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J. Electrochem. Soc., Vol. 138, No. 5, May 1991 9 The Electrochemical Society, Inc.
REFERENCES
1. E. Broszeit, "Metallurgical Aspects of Wear," E. Hornbogen and K.H. Z u m Gahr, Editors, pp. 183-205,
Deutsche Gesellschaft fur. Metallurde. E.V. (1981).
2. J. Ambrose and J. Kruger, This Journal, 121, 599
(1974).
3. J. C. Nelson, C. Kang, and A. Bronson, ibid., 136, 2948
(1989).
4. G. T. Burstein a n d D. H. Davies, ibid., 128, 33 (1981).
5. G. T. Burstein and G. W. Ashley, Corrosion, 40, 110
(1984).
6. G. T. Burstein and G. O. H. Willock, This Journal, 136,
1313 (1989).
7. F. P. Ford, G. T. Burstein, and T. P. Hoar, ibid., 127,
1325 (1980).
8. H. J. Pearson, G. T. Burstein, and R. C. Newman, ibid.,
128, 2297 (1981).
9. G. T. Burstein and A. J. Davenport, ibid., 136, 936
(1989).
10. F. Mansfeld, Corrosion, 44, 856 (1988).
11. R. C. Weast, Editor, "Handbook of Chemistry and
Physics," 50th ed., p. D-121, Chemical R u b b e r Co.,
Cleveland, OH (1969).
12. J. S. Newman, "Electrochemical Systems," pp. 297403, Prentice-Hall, Englewood Cliffs, NJ (1973).
13. N. Ibl, "Comprehensive Treatise of Electrochemistry,"
Vol. 6, E. Yeager, J. O'M. Bockris, B. E. Conway, and
S. Sarangapani, Editors, pp. 239-315, P l e n u m Press,
New York (1983).
14. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals,
Series and Products, Academic Press, New York
(1980).
Interpretation of Finite-Length-Warburg-Type Impedances in
Supported and Unsupported Electrochemical Cells with
Kinetically Reversible Electrodes
Donald R. Franceschetti*
Department of Physics, Memphis State University, Memphis, Tennessee 38152
J. Ross Macdonald*
Department of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill,
North Carolina 27599-3255
Richard P. Buck*
Department of Chemistry, The University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3290
ABSTRACT
The origin of finite-length-Warburg-type impedances in supported and unsupported systems is examined within a
c o m m o n framework and with reference to previous exact and approximate results. While close agreement is found between an approximate treatment based on bulk electroneutrality and an exact solution of the Nernst-Planck-Poisson
equation system for u n s u p p o r t e d systems of m a n y Debye length thicknesses with rapid electrode reaction kinetics, the
approximate treatment is unjustified when the electrode reaction is slow or the electrode separation is less than or comparable to the Debye length.
I n the years since Warburg (1) published his classic
study of diffusion u n d e r ac conditions in a supported electrolyte placed between kinetically reversible electrodes,
response functions of the Warburg type have been obtained by n u m e r o u s authors for a wide range of systems
involving both supported and unsupported electrolytes.
A m o n g recent theoretical treatments, especially those
dealing with finite-length effects, differences in notation,
approximations, and terminology have tended to obscure
both similarities and differences between the situations
considered and the results obtained. I n particular, published discussions of thin layer effects often assume that
while electrode separation is small compared to the effective diffusion length for the electrolyte, it is still large compared to the Debye length. While the latter condition is almost always met in systems with aqueous electrolytes, it
need not apply in solid materials and m e m b r a n e situations. Further, quite similar results have been obtained in
treatments which both include and explicitly neglect Poisson's equation. In the present brief communication we
compare three different approaches leading to a finitelength Warburg impedance of the form
Zw(r
= Zw(0) [tanh (iA)~
~
[1]
where A is proportional to the angular frequency o~.
Specifically e x a m i n e d are cases of (i) a supported electro* Electrochemical Society Active Member.
lyte placed between kinetically reversible parent ion electrodes, (ii) an u n s u p p o r t e d electrolyte with similar electrodes with electroneutrality assumed, and (iii) a
treatment of the latter case based on an exact solution of
the Nernst-Planck-Poisson equation system. We use a consistent notation throughout and c o m m e n t upon related
treatments to be found in the literature.
Supported Systems
Consider a supported electrochemical system of the
form
M IMZ§ supporting electrolyte IM
[2]
where the planar electrodes are parallel and a distance d
apart. The supporting electrolyte allows us to neglect the
migration term in the Nernst-Planck equation
Jp = -Dp(OCp/OX)+ p.pcpE
[3]
since the electric field E is very small within the electrolyte. Here Jp is the flux of the Mz§ ion, Dp the diffusion coefficient, for the ion, % its concentration, and ~p its electrical
mobility. On setting E = 0 in Eq. [3], we obtain Fick's first
law of diffusion. U n d e r small-signal ac conditions, the system variables may be separated into steady-state and timed e p e n d e n t parts, i.e.
cp = %o + %1 exp (icot)
[4a]
J. Electrochem. Soc., Vol. 138, No. 5, May 1991 9 The Electrochemical Society, Inc.
and
Jp = dp0 + Jpl e x p (loot)
[45]
On s u b s t i t u t i n g t h e s e f o r m s in Eq. [3] and e q u a t i n g t h e
t i m e - d e p e n d e n t terms, o n e obtains
Jpl = - Dp( d % , / d x )
[5]
I f M z§ does n o t participate in any b u l k c h e m i c a l reaction,
F i c k ' s s e c o n d law t h e n can be w r i t t e n in the small-signal
f o r m (2)
i(~Cpl = Dp(d2%1/dx 2)
[6]
It m a y be n o t e d that it follows f r o m t h e linearity of the first
t e r m in Eq. [3] that Eq. [5] and [6] h o l d e v e n w h e n %0 varies
w i t h position. T h e g e n e r a l solution of Eq. [6] m a y b e
written
%1(x) = A sinh [(ir176
+ B c o s h [(io)/Dp)~
[7]
w h e r e A and B are c o n s t a n t s w h i c h m u s t be selected to satisfy t h e b o u n d a r y conditions. F o r the s y s t e m u n d e r consideration h e r e one c o n s t a n t m a y be e l i m i n a t e d by a symm e t r y a r g u m e n t . P l a c e t h e c e n t e r of t h e s y s t e m at x = 0, so
that the e l e c t r o l y t e e x t e n d s b e t w e e n x = - d / 2 and x = d/2.
N o t e that t h e p h y s i c a l situation at any t i m e t will be the
m i r r o r i m a g e of t h a t at t i m e t - ~/r i.e., the electrodes will
h a v e r e v e r s e d polarity and t h e c u r r e n t will be e q u a l in
m a g n i t u d e b u t o p p o s i t e in direction. With t h e c o o r d i n a t e
s y s t e m chosen, this b e h a v i o r r e q u i r e s f r o m Eq. [4a] t h a t
%l(-X) = -%1(x) and t h u s that B = 0. T h e c o n s t a n t A can be
related to t h e m a g n i t u d e of t h e t i m e - d e p e n d e n t part of the
applied p o t e n t i a l difference, V~ppl,1 e x p (io~t). We a s s u m e
that this p o t e n t i a l d i f f e r e n c e is e v e n l y d i v i d e d b e t w e e n the
two interfaces and can be d e s c r i b e d by N e r n s t ' s e q u a t i o n
which, after linearization of t h e l o g a r i t h m i c term, yields (3)
V~ppl,1 = ( R T / z F % o ) [ % l ( - d / 2 ) - %1(d/2)]
[8]
w h e r e R is t h e gas constant, T t h e a b s o l u t e t e m p e r a t u r e ,
and F t h e Faraday. This q u a n t i t y is t h e limiting potential
d i f f e r e n c e for e x c h a n g e rates of p o t e n t i a l - d e t e r m i n i n g
species t h a t are r a p i d in c o m p a r i s o n w i t h m a s s transport.
It is c o n s i s t e n t w i t h t h e e l e c t r o d e kinetic m o d e l s of ButlerV o l m e r and Chang-Jaffd, for e x a m p l e . W h e n we calculate
t h e electrical c u r r e n t (per u n i t area), Ii = zFJp~, and divide
into Vappl,1, w e find t h e i m p e d a n c e (also per unit area)
Z(m) = [2RT/z2F2%o(io~Dp)~
t a n h [(ir176
gle e l e c t r o d e and D the diffusion coefficient of the disc h a r g e d species in t h e electrode.
O n e of t h e p r e s e n t a u t h o r s has ,extended the t r e a t m e n t
of t h e ac r e s p o n s e of t h e p a r e n t - m e t a l - t y p e cell to the case
w h e r e t h e e l e c t r o d e r e a c t i o n occurs w i t h arbitrary and different p o t e n t i a l - d e p e n d e n t rates at t h e t w o e l e c t r o d e s (2).
In this t r e a t m e n t , B u t l e r - V o l m e r r e a c t i o n kinetics w e r e ass u m e d and t h e d i s p l a c e m e n t c u r r e n t t h r o u g h the c o m p a c t
d o u b l e layer was explicitly included. T h e e x a c t i m p e d a n c e
result is c o n s i s t e n t w i t h Eq. [9] for identical electrodes and
w i t h t h e Llopis and C o l o m result (6) w h e n one electrode is
kinetically r e v e r s i b l e and the o t h e r not. Results for nonplanar e l e c t r o d e g e o m e t r i e s are g i v e n by S l u y t e r s - R e b a c h
and Sluyters (11).
Unsupported Systems, Eiectroneutrality Assumed
The t r e a t m e n t in this section is b a s e d on the a p p r o a c h of
B u c k (12). C o n s i d e r n o w an u n s u p p o r t e d s y s t e m
MIMX, solvent IM
[10]
w h e r e the salt M X is fully dissoc.iated into M z+ and X ~ions, w i t h c o n c e n t r a t i o n s d e n o t e d by cp(x) and cn(x), respectively. T h e t r a n s p o r t of cations is d e s c r i b e d by Eq. [3]
as before, w h i l e t h e t r a n s p o r t of anions is d e s c r i b e d by a
second Nernst-Planck equation
Jn = - Dn(Oc,/Ox) - ~ncnE
[11 ]
T h e mobilities ~p a n d ~, are generally t a k e n to b e related
to t h e c o r r e s p o n d i n g diffusion coefficients t h r o u g h the
E i n s t e i n relation ~i = D~zF/RT.
I f e l e c t r o n e u t r a l i t y is a s s u m e d to hold, it follows that
% = cn and t h a t Jp and Jn are e q u a l e x c e p t for a spatially invariant t e r m related to t h e faradaic c u r r e n t If
Jp - Jn = I f / z F
[12]
On setting cn = Cp in Eq. [3] and [11] and e l i m i n a t i n g the
c o n c e n t r a t i o n g r a d i e n t terms, one obtains t h e electric field
in t h e e l e c t r o n e u t r a l r e g i o n as
E -
RT
Jp/Dp - J J D n
F
2z%
[13]
Alternatively, on u s i n g Eq. [12] to e l i m i n a t e J , f r o m Eq. [11]
and t h e n c o m b i n i n g Eq. [11] and [3] to e l i m i n a t e E, one
finds
[9]
On c o m p a r i n g Eq. [9] w i t h Eq. [1] w e find that A =
oJ(d/2)2/Dp, w h i c h m a y also be w r i t t e n as A = (d/2)2/L~i~,
w h e r e Ld~ff-= (DpAo) ~ is t h e effective diffusion length, at the
f r e q u e n c y of m e a s u r e m e n t , for the e l e c t r o a c t i v e ion. The
r e a s o n that t h e s y s t e m l e n g t h appears as d i v i d e d by two is
t h e s y m m e t r y of t h e system, w h i c h r e q u i r e s cpL to be zero
at t h e center. T h e divisor of t w o was u n f o r t u n a t e l y o m i t t e d
by two of t h e p r e s e n t a u t h o r s in a r e c e n t r e v i e w (4). An exp r e s s i o n in the f o r m of Eq. [1] appears first to h a v e b e e n derived in a biological c o n t e x t by L a b e s and Lullies (5). An
e x p r e s s i o n of the s a m e form, b u t w i t h a factor d (rather
t h a n d/2) a p p e a r i n g in t h e a r g u m e n t of t h e t a n h f u n c t i o n
was o b t a i n e d by L l o p i s and C o l o m (6), w h o c o n s i d e r e d the
diffusion of a single e l e c t r o a c t i v e species t h r o u g h a N e r n s t
layer of t h i c k n e s s d e s t a b l i s h e d at a stationary e l e c t r o d e in
a stirred electrolyte. T h e r e is no s y m m e t r y of cp~ in this
case b e c a u s e Cp~ is zero at t h e solution side of the N e r n s t
layer. A similar result was o b t a i n e d by D r o s s b a c h and
S c h u l t z (7). S l u y t e r s (8) e x a m i n e d t h e ac b e h a v i o r of a thin
layer of s u p p o r t e d e l e c t r o l y t e w i t h identical, kinetically reversible, e l e c t r o d e s and b o t h r e d u c e d and oxidized species
diffusing w i t h i n t h e electrolyte, and o b t a i n e d a result
e q u i v a l e n t (2) to p l a c i n g two i m p e d a n c e s of the form of Eq.
[9], in series, one w i t h a diffusion coefficient e q u a l to that
of t h e r e d u c e d species, t h e o t h e r w i t h the diffusion coefficient of t h e o x i d i z e d species. A n i m p e d a n c e of the form of
Eq. [1] can also arise w h e n a neutral p r o d u c t species m u s t
diffuse t h r o u g h , or along t h e internal pores of, an e l e c t r o d e
to e x c h a n g e w i t h t h e a m b i e n t a t m o s p h e r e (9, 10). In this
case w e h a v e A = (~d~/D~, w h e r e d is t h e t h i c k n e s s of a sin-
1369
Jp = Ds(O%/Ox) + tpIf/zF
[14]
or, u s i n g Eq. [12] again and n o t i n g that c, = Cp
J~ = Ds(OCn/OX) - t f l f / z F
[15]
Here
D~ = 2 D , D p / ( D , + Dp)
[16]
is the c o u p l e d diffusion coefficient, and tp = D J ( D . + Dp)
and t~ = 1 - tp are t h e t r a n s p o r t n u m b e r s of t h e positive
and n e g a t i v e species, respectively.
As in t h e s u p p o r t e d case, t h e c o n c e n t r a t i o n and other
s y s t e m variables can be s e p a r a t e d into e q u i l i b r i u m and sin u s o i d a l l y v a r y i n g parts. Specifically, Eq. [4] can still be
a s s u m e d to h o l d and, since Ill does not v a r y w i t h position,
Eq. [6] and [7] b e c o m e
i(o%1 = D~(d2cpJdx 2)
[6]
and
%1 = A sinh [(ir176
[7]
w h e r e again a s y m m e t r y a r g u m e n t has b e e n u s e d to elimin a t e one c o n s t a n t of integration.
The p o t e n t i a l d r o p V~ppl,~across the s y s t e m is taken, in a
s e g m e n t e d potential m o d e l (12), as t h e s u m of the Nernstian potential difference, V~.i.t, created by t h e concentration p e r t u r b a t i o n s at the e l e c t r o d e s and g i v e n by Eq. [8],
and t h e p o t e n t i a l drop, V~.... across the e l e c t r o n e u t r a l region, w h i c h can be o b t a i n e d f r o m Eq. [13]. W h e n one t h e n
substitutes Eq. [14] and [15] into Eq. [13], linearizes, and
J. Electrochem. Soc., Vol. 138, No. 5, May 1991 9 The Electrochemical Society, Inc.
1370
then integrates across the thickness d of the electrolyte,
one finds
Yl en -- -
'
-
2zFcpo
-~
[%,(-d/2)
+ t,, I
RT
- %1(d12)] + - -
2z2F2cpo \Dp
If~d
[17]
Ifld
[18]
D,/
so that
V1 - - 4 R T t , A sinh [(iLolDs)~
zF
+
2 z2F2cpo
+
To evaluate A in the present case, one may set Jpt = If~/zF
at x = -+d/2, since the flow of negative charge is zero at the
electrodes, and thus from Eq. [14]
D~(dCpJdx) = tnIfl/ZF
[19]
so that A = -t~Irl/zF (io~D~)~ Then, on taking the ratio Z =
V~/If~, one obtains the i m p e d a n c e
4RTt~ 2
Z(~) = 2 2
0~ tanh [(ir176
z F cp0(icoD~)
+,
RTd
z2F2%o(D . + Dp)
[20]
where the first term is a finite-length Warburg i m p e dan ce
and the second t e r m m a y be recast, using the Einstein relations, as
R| = dl[zF(cpo~p + c~0~,)]
[21]
the bulk resistance of the electrolyte. In the low-frequency
limit, Eq. [20] reduces to Rp = d/zFcpo~p, the dc resistance of
this system. In the high-frequency limit, the first term vanishes and Eq. [20] reduces to R=.
In earlier work B u c k (12) obtained the approximate
result
Z(r
= [4RTtnlz~F2cpo(ic~D~) ~ tanh [(i~fD~)~
[22]
in a treat m en t wh i ch parallels the present one but ignores
(sets to zero) the last terms in Eq. [12], [14], [15], [17], and
[18]. This result is identical to the first term in Eq. [20], with
the single ex cep t i o n of the factor t~ which appears here to
the first p o wer rather than the second. While Eq. [22] reduces to Rp in the dc limit, it fails to yield R| in the limit of
high frequencies. Since Eq. [22] yields the proper behavior
in the dc limit, we will refer to the approximations employed as the "quasi-static" electroneutral approach.
Unsupported Systems,Exact Treatment
In 1973, J. R. Macdonald obtained an exact solution (14)
to a well-defined mathematical model of the small-signal
ac response of a slab of u n s u p p o r t e d electrolyte placed bet w e e n two identical plane parallel electrodes. The m o d el
included Nernst-Planck and continuity equations for both
negative and positive mobile charge, and involved full satisfaction of Poisson's equation and full consideration of
the Maxwell displacement current throughout the electrolyte. The system was assumed to be free of intrinsic space
charge layers in the absence of an applied potential difference, and allowance was m a d e for a possible uniform
background of i m m o b i l e charge. In later work Macdonald
and Franceschetti e x t e n d e d the model to allow the timed e p e n d e n t generation and recombination of stationary
and mobile charge (15). The electrode kinetics were incorporated through the assumption of boundary conditions of
the Chang-Jaff~ type
Jp = +_kp(cp - %0)
with a similar condition for negative charges.
Although the Chang-Jaffd boundary conditions do not
possess the same degree of physical realism as the equations of Butler-Volmer electrode kinetics, they lead to precisely the same form in the small-signal dc limit, and in the
limit considered in this paper where the rate constant kp
becomes infinite (15). By making the rate constant complex and frequency-dependent, it is possible to take into
account the occurrence of one or more adsorbed intermediate states for the discharging ion and to allow for the diffusion of a neutral product into the electrode or into the
electrolyte (8, 16).
The full solution obtained by Macdonald is quite complicated, and Macdonald devoted several subsequent papers
to exploring equivalent circuit representations of various
limiting cases (16, 18). A n important result (14, 18) in the
present context was the first expression for A which involved arbitrary valence numbers, z~ and zp, as well as arbitrary diffusion coefficients, Dn and Dp. In the condensed
notation of the earlier work, the expression is A =
M2bcoR|
It can be readily e x p a n d e d and becomes (19),
on using the Einstein relation and the bulk electroneutrality condition, znc~o = ZpCp0
[23]
I Dnzn + Dpzp 1
[ ~ n l + ~Pll [24]
A = (~d2/4) L ( ~ ) ~
+ ~ ) J = (")d2F/4RT) L-~; ~ + -z~
~ J
a result in full ag r eem en t with the A term implicit in
Eq. [22] w h e n zn =- Zp =- z.
Franceschetti and Macdonald (15) obtained an approximate hierarchical ladder network circuit which was
broadly applicable in the case in which only one of the mobile species reacts at the electrodes and the electrolyte
thickness includes m a n y D eb y e lengths. F o r the present
case of equal charges and very fast electrode kinetics, this
circuit reduces to three elements, the bulk resistance,
Eq. [21], in series with the finite-length Warburg element,
expressed in that work as
Zw(~) = (RJ~m) tanh (io~TDHN2)O'5/(iW'rDHN2)~
[25]
the combination being in parallel with the geometric capacitance of the electrolyte
Cg = ~/4~d
[26]
Here e is the dielectric constant of the electrolyte, ~r~ =
~n/~p = D,/Dp is the mobility ratio, TD = R|
is the dielectric relaxation time, and
HN =(d/4LD)(~ 1 + 2 + ~)0.5
where
L D is
[27]
the Debye length
LD = [eRT/8~rF2z2%o] ~
[28]
for the present fully dissociated equivalent case.
Although Eq. [25], taken together with Eq. [27] and [28]
bears no striking resemblance to the finite-length Warburg
i m p ed an ce derived in the previous section, we shall now
show that the result is identical. Using the Einstein relations for ~, and ~p and the definition ofD~, Eq. [16], we can
easily show that
HN2 = d2(Dp + Dn)/(8LD2D S)
[29]
and then using Eq. [21], [25], and the Debye relation to relate TDto L d, we find that
TDHN2
=
d2/4Ds
[30]
With this substitution in Eq. [24], expressing R= and ~m in
terms of Dn and Dp and a little further manipulation, one
finds that Zw(r is now identical to the first term in Eq. [20].
Thus, with the exception of the geometric capacitance,
which is frequently negligible in practice, the approximate
circuit developed by Macdonald and co-workers from the
full solution of the Nernst-Planck-Poisson equation syst em is, for a system of the type discussed here, with many
Debye lengths between the electrodes and kinetically reversible electrodes, identical to that obtained in the previous section.
J. Electrochem. Soc., Vol. 138, No. 5, May 1991 9 The Electrochemical Society, Inc.
2.0
UNSUPPORTED CONDITIONS
Q2
1.o
r'-d
I
0.0
0.0
. . . . . . . . . . . .
1.0
2.0
5.0
z'/R
Fig. 1. Complex plane plots for system impedance calculated from
"quasi-state" electroneutral approach (solid circles) and exact solution
to Nernst-Plank-Poisson System (open circles). Here Dn = 2Dp and impedance is plotted in units of R| Eq. [21 ]. The arrows indicate the frequency at which A = ~/10. Frequency increases by 10 ~ between
neighboring plotted points.
Discussion
Although the approximate equivalent circuit developed
from the exact solution of the unsupported case discussed
above coincides with the result developed assuming bulk
electroneutrality, it should be noted that the exact solution
applies also in cases in which the electrode separation is
less than or comparable to the Debye length. Bulk electroneutrality cannot then be assumed. Further, when the
electrode reaction is not very fast, the use of the Nernst
equation, as in Eq. [8], (which describes potential drops at
thermal equilibrium) is not allowed. In addition, the quasistatic electroneutral treatment leads to incorrect behavior
at high frequencies, as is shown in Fig. 1, where the impedance functions derived from the exact and quasi-static
electroneutral treatments are compared for Dn = 2Dp.
I n some very recent publications, Lorimer (20) and Pollard and Compte (21) discuss corrections to the electroneutral treatment arising from interactions between ions.
Lorimer considers the coupling of ion fluxes and the diffusion of ion pairs. Pollard and Compte consider in addition
the possible presence of an inert phase. While the corrections discussed by these authors may be important in
specific situations, since both treatments assume electroneutrality their results cannot be considered valid for systems with electrode separations comparable or less than
the Debye length. Further, the implication of Pollard and
Compte that all earlier analyses of diffusion effects which
use concentrations rather than activities, such as the present and nearly all earlier work, are only useful "in the limit
of an infinitely dilute, single-phase electrolyte," should be
put to the test of experiment. I n actuality, the dilute approximation, which uses concentrations, is excellent for a
great m a n y real situations up to quite high concentrations,
and it is rare in practice to find that a treatment using activities is indispensable.
Acknowledgments
The authors wish to acknowledge the following for support of this research: the Donors of the Petroleum Research Fund, administered by the American Chemical Society, Memphis State University for a Faculty Research
Grant, and the yon H u m b o l d t Foundation.
Manuscript submitted Oct. 15, 1990; revised manuscript
received Jan. 8, 1991.
Memphis State University assisted in meeting the publication costs of this article.
c~
Cg
LIST OF SYMBOLS
concentration of species i, moYcm 3
geometric capacitance of electrolyte per unit area,
F/cm 2
1371
d
Di
E
F
thickness of cell, cm
diffusion coefficient of species i, cm2/s
electric field, V/cm
Faraday constant 96,487 C/equiv
HN
dimensionless quantity related to mobility ratio,
Eq. [27]
I
electric current density, A/cm 2
Ji
flux density of species i, mol/cm 2 s
k~
electrode reaction rate constant, mol cm/s
Ldiff diffusion length, cm
LD
Debye length, cm
M
electrode metal
R
universal gas constant, 8.3144 J/mol K
R|
electrolyte resistance per unit area, t2 cm 2
t
time, s
t~
transference n u m b e r of species i
T
absolute temperature, K
V
electrical potential difference, V
x
position within cell, cm
Z
impedance per unit area, ~ crn 2
Greek letters
A
dimensionless multiple of angular frequency, Eq. [1]
~h
electrical mobility of species i, cm2/V s
9rm mobility ratio ~n/~p
TD
dielectric relaxation time, s
r
angular frequency, rad/s
Subscripts
f
faradaic
n
negative species, anion
p
positive species, cation
s
coupled, positive and negative
W
Warburg
0
steady-state c o m p o n e n t
1
sinusoidally varying component
appl applied
en
electroneutral
int
interfacial
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