n a n t l y electronic conductor. I n the
p r e s e n t case, there seems to be good evidence that
this is so. The high electronic conductivity of t u n g s t e n
bronze phases is well-established (20); values are
much higher t h a n ionic conductivities calculated from
previous or present m e a s u r e m e n t s of diffusion coefficients, although direct m e a s u r e m e n t s of partial ionic
conductivities have not been made.
A second a s s u m p t i o n implicit i n the derivation
given above is that the d r i v i n g force for diffusion is
only a gradient in composition, and that the electric
field in the electrode is negligible. This, of course,
will be true i n view of the high electronic conductivity
m e n t i o n e d above.
Thirdly, i n order for the solution of the diffusion
equations to be correct, it m u s t be assumed that the
system is linear. I n other words, the diffusion coefficient is assumed to be i n d e p e n d e n t of concentration
over the range of a l t e r n a t i n g voltage applied. I n practice, the applied voltage can be made very small, so
that the m e a s u r e d impedance becomes i n d e p e n d e n t of
amplitude. This test is very easy to carry out due to
the steady-state n a t u r e of the experiment. It seems
likely that the diffusion coefficient will change r a t h e r
slowly with composition, and hence voltage, except i n
very n a r r o w phases where rapid variation of the
t h e r m o d y n a m i c e n h a n c e m e n t factor m a y be expected
(16). S i m i l a r considerations of l i n e a r i t y also apply
of course at higher frequencies where the double
layer capacitance a n d charge transfer resistance might
also show voltage d e p e n d e n t behavior.
Kinetic properties of W03 thin film electrochromic
electrodes.--The p r i m a r y observation concerning the
kinetics of l i t h i u m incorporation into WO3 thin films
is that both diffusion and interface kinetics are i m portant. At long times (low frequencies), the i n t e r face is essentially at equilibrium, and diffusion of
n e u t r a l l i t h i u m i n a concentration gradient limits the
rate of coloration of the oxide. At times shorter t h a n
about 0.5 sec however, charge transfer kinetics limit
the rate of injection. The following considerations
are relevant.
(i) Firstly, the t h e r m o d y n a m i c and kinetic properties of WO8 thin films are very d e p e n d e n t on the
method of p r e p a r a t i o n (6), and in p a r t i c u l a r are dep e n d e n t on the degree of crystallinity of the films.
The results given here, although typical of films which
show r e l a t i v e l y rapid coloration on l i t h i u m injection,
must not be regarded as applying to all WO3 films. A
detailed study of the thermodynamics and kinetics of
different kinds of films is c u r r e n t l y u n d e r w a y a n d
the results will be published separately.
(it) Secondly, the experiments reported here are
small signal experiments. I n a practical device m u c h
larger voltages or c u r r e n t s will be applied to the
samples with several consequences. First, the reaction
resistance o will change with applied voltage due to
both the change in l i t h i u m concentration at the i n t e r face and the i n h e r e n t voltage dependence of the
heterogeneous charge transfer reaction. Secondly,
large voltage differences across the interface will have
i n . p o r t a n t effects on the rate of bleaching as discussed
b y F a u g h n a n et al. (21). If the potential of the surface layer of WO3 is increased beyond about 3V with
respect to lithium, then the surface will become depleted of l i t h i u m and, since the electronic conductivity
falls as l i t h i u m is removed, an electric field will develop i n the electrochromic which will tend to i n crease the rate of l i t h i u m removal. This might happen,
for example, when the WO3 is bleached u n d e r constant c u r r e n t conditions, when the diffusion process
cannot indefinitely support the current. A third effect
of higher voltages will occur u n d e r rapid coloration
conditions where the voltage falls to such low values
that a concentration d r i v e n phase change on the s u r face of the oxide might be irreversible. Mohapatra
and Wagner (9), for example, found that the chemical potential of l i t h i u m in amorphous LiyWO3 was
constant for y > 0.4 ( a p p r o x i m a t e l y 2V vs. Li). If
this constant activity corresponds to a two- or t h r e e phase region, then the n e w phase(s) formed m a y
have~ very slow or i r r e v e r s i b l e kinetics.
I n summary, therefore, small signal results should
not be t a k e n as a definitive measure of how a n electrochromic device would operate u n d e r large signal
conditions.
The charge transfer resistance of the interface was
found to be 100-150 t l / c m 2. As discussed i n the Results
section, an overlap of the diffusion response a n d the
charge transfer response prevents a n accurate e v a l u ation of this resistance and its voltage dependence.
The corresponding exchange c u r r e n t density is 8 X
10-~-1.3 X 10 -4 A / c m 2, a value which will depend
on the concentration of Li + in the liquid electrolyte
as well as electrode m a t e r i a l parameters. Mohapatra
(8) has suggested a value as high as 2 X 10 -8 A / c m 2
for l i t h i u m injection. Somewhat smaller values (6 X
10-6-3 X 10 -5 A / c m 2) have been found b y C r a n d a l l
and F a u g h n a n (22) for the injection of hydrogen into
amorphous WO8.
Chemical diffusion coefficients for low l i t h i u m concentrations are in good a g r e e m e n t with the value of
2 • 10 -12 cm2/sec suggested b y G r e e n (3). A n i n crease with increasing l i t h i u m concentration of about
one order of m a g n i t u d e is observed, b u t the values
are still much lower t h a n the 5 X 10 -9 cm2/sec r e ported by Mohapatra (8). The t h e r m o d y n a m i c results
given by Mohapatra and W a g n e r in Ref. (9), however,
were in very good agreement with the results of the
present study. More recent results b y G r e e n (10)
show a slightly smaller value for D of 4 X 10 -18 cm2/
sec on films which also have similar coulometric t i t r a tion curves. G r e e n also suggested that rapid diffusion
along grain boundaries is followed b y m u c h slower
diffusion into the b u l k of the grains, with an associated diffusion coefficient of about 6 X 10 -16 cm2/sec.
In the present study, no evidence for two diffusion
processes, either i n series or in p a r a l l e l has b e e n
found, as the form of the results is adequately explained by a single diffusion coefficient.
As discussed by W e p p n e r and Huggins (16), the
chemical diffusion coefficient/~ is related to the component diffusion coefficient Dk by the relation
d In aLI
= Dk ~
d In CLl
[43]
for the case where the electronic transference n u m b e r
is close to unity. I n this equation the activity and concentration terms refer to n e u t r a l lithium. Hence
dE zF
dy RT
[441
and since
Dk = kTb
[45]
and
trLi =
z2F2CLib
[46]
where b is the general mobility and ~ the partial
conductivity of Li + ions
~Li =
VM
\-~y--y/
[47]
Numerical values for Dk and ~Li are given in the table.
Like the chemical diffusion coefficient, these q u a n t i ties appear to increase with increasing l i t h i u m c o n -
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350
J. Electrochem. Soc.: E L E C T R O C H E M I C A L S C I E N C E A N D T E C H N O L O G Y
tent, the most rapid rate of increase occurring a r o u n d
y ----0.17.
I n summary, a general steady-state a-c method
has been described which allows the diffusion coefficient of a species in a n electronically conducting
t h i n film electrode to be determined. I n addition,
qualitative i n f o r m a t i o n about the kinetics of t h e
interface charge transfer reaction has been obtained,
and it is believed that in some situations detailed
interfacial kinetic data could be obtained. The method
has a n u m b e r of advantages over t r a n s i e n t c u r r e n t or
voltage step techniques, s t e m m i n g partly from the
method of analysis i n the f r e q u e n c y domain which
allows a clear distinction to be d r a w n b e t w e e n i n terface and b u l k processes.
Acknowledgment
The c o m p u t e r p r o g r a m for the low frequency m e a surements was conceived and w r i t t e n b y Dr. B. A.
Boukamp of this laboratory, to whom the authors are
most grateful.
Manuscript s u b m i t t e d J u l y 30, 1979; revised m a n u script received Aug. 28, 1979.
A n y discussion of this paper will appear in a Discussion Section to be published in the December 1980
JOURNAL. All discussions for the December 1980 Discussion Section should be submitted b y Aug. 1, 1980.
Publication costs of this article were assisted by
Stanyord University.
REFERENCES
1. H. N. Hersh, W. E. Kramer, and J. H. McGee, Appl.
Phys. Lett., 27, 646 (1975).
2. R. J. Colton, A. M. Guzman, and J. Wayne Rabalais,
J. Appl. Phys., 49, 409 (1978).
3. M. Green, W. C. Smith, and J. A. Weiner, Thin
February I980
Solid Films, 38, 89 (1976).
4. M. Green a n d K. S. Kang, ibid., 44, L19 (1977).
5. W. C. D a u t r e m o n t Smith, M. Green, and K. S. Kang,
Electrochim. Acta, 22, 751 (1977).
6. H. R. Zeller and H. U. Beyeler, J. Appl. Phys., 13,
231 (1977).
7. T. J. Knowles, Appl. Phys. Lett., 31, 817 (1978).
8. S. K. Mohapatra, This Journal, 125, 284 (1978).
9. S. K. Mohapatra and S. Wagner, ibid., 125, 1603
(1978).
10. M. Green, Thin Solid Films, 50, 145 (1978).
11. M. S l u y t e r s - R e h b a c h and J. H. Sluyters, in "Electroanalytical Chemistry," Vol. 4, A. J. Bard, Editor, pp. 1-128, Marcel Dekker, New York (1970).
12. D. D. Macdonald, "Transient Techniques in Electrochemistry," p. 229, P l e n u m Press, New York
(1977).
13. D. E. Smith, in "Electroanalytical Chemistry," Vol.
1, A. J. Bard, Editor, pp. 1-155, Marcel Dekker,
New York (1966).
14. J. E. B. Randles, Discuss. Faraday Soc., 1, 11 (1947).
15. E. Warburg, Ann. Physik, 67, 493 (1899); 6, 125
(1901).
16. W. Weppner and R. A. Huggins, This Journal, 124,
1569 (1977).
17. W. W e p p n e r and R. A. Huggins, Annual Review of
Materials Science, R. A. Huggins, Editor, p. 269,
A n n u a l Reviews Inc., Palo Alto (1978).
18. H. S. Carslaw and J. C. Jaeger. "Conduction of
Heat i n Solids," 2nd Ed., p. 105, Oxford Univ.
Press, Oxford (1959).
19. J. P. Randin, J. Electron. Mater., 7, 47 (1978).
20. P. J. Wiseman and P. G. Dickens, in "MTP I n t e r national Review of Science." Inorganic Chemistry. Series II. Vol. I0, L. E. J. Roberts, Editor,
p. 211, London (1975).
21. B. W. Fau~hnan. R. S. Crandall. and M. A. Lampert,
Appl. Phys. Lett., 27, 275 (1975).
22. R. S. Crandall and B. W. F a u g h n a n , Appl. Phys.
Lett., 28, 95 (1976).
Development of Sulfur-Tolerant Components
for the Molten Carbonate Fuel Cell
A. F. Sammells,* S. B. Nicholson, and P. G. P. Ang*
Institute of Gas Technology, Chicago, I~linois 60616
ABSTRACT
The sulfur tolerance of candidate anode and anode c u r r e n t collector m a t e r i a l s for the m o l t e n carbonate fuel cell were evaluated in an electrochemical
half-cell using both steady-state and transient potentiostatic techniques. Hydrogen sulfide was introduced into the fuel at concentrations of 50 and 1000
ppm. At the higher sulfur concentration using low BTU fuel, both nickel and
cobalt were observed to undergo a negative shift in their open-circuit potentials, and high anodic and cathodic currents were observed compared with clean
fuels. Exchange currents measured using the transient potentiostatic technique
were not greatly affected by 50 ppm H2S introduced into the fuel. However,
at higher sulfur concentrations, higher apparent exchange currents-were observed, indicating a probable sulfidation reaction. Of the new anode materials
evaluated, Mg0.05La0.95CrO3
and TiC showed
good stability in the anodic
region. With the former material, exchange current densities in low BTU fuel
were calculated to be ~-~8 mA/cm 2 at 650~
lower values than found for
either nickel or cobalt anodes under similar conditions. Of the anode current
collector materials evaluated, high stabilities were found for 410 and 310 stainless steels. The implications and relevance of these results on fuel cell performance are discussed.
The essential components which comprise the molten
carbonate fuel cell are a porous nickel or cobalt anode,
and a porous nickel oxide cathode, which are separated
b y an ionically conducting m o l t e n carbonate m i x t u r e
* Electrochemical Society Active Member.
Key words: sulfur-tolerant anode, current collectors, molten
carbonate fuel cell.
supported on a l i t h i u m a l u m i n a t e matrix. These components together are commonly referred to as the tile.
To date, these fuel cell components have shown g o o d
electrochemical performance and corrosion stability
u n d e r cell operating conditions over several thousand
hours in the absence of s u l f u r - c o n t a i n i n g species in t h e
fuel and oxidant. However, commercialization of this
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