Department of Physics and Astronomy, University of North Carolina, Chapel Hill, North Carolina 27514. The identification and characterization of the pro-.
Small-Signal A-C ResponseTheory for ElectrochromicThin Films Donald R. Franceschetti* Department of Physics, Memphis State University, Memphis, Tennessee 38152
and J. Ross Macdonald* Department of Physics and Astronomy, University of North Carolina, Chapel Hill, North Carolina 27514 The identification and characterization of the processes responsible for the electrochromic properties of thin transition m e t a l oxide films are m a t t e r s of high c u rr en t interest. S e v e r a l authors (1-3) have applied small-signal a-c techniques in this area. Ho et al. (2) have analyzed their a-c data on WO3 with injected Li using the standard Randles (4) e q u i v a l e n t circuit, but with a modified (finite length) W a r b u r g element. Glar u m and Marshall (3) have devised a slightly different circuit from their data on IrO2 with injected protons. Both sets of authors have given some discussion of the t h e o r y u n d e r l y i n g the use of these circuits. In a somew h a t ear l i er paper (5) the present authors d e r i v ed an e q u i v a l e n t circuit for an electrochemical system characterized by an electrode adsorption-reaction-diffusion sequence that yields the circuits m e n t i o n e d above, or parts of them, as limiting cases. Much of this a n a l y sis has been r e c e n t l y republished i n d e p e n d e n t l y by Br au n s h t ei n et al. (6). In the present paper we discuss our earlier t r e a t m e n t as it m i g h t be applied to an electrochromic system. Our t r e a t m e n t leads to an e q u i v alent circuit which, we believe, may_ be useful in the analysis of impedance or admittance data on electrochromic thin films, p a r t i c u l a r l y if the injection of atoms into the film involves an adsorbed intermediate. We consider an electrochemical cell consisting of an inert electronic conductor, a thin l a y e r of electrochromic m a t e r i a l AyB, a liquid electrolyte with mobile A + ions, and an electrode of solid A metal, or if A represents hydrogen, a h y d r o g e n electrode. We shall assume that c u r r e n t flow th r o u g h the system is effectively one-dimensional, at least over the region in which a significant potential drop occurs. We also assume that AyB is a sufficiently good electronic conductor that the transport of A w i t h i n the l a y e r of A~B occurs p u r e l y by diffusion. We assume that the system has been allowed to come to e q u i l i b r i u m u n d e r a steady applied potential difference. T h e n the AyB la y e r has a spatially u n i f o r m composition and the potential drop falls essentially b etween the surface of the A~B la y e r in contact with the electrolyte and the A electrode. We ass~,me that an A-+ion combines w i t h an electron f r o m the conduction band to f o r m an adsorbed i n t e r m e d i a t e before e n t e r ing the AyB film. Adopting the notation of our ear l i er w o r k (5), we let PR denote the concentration of the A + ions at the point of closest approach to the AyB film, let r denote the concentration of the adsorbed intermediate, and let bL denote the concentration of A just inside the surface of the AuB film. Then for any deviation f r o m the e q u i l i b r i u m potential difference the equations g o v e r n i n g the b e h a v i o r of the r e a c t a n t species at the A~B/liquid interface m a y be w r i t t e n (5, 7) /pR ~--- evl (PR, F, ~])
[1]
d r / d t = v l (PR, ]?, ~l) -- V2 (D, bL)
[2]
Jb~ = v~ (r, bL)
[3]
and w h e r e IpR is the faradaic current, JbL is the flux of A into the AyB layer, vl and v2 are as yet unspecified rate functions, and ~l is the additional potential drop a c r o s s * Electrochemical Society Active Member. Key words: films, impedance, diffusion.
the compact layer b e t w e e n the AyB film and the liquid electrolyte. U n d e r s m a l l - s i g n a l a-c conditions we m a y separate e a f h of the variab es in Eq. [1]-[3] into an e q u i l i b r i u m part and a sinuso~dal perturbation, e.g., PR = P0R --~ PiR exp (i~,t). On m a k i n g an appropriate Taylor series expansion of the reaction rates about their e q u i l i b r i u m values, we obtain
Ipl• ~- e[klfplR -- k~hrl + (e~ll/kT)71fPoa] i~rl = IplR/e
--
k~frl
+
k3bbla
[4] [5]
and Jbla :
k 3 f r i - - k3blI.
[6]
w h e r e each of the k's and 71~ represents a p ar t i a l der i v a t i v e of the rate functions vl and vf. We assume that within the AyB l ay er the transport of A is g o v e r n e d by Fick's laws, with diffusion constant Die. In this note we sha!l assume that the A atoms are completely blocked at the interface b e t w e e n the AyB l a y e r and the i ne r t electronic conductor, a physically reasonable assumption for the e x p e r i m e n t a l a r r a n g e m e n t s that have been employed. In this case, the result obtained in Ref. (5) m a y be w r i t t e n as IplR = e [ k i * P l R + (e~l/kT)71*poR]
[7]
w h er e kl* = flkif and 71" ---- flTlf, with fl -- {1 ~- k l b / [ i a ~- k s f / ( 1 + F1 ( w ) ) ] } - - i
[8]
and F l ( ~ ) --
kab
ctnh (leslie~Die)
[9]
w h e r e le is t h e thickness of the AyB film. The qua nt i ties kl* and 71" m a y be considered to be complex, f r e q u e n c y - d e p e n d e n t rate constants, a notion first i n t r o duced by L~nyi (8). If Re is a constant n o r m a l i z i n g r e sistance, it m ay readily be shown that RcFl(w) is t h e i m p ed an ce of a length l~ of distributed transmission line of characteristic fmpedance Rck3b/(i~D~e) 1/2 w i t h series resistance per unit l e n g t h R~e~ =- Rckab/Dle and shunt capacitance p er unit l e n g t h C~n = i~/k~bRc, t e r m i n at ed by an infinite resistance. I f the liquid electrolyte e m p l o y e d in the e x p e r i m e n t a l system is fairly concentrated ( > 1M) and assuming that the A § ions are appreciably m o r e mobile in the solution than A atoms are in the solid AyB one m a y neglect PlR in Eq. [7] and then define an i n t e r facial admittance I~IR efPoR --
-
~i
-
kT
~'1"
[10]
w h i ch is re~resented ex act l y by the e q u i v a l e n t circuit of Fig. 1. The circuit elements are the charge t r a n s f e r resistance RR : k T / ( e 2 p o R ~ l f ) [11] the adsorption capacitance
CA = 1/(RRklb)
[12]
an adsorption related resistance RA = R R k i b / k 3 f
[13]
and a distributed capacitative e l e m e n t v d t h i m p e da nc e
1754
Vol. 129, No. 8
I toA ZD RA Fig. 1. Equivalent circuit representing the interfacial impedance. See Eq. 1-10]-1-15].
ZD
=
1755
ELECTROCtIROMIC THIN FILMS
[14]
ZDo c t n h (~iwLe2/Dle)/~i~le2/Dle
with
RRklbZe/k3fDle
ZD O :
[15]
W h e n the W a r b u r g e l e m e n t and charge transfer resistance in the Randles circuit (2, 4) are replaced by the circuit segment shown in Fig. 1, one obtains the equivalent circuit appropriate for the system considered in this note.
Some impedance plane plots for this generalized Randles circuit are shown in Fig. 2. In Fig. 2(a) we have set RA and CA equal to zero so that our circuit reduces to that of Ha et al. (2), consisting of a b u l k (liquid electrolyte) resistance Re, double layer capacitance CD, charge transfer resistance RR, and the distributed capacitative e l e m e n t ZD. The figure shows a single semicircular arc, associated with RR and CD, and a straight segment, with 45 ~ slope which curves to approach a vertical asymptote, characteristic of Z#. I n Fig. 2(b) RA and CA have been given values so that RACA > > R•CD, and two semicircular arcs are apparent, the one at lower frequencies being associated with RA and CA. In Fig. 2(c), RACA ~ RRCD and only a single, approximately semicircular arc is apparent. In fact, the impedance curves of Fig. 2(c) and (a) a r e almost indistinguishable in shape, even though they represent two d~stinctly different sets of circuit parameters. We are thus led to suggest thai any determination of circuit parameters by graphical analysis of impedance plane curves be confirmed by nonlinear least-squares fitting of the data as a function of frequency to the circuit concerned (9).
(a) lO'SHz
c~
A
% A
(c)
N
E
m
i
10-SHz, 6
_
s
N
4
,
3
]
0
I
2
3
4
5
6
7
8
Re(Z) (104,o.) 9
84
(b)
8 7
10-SHz
1"-4 4
E 5
,
2 lOHz
I0" Hz / , , ~ ~ ' '
" ~ 1 H 0
.....
\
I
2
3
4
5
6
7
8
9
(104~)
Fig. 2. Impedance plane plats for Randles equivalent circuit with charge transfer resistance and Warburg impedance replaced by circuit of Fig. 1. The bulk resistance, R~ z 1~, double layer capacitance, CD = 1 /~F, and distributed capaeitative element ZDo ---- 1000~, le2/Dle ~ 250 sec, are the same for all plots. (a) RH ---40,000Q,, CA ~ 0, RA = 0. (b) RR ---- 15,000_O., CA ~- 100 ~F, RA = 25,000~. (c) RR = 15,000~, CA z 1 ~F, RA = 25,00&Q.
5
I
'
10-4H z
Re(Z)
v
I
~
0
6
o
2
z
/'~/
2 Hz / / "
" e " ~ l 0"3 H Z
I
I
I
1
I
!
I
I
I
2
3
4
5
6
7
8
Re(Z) (1049,)
1756
J. Elec$rochem. 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
Acknowledgment This w o r k was s u p p o r t e d in p a r t b y the N a t i o n a l Science F o u n d a t i o n u n d e r G r a n t DMR-80-05236, b y the U.S. A r m y Research Office, b y Research C o r p o r a tion G r a n t 1696, and b y a Memphis State U n i v e r s i t y F a c u l t y Research Grant. M a n u s c r i p t s u b m i t t e d June 29, 1981; revised m a n u script received Jan. 23, 1982. A n y discussion of this p a p e r will a p p e a r in a Discussion Section to be p u b l i s h e d in the J u n e 1963 JOURNAL. All discussions for the J u n e 1983 Discussion Section should be s u b m i t t e d b y Feb. 1, 1983.
Publication costs o~ this article were assisted by Memphis State University.
August 1982
REFERENCES 1. B. R e i c h m a n and A. J. Bard, This Journal, 126, 583 (1979). 2. C. Ho, 1. D. Raistrick, and R. A. Huggins, ibid., 127, 343 (1980). 3. S. H. G t a r u m and J. H. Marshall, ibid., 127, 1467 (1980). 4. J. E. B. Randles, Disc. Faraday Soc., 1, 11 (1947). 5. D. R. F r a n c e s c h e t t i and J. R. Macdonald, J. Electroanal. Chem. lnter]acial Electrochem., 101, 307 (1979). 6. D. Braunshtein, D. S. Tannhauser, and I. Riess, This Journal, 128, 82 (1981). 7. D. R. F r a n c e s c h e t t i and J. R. Macdonald, J. ElectroanaL. Chem. Interracial Electrochem., 82, 271 (1977). 8. S. Lanyi, Solid State Commun., 27,743 (1978). 9. J. R. M a c d o n a l d and J. A. Garber, This Journal, 124, 1022 (1977).
