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The impedance of two-terminal electrochemical cells
Surface Technology, 13 (1981) 339 - 348
339
THE I M P E D A N C E OF TWO-TERMINAL ELECTROCHEMICAL CELLS
S. A. G. R. KARUNATHILAKA and N. A. HAMPSON Department of Chemistry, University of Technology, Loughborough, Leics. LE11 3TU (Gt. Britain) R. LEEK Electronic and Electrical Engineering Department, University of Technology, Loughborough, Leics. LE11 3TU (Gt. Britain) (Received December 1, 1980)
Summary The i m p e d a n c e b e h a v i o u r of cells c o m p o s e d of simple c o m b i n a t i o n s of e l e c t r o d e s is c a l c u l a t e d . The v a r i a t i o n in the i m p e d a n c e for some complex t y p e s of e l e c t r o d e b e h a v i o u r is also c a l c u l a t e d . The results are c o m p a r e d w i t h e x p e r i m e n t a l data. The W a r b u r g i m p e d a n c e u l t i m a t e l y d o m i n a t e s e v e r y s i t u a t i o n if the f r e q u e n c y is r e d u c e d sufficiently.
1. I n t r o d u c t i o n An i m p o r t a n t w a y of assessing the s u r f a c e c h a r a c t e r i s t i c s of a single e l e c t r o d e is to s t u d y its i m p e d a n c e . The e l e c t r o d e c a p a c i t a n c e m e a s u r e d in an indifferent e l e c t r o l y t e is a m e a s u r e of the t r u e electrode a r e a and of the p h y s i c a l and e l e c t r i c a l p r o p e r t i e s of the i n t e r p h a s e . A c l e a n electrode in the p o l a r i z a b l e r e g i o n yields the d o u b l e - l a y e r c a p a c i t a n c e Cc, and the v a r i a t i o n in the Cc E c u r v e o v e r a r a n g e of c o n c e n t r a t i o n in w h i c h diffuse-layer c h a r a c t e r i s t i c s p r e d o m i n a t e yields the p o i n t of zero change. In an e l e c t r o l y t e s o l u t i o n c o n t a i n i n g an e l e c t r o a c t i v e species, m e a s u r e m e n t of the i m p e d a n c e at a series of f r e q u e n c i e s enables the k i n e t i c c o n s t a n t s of the e l e c t r o d e r e a c t i o n to be d e t e r m i n e d . T h e s e effects h a v e been r e v i e w e d e l s e w h e r e [1]. In a c o n v e n t i o n a l e l e c t r o c h e m i c a l e x p e r i m e n t using a t h r e e - t e r m i n a l cell the e l e c t r o d e of i n t e r e s t ( w o r k i n g electrode) has an effective i m p e d a n c e of a b o u t a f a c t o r of 102 g r e a t e r t h a n the c o u n t e r e l e c t r o d e has, so t h a t w h e n a small a.c. p o t e n t i a l (usually less t h a n 7.5 m V m e a s u r e d with r e s p e c t to a r e f e r e n c e ) is applied to the cell the c u r r e n t flow is d o m i n a t e d by this e l e c t r o d e and the cell i m p e d a n c e is t h a t of the w o r k i n g electrode. U n d e r t h e s e c o n d i t i o n s the e x p e r i m e n t a l set-up reflects the b e h a v i o u r of one electrode. 0376-4883/81/0000-0000/$02.50
J(' Elsevier Sequoia/Printed in The Netherlands
340
The i m p e d a n c e of small e l e c t r i c a l s t o r a g e cells is of i n t e r e s t not only for the p u r p o s e of p r e d i c t i n g t h e i r b e h a v i o u r in e l e c t r o n i c circuits but also b e c a u s e it is a useful p a r a m e t e r for a s s e s s i n g t h e i r q u a l i t y [2]. We h a v e s h o w n [3] t h a t in some of these s y s t e m s one e l e c t r o d e d o m i n a t e s the cell b e h a v i o u r and t h a t the cell can be effectively t r e a t e d as the i m p e d a n c e of a single c o n t r o l l i n g e l e c t r o d e [3]. H o w e v e r , some cells do not c o n f o r m to this simple b e h a v i o u r a n d b o t h e l e c t r o d e s m u s t be considered. In this p a p e r we p r e s e n t some t h e o r e t i c a l p r e d i c t i o n s of the b e h a v i o u r expected for s i t u a t i o n s c o m m o n l y e n c o u n t e r e d in such systems.
2. P r o c e d u r e
In o r d e r to o b t a i n the i m p e d a n c e f r e q u e n c y b e h a v i o u r c o r r e s p o n d i n g to s y s t e m s of i n t e r e s t it. was c o n s i d e r e d t h a t the c o m p l e x p l a n e r e p r e s e n t a t i o n would allow the m a x i m u m of i n f o r m a t i o n to be m o s t r e a d i l y presented. In this m e t h o d the real and i m a g i n a r y p a r t s of the t o t a l i m p e d a n c e are plotted a g a i n s t e a c h o t h e r p a r a m e t r i c a l l y u s i n g f r e q u e n c y as the variable. This m e t h o d is well k n o w n , a n d this t y p e of plot is c o m m o n l y referred to as a S l u y t e r s plot. D a t a for these r e p r e s e n t a t i o n s w e r e o b t a i n e d by u s i n g the v e c t o r e q u a t i o n for e a c h system. The c a l c u l a t i o n and i s o l a t i o n of the real Z' and i m a g i n a r y - Z " p a r t s of the i m p e d a n c e were p e r f o r m e d u s i n g the complex n u m b e r h a n d l i n g c a p a b i l i t y of a P R I M E 400 c o m p u t e r . As an e x a m p l e we c o n s i d e r the n e t w o r k s h o w n in Fig. 1. The impeda n c e s of the c o m p o n e n t p a r t s a r e g i v e n by
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Fig. 1. Circuit representing a charge transfer and diffusion process with active adsorption of the intermediate species and an accompanying or preceding chemical reaction.
341
and the e q u i v a l e n t i m p e d a n c e at an a n g u l a r f r e q u e n c y (~ is given by
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The real and i m a g i n a r y parts of Z are o b t a i n e d from the c o m p u t e r and printed out g r a p h i c a l l y .
3. R e s u l t s a n d d i s c u s s i o n F i g u r e 2 shows the familiar plot of the impedance c o m p o n e n t s for an electrode at w h i c h an e l e c t r o n t r a n s f e r r e a c t i o n is followed by diffusion in solution. The m a g n i t u d e s of the c o m p o n e n t values were chosen to represent the b e h a v i o u r of a zinc electrode typical of a small p r i m a r y cell (e.g. 2000 mA h of the a l k a l i n e Z n - M n O 2 or a l k a l i n e Z n - H g O type).
100
CL
0
. . . . . . . ,&. . . . . . . . . . . . . . . . . . . 1'00
(a)
Fig. 2. (a) Randles circuit: C1, = 1157 I~F; 0 plot correspofiding to (a).
(b)
~. . . . . . . . . 1,S0
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0.055 £1; a = 0.0567 gl s 1'2. (b) Complex plane
F i g u r e 3(a) shows the a n a l o g u e for two electrodes in series (in w h a t follows the s o l u t i o n r e s i s t a n c e R~ is omitted ; if present it is r e p r e s e n t e d by the i n t e r c e p t on the real axis at ~ = ~2) and Fig. 3(b) shows the cell b e h a v i o u r of the a r r a n g e m e n t displayed in the complex plane. The shape of the locus is identical with t h a t of a single electrode ; however, the m a g n i t u d e s of the s h a p e are c o n s i d e r a b l y increased. The W a r b u r g c o n t r i b u t i o n still has a g r a d i e n t of 4 5 . F i g u r e 3(c) shows how the high f r e q u e n c y semicircle is distorted by c h a n g i n g the double-layer and c h a r g e t r a n s f e r resistances of one electrode in a w a y e q u i v a l e n t to r e d u c i n g the area of this electrode by 50%. W h e n the i m b a l a n c e b e t w e e n the electrode areas becomes gross for identical c h a r g e t r a n s f e r c o m p o n e n t s a s e c o n d semicircle at high f r e q u e n c y with an identical radius to the first develops f u r t h e r a l o n g the real axis (Fig.
342
3(d)). When the charge transfer components are significantly different in addition to having different area factors (and therefore different doublelayer capacitances), this is reflected in the relative magnitudes of the two high frequency semicircles (Fig. 3(e)). However, the Warburg part of the complex plane representation remains the same for all these types of behaviour.
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a 0.'0 0.?C 0.3,3 0"~*, 0 "~(' 150 200 2 %0 3,00 (e) R, 1C2L Fig. 3. (a) Cell r e p r e s e n t a t i o n c o m p r i s i n g t w o R a n d l e s circuits in series: (!I~ = 1157 t.d~':
0
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( c ) Ci~ = 0 . 5 C L : d' = 0 . 5 0 .
2 ~!
343
3.1. The effect of adsorption at the electrodes A d s o r p t i o n effects at e l e c t r o d e s c a n arise from a n u m b e r of causes. T h e simplest effect is w h e n a l a y e r of i n a c t i v e o r g a n i c i m p u r i t y is deposited o n t o t h e e l e c t r o d e and simply r e d u c e s the c a p a c i t a n c e e v e r y w h e r e to a low value. H o w e v e r , o t h e r a d s o r p t i o n effects arise as a c o n s e q u e n c e of the electroc h e m i s t r y . An o b v i o u s effect is w h e n the electrode p r o d u c t s and r e a c t a n t s u n d e r g o a r e a c t i o n in the a d s o r b e d s t a t e at the e l e c t r o d e w h i c h m a y or m a y not be coupled w i t h a diffusion process in solution. F o r example, an o r g a n i c c o m p o u n d m a y u n d e r g o a r e d o x r e a c t i o n in the a d s o r b e d s t a t e w i t h the p r o d u c t s r e m a i n i n g at the e l e c t r o d e ; altern a t i v e l y , the r e a c t i o n p r o d u c t s m a y b e c o m e desorbed a f t e r the r e a c t i o n and pass into solution. T h e s e processes m a y o c c u r t o g e t h e r and p r o v i d e b o t h a p a r a l l e l r e a c t i o n p a t h w h i c h is r e p r e s e n t e d by a p a r a l l e l RyCy b r a n c h and an a d d i t i o n a l step in the o v e r a l l r e a c t i o n w h i c h is r e p r e s e n t e d by an a d s o r p t i o n c o m p o n e n t R× Cx in the e l e c t r o d e a n a l o g u e . The m o s t f a m i l i a r process is t h a t w h e n the c h e m i c a l r e a c t i o n w h i c h precedes or a c c o m p a n i e s t h e e l e c t r o d e p o s i t i o n is a m e t a l l u r g i c a l step in a m e t a l d e p o s i t i o n or d i s s o l u t i o n r e a c t i o n , e.g. Mlattic ~ ~
Maa ~
z-V Msolution
(6)
w h i c h is the w e l l - k n o w n a d a t o m m e c h a n i s m for m e t a l ion e x c h a n g e reactions. The e l e c t r i c a l a n a l o g u e s h o w n in Fig. 4(a) was first s u g g e s t e d by G e r i s c h e r [4] a n d differs slightly from a r a t h e r m o r e complex a n a l o g u e p r o p o s e d by F l e i s c h m a n n et al. [5] in w h i c h an a d d i t i o n a l c o m p o n e n t e x p r e s s i n g s u r f a c e diffusion is included in the resistive c o m p o n e n t of the c r y s t a l l i z a t i o n i m p e d a n c e . The i m p e d a n c e locus for the simpler process is s h o w n in Fig. 4(b) for the case of a r e l a t i v e l y small a d a t o m c o n c e n t r a t i o n a n d a r e l a t i v e l y fast s u r f a c e diffusion process. It h a s a v e r y e l o n g a t e d s h a p e w h i c h is formed from two semicircles of differing radii. F o r the case of an e v e n f a s t e r s u r f a c e diffusion process the well-defined t w i n semicircles o v e r l a p to form an e l o n g a t e d single semicircle. W h e n the surface conc e n t r a t i o n is i n c r e a s e d the t w i n s e m i c i r c u l a r s h a p e is a g a i n evident as s h o w n in Fig. 4(d). In all these cases the W a r b u r g p a r t of the locus r e m a i n s a n d h a s a c o n s t a n t g r a d i e n t of 4 5 . We can c o n c l u d e from these s h a p e s t h a t w h a t e v e r the m e c h a n i s m of the e l e c t r o c h e m i c a l r e a c t i o n at the electrodes the s o l u t i o n process u l t i m a t e l y d o m i n a t e s , and t e c h n o l o g i c a l l y in a d.c. e x p e r i m e n t the "fine s t r u c t u r e " of the e l e c t r o d e r e a c t i o n is rapidly overs h a d o w e d by the s o l u t i o n processes w h i c h u l t i m a t e l y control the c u r r e n t . T h e m o s t g e n e r a l case of a d s o r p t i o n at the e l e c t r o d e s occurs w h e n t h e r e is a r e a c t i o n in the a d s o r b e d s t a t e at the electrode in addition to the c h a r g e t r a n s f e r r e a c t i o n and a p r e c e d i n g or a c c o m p a n y i n g c h e m i c a l reaction. This s i t u a t i o n could arise for the case of an electrode at w h i c h an o r g a n i c r e d o x r e a c t i o n was o c c u r r i n g in p a r a l l e l w i t h a c h a r g e t r a n s f e r r e a c t i o n (metal deposition) w i t h a d a t o m diffusion s u c h as m i g h t be env i s a g e d at the n e g a t i v e e l e c t r o d e of a cell in the p r e s e n c e of an o r g a n i c e x p a n d e r . T h e e l e c t r o d e a n a l o g u e of this is s h o w n in Fig. 1.
344
Y o 1 5C,
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(a)
(b) ci
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R,,lc a ......................................
9
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1.oo
~o,¢
0.50
0 SO
I O0
1.60
200
0 50
(d)
1.00
I SO
2 30
h','lO i i
Fig. 4. (a) Electrode representation for the case when an electron transfer reaction with diffusion in solution precedes or accompanies the eleetrodeposition. (b) CA = 10CI. : R A = 0. (c) Ca = 10CI~; RA = 0.50. (d) ('a = 30CL: R,\ = 0.750.
F i g u r e 5(a) s h o w s t h a t t h e i m p e d a n c e l o c u s f o r t h e c o m p l e t e c i r c u i t is q u i t e s i m p l e a n d m i g h t e a s i l y be t a k e n for a s i m p l e c a s e o f c h a r g e t r a n s f e r and diffusion. There are relatively minor distortions of the semicircular shape at high frequencies and at two intermediate frequencies which i n d i c a t e t h a t t h e e l e c t r o d e is c o m p l e x . T h e W a r b u r g c o n t r i b u t i o n h a s a g r a d i e n t o f 45 ~s e x p e c t e d . If t h e r e is n o a c c o m p a n y i n g o r p r e c e d i n g e l e c t r o d e r e a c t i o n , t h e s i m p l e c h a r g e t r a n s f e r r e a c t i o n is c o m b i n e d w i t h h p a r a l l e l r e a c t i o n in t h e a d s o r b e d s t a t e as d i s c u s s e d o r i g i n a l l y b y L a i t i n e n a n d R a n d l e s [6]. F i g u r e 5(b) s h o w s t h a t t w o s e m i c i r c l e s a r e p r e s e n t for r e l a t i v e l y w e a k a d s o r p t i o n . T h e s e a r e r e p l a c e d by a m o r e c o m p l e x s h a p e as t h e a d s o r p t i o n ( h i g h e r p a r a l l e l r e s i s t a n c e ) b e c o m e s m o r e i n t e n s e (Fig. 5((:)).
345
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F i g . 5. (a) R y = R x = 0, Cy = C x = CL ; (b) R x = 0, R y = 0, C y = 10CL ; (c) R x = 0, R y = 0, C y = 100CL; (d) R x = 0, R y = 100, Cy = 10CL; (e) Rx = 0, Ry = 100, C y : C L ; ( f ) R x = 0,
R y = 100, Cy = 50CL.
346
If the c u r r e n t going into the p a r a l l e l a d s o r p t i o n r e a c t i o n is less (higher p a r a l l e l r e a c t i o n r e s i s t a n c e ) the r e a c t i o n a p p r o x i m a t e s to the simple c h a r g e t r a n s f e r and diffusion curve, a n d this r e s e m b l a n c e is e v e n s t r o n g e r w h e n the e x t e n t of a d s o r p t i o n is d e c r e a s e d f u r t h e r as s h o w n in Fig. 5(c). In all the c u r v e s the W a r b u r g c o n t r i b u t i o n is r e l a t i v e l y u n d i s t o r t e d ; it is possible to o b t a i n an a p p a r e n t d i s t o r t i o n at the h i g h f r e q u e n c y end by h a v i n g a l a r g e a m o u n t of a d s o r p t i o n (high Cy) and a r e l a t i v e l y slow r e a c t i o n r a t e (high Ry). This is a r a t h e r u n l i k e l y c i r c u i t ; h o w e v e r , it is of some i n t e r e s t for it is one of the r e l a t i v e l y few cases w h i c h p r o d u c e s the s h a p e s h o w n in Fig. 5(f). As r e m a r k e d earlier, all c u r v e s showed a n o r m a l W a r b u r g s h a p e at the low f r e q u e n c y end.
3.2. Experimental data E x a m p l e s of the t h e o r e t i c a l i m p e d a n c e loci discussed in Section 3.1 c a n be f o u n d in the field of p r i m a r y cells. T h e i m p e d a n c e locus of a new a l k a l i n e Zn M n O 2 cell o b i a i n e d i m m e d i a t e l y a f t e r m a n u f a c t u r e is s h o w n in Fig. 6; the c h a r g e t r a n s f e r r e a c t i o n s a r e e x t r e m e l y fast a n d all t h a t we see is a W a r b u r g s h a p e in w h i c h the p o r o s i t y of the zinc e l e c t r o d e h a s caused a r e d u c t i o n in the slope to below 45 .
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Fig. 6. New alkaline Zn MnO: cell immediately after manufacture. Fig. 7. 10% discharged Leclanch~ cell. F i g u r e 7 shows the i m p e d a n c e s h a p e of a L e c l a n c h ~ cell in w h i c h the well-defined ldcus shows t h a t - t h e r e is a c h a r g e t r a n s f e r process at the zinc e l e c t r o d e in addition to diffusion in the solution. F i g u r e 8 r e p r e s e n t s a. 50% d i s c h a r g e d a l k a l i n e Z n - M n O 2 cell. The MnO2 e l e c t r o d e c l e a r l y h a s a c h a r g e t r a n s f e r r e s i s t a n c e of the s a m e o r d e r as t h a t of the zinc e l e c t r o d e ; h o w e v e r , it h a s a l a r g e r s u r f a c e area. F i g u r e 9 c o r r e s p o n d s to an a l k a l i n e
2"50
347
Zn H g O cell at a 60% d i s c h a r g e d s t a t e a n d a d i s t o r t i o n of the initial p a r t of the W a r b u r g slope s u g g e s t s the p r e s e n c e of a p a r a l l e l r e a c t i o n in the a d s o r b e d state• T h u s it c a n be seen t h a t the i m p e d a n c e s of c o m m o n t w o - t e r m i n a l p r i m a r y cells b e h a v e in the w a y e x p e c t e d from r e l a t i v e l y simple types of e l e c t r o d e b e h a v i o u r . T h e s e types of b e h a v i o u r are i m p o r t a n t in p r e d i c t i n g the quality, the s t a t e of c h a r g e and the r e a c t i v i t y of e l e c t r i c a l s t o r a g e devices; it is c l e a r t h a t t h e y are sufficiently different to be a basis for the s a t i s f a c t o r y a s s e s s m e n t of these c h a r a c t e r i s t i c s .
4. C o n c l u s i o n s (1) The e x p e c t e d i m p e d a n c e loci o b t a i n e d for simple cells c a n be o b s e r v e d in a c t u a l s i t u a t i o n s . (2) The W a r b u r g i m p e d a n c e finally d o m i n a t e s the i m p e d a n c e b e h a v i o u r of all s y s t e m s as the f r e q u e n c y is reduced. (3) The m e c h a n i s m of the cell r e a c t i o n s and the r a t e - c o n t r o l l i n g e l e c t r o d e c a n be e s t i m a t e d from the i m p e d a n c e spectra.
3.00
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." °o
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;
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i
,
i
i
i
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i
i
i
i
,
,
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Fig. 8. 50% discharged alkaline Zn- MnO 2 cell. Fig. 9. 60°Z, discharged alkaline Zn HgO cell.
References 1 M. Sluyters-Rehback and J. H. Sluyters, in A. J. Bard (ed.), E/ectroanaly.tical Chemistry, Vol. 4, Dekker, New York, 1970, p. 1.
348
2 3 4 5 6
S . A . G . R . K a r u n a t h i l a k a , N. A. H a m p s o n a n d R. Leek, d. Appl. Electrochem., 10 (1980) 3. S . A . G . R . K a r u n a t h i l a k a , N. A. Hampson, R. Leek a n d T. J. Sinclair, d. Appl. Electrochem., I0 (1980) 357;603; 709. H. Gerischer, Z. Elektrochem., 62 (1958) 256. M. F l e i s c h m a n n , S. K R a n g a r a j a n and H. R. Thirsk, Trans. Faraday Sot., 63 (1967) 1240, 1251, 1256. H . A . L a i t i n e n and J. E. B. Randles. Trans. Faraday Soc., 51 (1955) 54.
339
THE I M P E D A N C E OF TWO-TERMINAL ELECTROCHEMICAL CELLS
S. A. G. R. KARUNATHILAKA and N. A. HAMPSON Department of Chemistry, University of Technology, Loughborough, Leics. LE11 3TU (Gt. Britain) R. LEEK Electronic and Electrical Engineering Department, University of Technology, Loughborough, Leics. LE11 3TU (Gt. Britain) (Received December 1, 1980)
Summary The i m p e d a n c e b e h a v i o u r of cells c o m p o s e d of simple c o m b i n a t i o n s of e l e c t r o d e s is c a l c u l a t e d . The v a r i a t i o n in the i m p e d a n c e for some complex t y p e s of e l e c t r o d e b e h a v i o u r is also c a l c u l a t e d . The results are c o m p a r e d w i t h e x p e r i m e n t a l data. The W a r b u r g i m p e d a n c e u l t i m a t e l y d o m i n a t e s e v e r y s i t u a t i o n if the f r e q u e n c y is r e d u c e d sufficiently.
1. I n t r o d u c t i o n An i m p o r t a n t w a y of assessing the s u r f a c e c h a r a c t e r i s t i c s of a single e l e c t r o d e is to s t u d y its i m p e d a n c e . The e l e c t r o d e c a p a c i t a n c e m e a s u r e d in an indifferent e l e c t r o l y t e is a m e a s u r e of the t r u e electrode a r e a and of the p h y s i c a l and e l e c t r i c a l p r o p e r t i e s of the i n t e r p h a s e . A c l e a n electrode in the p o l a r i z a b l e r e g i o n yields the d o u b l e - l a y e r c a p a c i t a n c e Cc, and the v a r i a t i o n in the Cc E c u r v e o v e r a r a n g e of c o n c e n t r a t i o n in w h i c h diffuse-layer c h a r a c t e r i s t i c s p r e d o m i n a t e yields the p o i n t of zero change. In an e l e c t r o l y t e s o l u t i o n c o n t a i n i n g an e l e c t r o a c t i v e species, m e a s u r e m e n t of the i m p e d a n c e at a series of f r e q u e n c i e s enables the k i n e t i c c o n s t a n t s of the e l e c t r o d e r e a c t i o n to be d e t e r m i n e d . T h e s e effects h a v e been r e v i e w e d e l s e w h e r e [1]. In a c o n v e n t i o n a l e l e c t r o c h e m i c a l e x p e r i m e n t using a t h r e e - t e r m i n a l cell the e l e c t r o d e of i n t e r e s t ( w o r k i n g electrode) has an effective i m p e d a n c e of a b o u t a f a c t o r of 102 g r e a t e r t h a n the c o u n t e r e l e c t r o d e has, so t h a t w h e n a small a.c. p o t e n t i a l (usually less t h a n 7.5 m V m e a s u r e d with r e s p e c t to a r e f e r e n c e ) is applied to the cell the c u r r e n t flow is d o m i n a t e d by this e l e c t r o d e and the cell i m p e d a n c e is t h a t of the w o r k i n g electrode. U n d e r t h e s e c o n d i t i o n s the e x p e r i m e n t a l set-up reflects the b e h a v i o u r of one electrode. 0376-4883/81/0000-0000/$02.50
J(' Elsevier Sequoia/Printed in The Netherlands
340
The i m p e d a n c e of small e l e c t r i c a l s t o r a g e cells is of i n t e r e s t not only for the p u r p o s e of p r e d i c t i n g t h e i r b e h a v i o u r in e l e c t r o n i c circuits but also b e c a u s e it is a useful p a r a m e t e r for a s s e s s i n g t h e i r q u a l i t y [2]. We h a v e s h o w n [3] t h a t in some of these s y s t e m s one e l e c t r o d e d o m i n a t e s the cell b e h a v i o u r and t h a t the cell can be effectively t r e a t e d as the i m p e d a n c e of a single c o n t r o l l i n g e l e c t r o d e [3]. H o w e v e r , some cells do not c o n f o r m to this simple b e h a v i o u r a n d b o t h e l e c t r o d e s m u s t be considered. In this p a p e r we p r e s e n t some t h e o r e t i c a l p r e d i c t i o n s of the b e h a v i o u r expected for s i t u a t i o n s c o m m o n l y e n c o u n t e r e d in such systems.
2. P r o c e d u r e
In o r d e r to o b t a i n the i m p e d a n c e f r e q u e n c y b e h a v i o u r c o r r e s p o n d i n g to s y s t e m s of i n t e r e s t it. was c o n s i d e r e d t h a t the c o m p l e x p l a n e r e p r e s e n t a t i o n would allow the m a x i m u m of i n f o r m a t i o n to be m o s t r e a d i l y presented. In this m e t h o d the real and i m a g i n a r y p a r t s of the t o t a l i m p e d a n c e are plotted a g a i n s t e a c h o t h e r p a r a m e t r i c a l l y u s i n g f r e q u e n c y as the variable. This m e t h o d is well k n o w n , a n d this t y p e of plot is c o m m o n l y referred to as a S l u y t e r s plot. D a t a for these r e p r e s e n t a t i o n s w e r e o b t a i n e d by u s i n g the v e c t o r e q u a t i o n for e a c h system. The c a l c u l a t i o n and i s o l a t i o n of the real Z' and i m a g i n a r y - Z " p a r t s of the i m p e d a n c e were p e r f o r m e d u s i n g the complex n u m b e r h a n d l i n g c a p a b i l i t y of a P R I M E 400 c o m p u t e r . As an e x a m p l e we c o n s i d e r the n e t w o r k s h o w n in Fig. 1. The impeda n c e s of the c o m p o n e n t p a r t s a r e g i v e n by
J (OCD
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Fig. 1. Circuit representing a charge transfer and diffusion process with active adsorption of the intermediate species and an accompanying or preceding chemical reaction.
341
and the e q u i v a l e n t i m p e d a n c e at an a n g u l a r f r e q u e n c y (~ is given by
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The real and i m a g i n a r y parts of Z are o b t a i n e d from the c o m p u t e r and printed out g r a p h i c a l l y .
3. R e s u l t s a n d d i s c u s s i o n F i g u r e 2 shows the familiar plot of the impedance c o m p o n e n t s for an electrode at w h i c h an e l e c t r o n t r a n s f e r r e a c t i o n is followed by diffusion in solution. The m a g n i t u d e s of the c o m p o n e n t values were chosen to represent the b e h a v i o u r of a zinc electrode typical of a small p r i m a r y cell (e.g. 2000 mA h of the a l k a l i n e Z n - M n O 2 or a l k a l i n e Z n - H g O type).
100
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0
. . . . . . . ,&. . . . . . . . . . . . . . . . . . . 1'00
(a)
Fig. 2. (a) Randles circuit: C1, = 1157 I~F; 0 plot correspofiding to (a).
(b)
~. . . . . . . . . 1,S0
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0.055 £1; a = 0.0567 gl s 1'2. (b) Complex plane
F i g u r e 3(a) shows the a n a l o g u e for two electrodes in series (in w h a t follows the s o l u t i o n r e s i s t a n c e R~ is omitted ; if present it is r e p r e s e n t e d by the i n t e r c e p t on the real axis at ~ = ~2) and Fig. 3(b) shows the cell b e h a v i o u r of the a r r a n g e m e n t displayed in the complex plane. The shape of the locus is identical with t h a t of a single electrode ; however, the m a g n i t u d e s of the s h a p e are c o n s i d e r a b l y increased. The W a r b u r g c o n t r i b u t i o n still has a g r a d i e n t of 4 5 . F i g u r e 3(c) shows how the high f r e q u e n c y semicircle is distorted by c h a n g i n g the double-layer and c h a r g e t r a n s f e r resistances of one electrode in a w a y e q u i v a l e n t to r e d u c i n g the area of this electrode by 50%. W h e n the i m b a l a n c e b e t w e e n the electrode areas becomes gross for identical c h a r g e t r a n s f e r c o m p o n e n t s a s e c o n d semicircle at high f r e q u e n c y with an identical radius to the first develops f u r t h e r a l o n g the real axis (Fig.
342
3(d)). When the charge transfer components are significantly different in addition to having different area factors (and therefore different doublelayer capacitances), this is reflected in the relative magnitudes of the two high frequency semicircles (Fig. 3(e)). However, the Warburg part of the complex plane representation remains the same for all these types of behaviour.
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343
3.1. The effect of adsorption at the electrodes A d s o r p t i o n effects at e l e c t r o d e s c a n arise from a n u m b e r of causes. T h e simplest effect is w h e n a l a y e r of i n a c t i v e o r g a n i c i m p u r i t y is deposited o n t o t h e e l e c t r o d e and simply r e d u c e s the c a p a c i t a n c e e v e r y w h e r e to a low value. H o w e v e r , o t h e r a d s o r p t i o n effects arise as a c o n s e q u e n c e of the electroc h e m i s t r y . An o b v i o u s effect is w h e n the electrode p r o d u c t s and r e a c t a n t s u n d e r g o a r e a c t i o n in the a d s o r b e d s t a t e at the e l e c t r o d e w h i c h m a y or m a y not be coupled w i t h a diffusion process in solution. F o r example, an o r g a n i c c o m p o u n d m a y u n d e r g o a r e d o x r e a c t i o n in the a d s o r b e d s t a t e w i t h the p r o d u c t s r e m a i n i n g at the e l e c t r o d e ; altern a t i v e l y , the r e a c t i o n p r o d u c t s m a y b e c o m e desorbed a f t e r the r e a c t i o n and pass into solution. T h e s e processes m a y o c c u r t o g e t h e r and p r o v i d e b o t h a p a r a l l e l r e a c t i o n p a t h w h i c h is r e p r e s e n t e d by a p a r a l l e l RyCy b r a n c h and an a d d i t i o n a l step in the o v e r a l l r e a c t i o n w h i c h is r e p r e s e n t e d by an a d s o r p t i o n c o m p o n e n t R× Cx in the e l e c t r o d e a n a l o g u e . The m o s t f a m i l i a r process is t h a t w h e n the c h e m i c a l r e a c t i o n w h i c h precedes or a c c o m p a n i e s t h e e l e c t r o d e p o s i t i o n is a m e t a l l u r g i c a l step in a m e t a l d e p o s i t i o n or d i s s o l u t i o n r e a c t i o n , e.g. Mlattic ~ ~
Maa ~
z-V Msolution
(6)
w h i c h is the w e l l - k n o w n a d a t o m m e c h a n i s m for m e t a l ion e x c h a n g e reactions. The e l e c t r i c a l a n a l o g u e s h o w n in Fig. 4(a) was first s u g g e s t e d by G e r i s c h e r [4] a n d differs slightly from a r a t h e r m o r e complex a n a l o g u e p r o p o s e d by F l e i s c h m a n n et al. [5] in w h i c h an a d d i t i o n a l c o m p o n e n t e x p r e s s i n g s u r f a c e diffusion is included in the resistive c o m p o n e n t of the c r y s t a l l i z a t i o n i m p e d a n c e . The i m p e d a n c e locus for the simpler process is s h o w n in Fig. 4(b) for the case of a r e l a t i v e l y small a d a t o m c o n c e n t r a t i o n a n d a r e l a t i v e l y fast s u r f a c e diffusion process. It h a s a v e r y e l o n g a t e d s h a p e w h i c h is formed from two semicircles of differing radii. F o r the case of an e v e n f a s t e r s u r f a c e diffusion process the well-defined t w i n semicircles o v e r l a p to form an e l o n g a t e d single semicircle. W h e n the surface conc e n t r a t i o n is i n c r e a s e d the t w i n s e m i c i r c u l a r s h a p e is a g a i n evident as s h o w n in Fig. 4(d). In all these cases the W a r b u r g p a r t of the locus r e m a i n s a n d h a s a c o n s t a n t g r a d i e n t of 4 5 . We can c o n c l u d e from these s h a p e s t h a t w h a t e v e r the m e c h a n i s m of the e l e c t r o c h e m i c a l r e a c t i o n at the electrodes the s o l u t i o n process u l t i m a t e l y d o m i n a t e s , and t e c h n o l o g i c a l l y in a d.c. e x p e r i m e n t the "fine s t r u c t u r e " of the e l e c t r o d e r e a c t i o n is rapidly overs h a d o w e d by the s o l u t i o n processes w h i c h u l t i m a t e l y control the c u r r e n t . T h e m o s t g e n e r a l case of a d s o r p t i o n at the e l e c t r o d e s occurs w h e n t h e r e is a r e a c t i o n in the a d s o r b e d s t a t e at the electrode in addition to the c h a r g e t r a n s f e r r e a c t i o n and a p r e c e d i n g or a c c o m p a n y i n g c h e m i c a l reaction. This s i t u a t i o n could arise for the case of an electrode at w h i c h an o r g a n i c r e d o x r e a c t i o n was o c c u r r i n g in p a r a l l e l w i t h a c h a r g e t r a n s f e r r e a c t i o n (metal deposition) w i t h a d a t o m diffusion s u c h as m i g h t be env i s a g e d at the n e g a t i v e e l e c t r o d e of a cell in the p r e s e n c e of an o r g a n i c e x p a n d e r . T h e e l e c t r o d e a n a l o g u e of this is s h o w n in Fig. 1.
344
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Fig. 4. (a) Electrode representation for the case when an electron transfer reaction with diffusion in solution precedes or accompanies the eleetrodeposition. (b) CA = 10CI. : R A = 0. (c) Ca = 10CI~; RA = 0.50. (d) ('a = 30CL: R,\ = 0.750.
F i g u r e 5(a) s h o w s t h a t t h e i m p e d a n c e l o c u s f o r t h e c o m p l e t e c i r c u i t is q u i t e s i m p l e a n d m i g h t e a s i l y be t a k e n for a s i m p l e c a s e o f c h a r g e t r a n s f e r and diffusion. There are relatively minor distortions of the semicircular shape at high frequencies and at two intermediate frequencies which i n d i c a t e t h a t t h e e l e c t r o d e is c o m p l e x . T h e W a r b u r g c o n t r i b u t i o n h a s a g r a d i e n t o f 45 ~s e x p e c t e d . If t h e r e is n o a c c o m p a n y i n g o r p r e c e d i n g e l e c t r o d e r e a c t i o n , t h e s i m p l e c h a r g e t r a n s f e r r e a c t i o n is c o m b i n e d w i t h h p a r a l l e l r e a c t i o n in t h e a d s o r b e d s t a t e as d i s c u s s e d o r i g i n a l l y b y L a i t i n e n a n d R a n d l e s [6]. F i g u r e 5(b) s h o w s t h a t t w o s e m i c i r c l e s a r e p r e s e n t for r e l a t i v e l y w e a k a d s o r p t i o n . T h e s e a r e r e p l a c e d by a m o r e c o m p l e x s h a p e as t h e a d s o r p t i o n ( h i g h e r p a r a l l e l r e s i s t a n c e ) b e c o m e s m o r e i n t e n s e (Fig. 5((:)).
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F i g . 5. (a) R y = R x = 0, Cy = C x = CL ; (b) R x = 0, R y = 0, C y = 10CL ; (c) R x = 0, R y = 0, C y = 100CL; (d) R x = 0, R y = 100, Cy = 10CL; (e) Rx = 0, Ry = 100, C y : C L ; ( f ) R x = 0,
R y = 100, Cy = 50CL.
346
If the c u r r e n t going into the p a r a l l e l a d s o r p t i o n r e a c t i o n is less (higher p a r a l l e l r e a c t i o n r e s i s t a n c e ) the r e a c t i o n a p p r o x i m a t e s to the simple c h a r g e t r a n s f e r and diffusion curve, a n d this r e s e m b l a n c e is e v e n s t r o n g e r w h e n the e x t e n t of a d s o r p t i o n is d e c r e a s e d f u r t h e r as s h o w n in Fig. 5(c). In all the c u r v e s the W a r b u r g c o n t r i b u t i o n is r e l a t i v e l y u n d i s t o r t e d ; it is possible to o b t a i n an a p p a r e n t d i s t o r t i o n at the h i g h f r e q u e n c y end by h a v i n g a l a r g e a m o u n t of a d s o r p t i o n (high Cy) and a r e l a t i v e l y slow r e a c t i o n r a t e (high Ry). This is a r a t h e r u n l i k e l y c i r c u i t ; h o w e v e r , it is of some i n t e r e s t for it is one of the r e l a t i v e l y few cases w h i c h p r o d u c e s the s h a p e s h o w n in Fig. 5(f). As r e m a r k e d earlier, all c u r v e s showed a n o r m a l W a r b u r g s h a p e at the low f r e q u e n c y end.
3.2. Experimental data E x a m p l e s of the t h e o r e t i c a l i m p e d a n c e loci discussed in Section 3.1 c a n be f o u n d in the field of p r i m a r y cells. T h e i m p e d a n c e locus of a new a l k a l i n e Zn M n O 2 cell o b i a i n e d i m m e d i a t e l y a f t e r m a n u f a c t u r e is s h o w n in Fig. 6; the c h a r g e t r a n s f e r r e a c t i o n s a r e e x t r e m e l y fast a n d all t h a t we see is a W a r b u r g s h a p e in w h i c h the p o r o s i t y of the zinc e l e c t r o d e h a s caused a r e d u c t i o n in the slope to below 45 .
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Fig. 6. New alkaline Zn MnO: cell immediately after manufacture. Fig. 7. 10% discharged Leclanch~ cell. F i g u r e 7 shows the i m p e d a n c e s h a p e of a L e c l a n c h ~ cell in w h i c h the well-defined ldcus shows t h a t - t h e r e is a c h a r g e t r a n s f e r process at the zinc e l e c t r o d e in addition to diffusion in the solution. F i g u r e 8 r e p r e s e n t s a. 50% d i s c h a r g e d a l k a l i n e Z n - M n O 2 cell. The MnO2 e l e c t r o d e c l e a r l y h a s a c h a r g e t r a n s f e r r e s i s t a n c e of the s a m e o r d e r as t h a t of the zinc e l e c t r o d e ; h o w e v e r , it h a s a l a r g e r s u r f a c e area. F i g u r e 9 c o r r e s p o n d s to an a l k a l i n e
2"50
347
Zn H g O cell at a 60% d i s c h a r g e d s t a t e a n d a d i s t o r t i o n of the initial p a r t of the W a r b u r g slope s u g g e s t s the p r e s e n c e of a p a r a l l e l r e a c t i o n in the a d s o r b e d state• T h u s it c a n be seen t h a t the i m p e d a n c e s of c o m m o n t w o - t e r m i n a l p r i m a r y cells b e h a v e in the w a y e x p e c t e d from r e l a t i v e l y simple types of e l e c t r o d e b e h a v i o u r . T h e s e types of b e h a v i o u r are i m p o r t a n t in p r e d i c t i n g the quality, the s t a t e of c h a r g e and the r e a c t i v i t y of e l e c t r i c a l s t o r a g e devices; it is c l e a r t h a t t h e y are sufficiently different to be a basis for the s a t i s f a c t o r y a s s e s s m e n t of these c h a r a c t e r i s t i c s .
4. C o n c l u s i o n s (1) The e x p e c t e d i m p e d a n c e loci o b t a i n e d for simple cells c a n be o b s e r v e d in a c t u a l s i t u a t i o n s . (2) The W a r b u r g i m p e d a n c e finally d o m i n a t e s the i m p e d a n c e b e h a v i o u r of all s y s t e m s as the f r e q u e n c y is reduced. (3) The m e c h a n i s m of the cell r e a c t i o n s and the r a t e - c o n t r o l l i n g e l e c t r o d e c a n be e s t i m a t e d from the i m p e d a n c e spectra.
3.00
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Fig. 8. 50% discharged alkaline Zn- MnO 2 cell. Fig. 9. 60°Z, discharged alkaline Zn HgO cell.
References 1 M. Sluyters-Rehback and J. H. Sluyters, in A. J. Bard (ed.), E/ectroanaly.tical Chemistry, Vol. 4, Dekker, New York, 1970, p. 1.
348
2 3 4 5 6
S . A . G . R . K a r u n a t h i l a k a , N. A. H a m p s o n a n d R. Leek, d. Appl. Electrochem., 10 (1980) 3. S . A . G . R . K a r u n a t h i l a k a , N. A. Hampson, R. Leek a n d T. J. Sinclair, d. Appl. Electrochem., I0 (1980) 357;603; 709. H. Gerischer, Z. Elektrochem., 62 (1958) 256. M. F l e i s c h m a n n , S. K R a n g a r a j a n and H. R. Thirsk, Trans. Faraday Sot., 63 (1967) 1240, 1251, 1256. H . A . L a i t i n e n and J. E. B. Randles. Trans. Faraday Soc., 51 (1955) 54.