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Simplified impedance/frequency‐response results for intrinsically conducting solids
and liquids
J. Ross Macdonald
Citation: The Journal of Chemical Physics 61, 3977 (1974); doi: 10.1063/1.1681691
View online: http://dx.doi.org/10.1063/1.1681691
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Simplified impedance/frequency-response results for
intrinsically conducting solids and liquids
J. Ross Macdonald*
Texas Instruments Incorporated, Dallas, Texas 75222
(Received 21 June 1974)
The small-signal ac response is considered of a system containing a single species of positive charge
and a single species of negative charge. The charge carriers may be of many different types (ions,
electrons, vacancies, etc.) and are assumed to have arbitrary mobilities and valences. Quite general
boundary conditions are considered which encompass the range from complete blocking to zero
blocking (infinite reaction rate at the electrodes) for positive and negative charges separately. The
present paper deals primarily with approximations to an earlier exact solution of the problem which,
in general, lead to an equivalent circuit made up of three parallel Recombinations in series. The
elements of one of these parallel circuits, associated only with bulk effects, are frequency
independent, and those of another, which are associated with nonzero blocking, may often be well
approximated as independent. The third Resection arises from diffusion effects, involves frequency
dependent elements, and exhibits approximate Warburg frequency response over a considerable
frequency range. In general, Cole-Cole or Nyquist impedance plane plots show three connected arcs,
two of which are frequently good semicircles. These arcs are directly associated with the three R C
sections. Under many conditions, only two of the three arcs may appear and melding of arcs into
each other can also occur. Very simple, as well as less simple, approximations are developed for the
impedances of the individual Resections as well as the over-all impedance of the system. The
accuracy of these approximations is evaluated, and it is shown how they may be used in
unambiguous cases to analyze frequency response data to yield estimates of mobilities, valences,
electrode reaction rate parameters, and bulk charge concentrations. Surprisingly, it is found that for a
certain mobility ratio range, the center are, associated with electrode reactions, well approximates a
depressed semicircle of the Cole-Cole relaxation time distribution type, yet no distribution of
relaxation times is present. Further, in the completely blocking case, where the center approximate
semicircle is not depressed but has an infinite radius, the lower frequency portion does not
necessarily begin with a vertical segment as frequency decreases but may be curved away from
vertical over a considerable frequency range. General impedance results for the present unsupported
conduction situation are found to be quite different in some ways from those following from
supported electrolyte treatments. Such treatments should thus not be used to analyze unsupported
situations. Finally, it is found that the parallel capacitance of the system, which may be far greater
than ordinary double layer values, can exhibit appreciable regions of w- m frequency response with
O:s; m :s; 2, and with m =0.5, I, 1.5, and 2 values especially prominent.
I. INTRODUCTION
In a previous paper, 1 an exact, closed-form solution
was presented for the small-signal impedance of a twoelectrode, intrinsic-extrinsic conduction system containing mobile positive and negative charge species of
arbitrary valence numbers zp and zn, arbitrary mobilities, fJ.p and fJ. n , and quite general boundary condition
parameters, rp and r n , themselves simply related to ordinary effective rate constants. 2 Because of its generality, this solution should often be useful in analyzing
experimental impedance-frequency results for semiconductors and insulators, aqueous and other liquid electrolytes, solid electrolytes such as superionic conductors, and fused salts. But, because of the extreme
complexity of the exact solution, which depends on seven
parameters even in its normalized form, it has, thus
far, only been explored in detail for an intrinsic conduction situation. 2 This exploration was based on accurate
computer calculations of the consequences of the exact
theory. Further, it should be mentioned that the exact
theory is only applicable in the extrinsic case for full
impurity dissociation.
Although numerous explicit formulas were given2 for
impedance and admittance components as functions of
frequency in restricted frequency ranges, one particular
approach toward unifying and simplifying the results was
discovered and briefly mentioned in the previous paper. 2
It was based on the finding that in the most general situation three connected arcs appeared when the imaginary
and real parts of the impedance were plotted parametrically in the impedance plane as circle diagrams or
Cole-Cole plots. 3 Under favorable conditions the two
highest frequency arcs were excellent semicircles with
origins on the real axis, while the lowest frequency arc
was a distorted semicircle, showing typical Warburg/
diffusion type behavior. 2,4 In general, anywhere from
one to three of these arcs could appear with or without
appreciable melding of one arc into another.
The above results suggested that the total impedance
of the system, ZT' might be usefully represented by the
sum of three separate impedances in series, each made
up of a conductance and capacitance in parallel. 2 Let the
impedances of the three parallel circuits, each representing one of the possible arcs, be Zl =Yi\ Z2 =Yi\
and Z3 =y;l, where Zl applies at the low end of the frequency spectrum and Z3 at the high end. The arcs will
be correspondingly identified: thus, arc 1 is the lowfrequency one. Two equivalent forms of the exact equivalent circuit have already been discussed. 1,2 An alter-
The Journal of Chemical Physics, Vol. 61, No. 10, 15 November 1974
Copyright © 1974 American Institute of Physics
3977
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3978
J. R ass MacDonald: I mpedance/frequency response for sol ids and I iqu ids
aid one to distinguish between them. Finally, although
main emphasis here will be on intrinsic conduction situations, many of the formulas of the present work will be
shown to apply to fully dissociated extrinsic conditions
as well.
II. GLOSSARY OF SYMBOLS
A. Major subscripts
G.,
(bl
FIG. 1. (a) Possible exact series-parallel equivalent circuit.
(h) Form of equivalent circuit for identical, plane-parallel
electrodes.
n
p
N
nate form is that represented in Fig. 1(a). To the degree that arcs 2 and 3 are perfect semicircles, the capacitive and conductive elements of Ya and Y 3 are frequency independent. Further, Y l is essentially intensive
(an interface quantity) when the frequency of the applied
potential is not too low; Ya is intensive for all conditions
of interest; and Y 3 , which represents only the bulk properties of the basic material, is always extensive. The
equivalent circuit can then be expanded to that of Fig.
l(b), applying for the situation of identical plane-parallel
electrodes with identical boundary condition parameters.
The two parallel circuits on the left are thus associated
with conditions at and within a Debye length or two of the
left electrode except at such low frequencies that Y l becomes extensive. Here we have made the identification
of the extensive Y 3 circuit elements as C3 =C" the geometric capacitance of the system in the absence of mobile charge, and G3 =Goo, its bulk or solution conductance. These important frequency-independent quantities lead to the intensive time constant TD C,/G.,
C,Roo, the dielectric relaxation time. A symbol glossary is presented in the next section.
=
=
In the present work, the above series-parallel approach is further investigated, generalized, and evaluated for accuracy in a variety of situations of possible
interest. Both simple and somewhat more complicated
results useful in analyzing experiments are developed
and compared with corresponding supported electrolyte
expressions. All circuit elements will be taken as specific or unit area, quantities, and identical, plane-par- .
allel electrodes and boundary conditions will usually be
assumed. The results may be readily generalized to the
situation of different electrodes and boundary conditions
and to that where there is a Single dominant working
electrode with which is associated virtually all the intensive circuit elements (interface quantities) of importance. The results apply accurately in electrochemical
situations only to the unsupported electrolyte case, but
some of them apply, at least approximately, to the supported case as well. Further restrictions have been
discussed earlier. l,a It is important to note that the
present three arcs and corresponding three series-parallel circuits appear here for a single homogeneous material, not a three layer model, which can also lead to
three parallel GC circuits in series. 5 The two situations
should not be confused, and the present results should
p
S
T
o
00
Designates an intrinsic or series "interface"
quantity; also used as index with i =n or p
DeSignates quantity associated with negative
mobile charged species
Designates quantity associated with positive
mobile charged species
Normalization of impedances and resistances
with R." of admittances and conductances
with G., =R:l , and of capacitances with Cg
A parallel quantity
A series quantity; also indicates a plateau region quantity
Stands for "total"
DeSignates either a static quantity or the zerofrequency limit of a frequency-dependent
quantity
The value of a quantity in the limit of high frequencies (i. e.. n» 1)
B. Major symbols in text
Numbers in parentheses indicate equations where the
symbol is used or defined.
A
(4GWPNCWS N)-l/2; normalized Warburg param-
eter; (20)
Warburg parameter; (21)
Double layer capacitance; essentially MCg in
two-electrode situation
Geometric capaCitance/unit area; E/4rrl
"Interface" capacitance unit/area; the series
capacitance associated with Z I
"Interface" capacitance/unit area in the plateau
region
Capacitative components of Y J ; j =1,2,3
Total parallel capacitance/unit area; associated with Y T; Note; Cpo =Co =C IO + C,
Total parallel plateau capaCitance/unit area
(Av'ITtlC,; (A33)
Diffusion coefficient; for positive carriers i = p;
for negative i =n
The Faraday
Frequency-independent parallel discharge conductance/unit area; G., - G E ; (3); see Fig. 2
(gsGoo/gpfJ,) (Goo - GD ) =R'jf; see Fig. 2
Conductive components of Y J ; j =1, 2, 3.
Total parallel conductance/unit area; associatedwithYT ; Note; GpO=G D ; Gp.,=G.,
Bulk conductance/unit area; R:l
Debye length in the present intrinsic case
=
(l/2L D )
Value of Re(Z T) at cusp between arcs 1 and 2;
(Re +Roo); see Fig. 3 and Table I
Percent accuracy of approximation for Im(ZTN);
(36)
J. Chem. Phys., Vol. 61, No. 10, 15 November 1974
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J. Ross MacDonald: I mpedance/frequency response for solids and liquids
Percent accuracy of approximation for Re(ZTN);
(35)
R
RCl
Gas constant
Negative resistance in the Warburg region;
(A26)
Gii; (3)
Frequency-independent series resistance-unit
area; Gil; (4)
"Interface" series resistance-unit area; associated with Z I
(R C1 +P2 ); (A36)
(Rws +RR); (A34)
ARJIfi; (A35)
Equilibrium charge transfer resistance-unit
area; (10), (13), (14)
R~
G;}
T
Absolute temperature
Yj
Zi 1
YJ
Zjl
YT
Yw
Total admittance/unit area of the system
zwt
Y 12
Zi~; (2)
Y 1N1
Approximate expression for YiN; (A20) and Appendix E.
Approximate expression for Y2N ; (A14)
Y 2N1
Zj
ZJ
ZT
ZTNl
ZTN2
"Interface" impedance-unit area
Additive components of ZT; j =1, 2, 3; (1)
y~l; (1)
Approximate expression for ZTN; (A27)
Approximate expression for ZTN; less accurate
than ZTN1; (34)
A o(1- i)/fW; Warburg impedance; (18), (19)
(Zl +Z2)==(ZT- Z S)
a
(O:/Ep) + (O~/En)
b
onop/ EnEp
c
(On/En) - (Op/ Ep)
d
(On/En) + (Op/ Ep) = a + b
1 +(rj/2); i=e (rn=rp), n, or p
gnEp +gpEn
gnop +gpon
n
p
Effective standard heterogeneous rate constants
for the supported case; i =n or p
Distance of separation of plane, parallel electrodes
Concentration of negative mobile charges
Concentration of positive mobile charges; also
p2==1+iO
M[coth(M)]
Designation of common value of rp and rn when
they are equal
Dimensionless discharge parameter for negative charges
Dimensionless discharge parameter for positive charges
rp/rn; unity when rp =rn =0
(YJ -1); j =1,2; (A10)
Valence number for negative mobile charges
Valence number for positive mobile charges
3979
Normalized radial frequency; WT D
Value of 0 at peak of arc j; j = 1, 2, 3
(a - 1)-1
Equilibrium transfer coefficient; also parameter in Cole-Cole dispersion formula
Angle of approximate Warburg response line;
(A38)
(MBJ)coth(MB J); (A9)
(1 +7T;1)"1 ==z./(zn +zp)
(1 + 7T.)"1 == Zp/(Zn + Zp)
op for 7Tm « 1; on for 7Tm » 1
Dielectric constant of the bulk material
(1 +7T;,1)"1 == Iln/(Iln + IIp)
(1 + 7Tm )"l == IIp/(Il n + IIp)
The peak height of arc 1; see Fig. 3
Eigenvalues; j =1, 2; (A1), (A2)
The arc 1 length (R D - P 2 ); see Fig. 3
Mobility; i =n or p
(Dj/l)rj; i =n or p; effective rate constants;
(11), (12)
Designation of common value of 1Tm and 1T. when
they are equal
Mobility ratio; Iln / IIp
Valence number ratio; zn/zp
Radius of arc 2; Re/2; see Fig. 3 and Table I
Dielectric relaxation time; CgR~
Radial frequency of the applied Sinusoidal
voltage
III. BACKGROUND
A normalized impedance situation is fully defined in
the intrinsic conduction case by the following six parameters, (rp, r.; 7Tm , 7T.; 0, M) and O. Here 7Tm == I.J. n/ IIp;
7T. ==z./zp; M==l/2L D ; and the normalized frequency is
0== WT D' The range of interest of 0 is 0.$0.$1. In these
expressions, l is the electrode separation, L D the intrinsic Debye length, and W the radial frequency. Electroneutrality holds in the bulk of the material when M
»1. When rp =r. =0, the electrodes are completely
blocking for both mobile charge carriers. Alternatively, when rp =0, rn =00, the negative carrier is completely
unblocked and reacts at an electrode with an infinite effective rate constant. It is important to note that because of symmetry if the quantities (rp , r.; 7Tm , 7T.; 0,
M), which define a given Situation, represent specific
values, then the situation (rn, rp; 7T;,1, 7T;1; 0, M) is fully
equivalent as far as its normalized total impedance ZT/
R~ is concerned. 1
Let a subscript "N" denote normalization of impedances and resistances with R~, admittances and conductances with G~, and capacitances with CI • Then we may
write
(1 )
where ZTN is the exact normalized impedance of the sys·
tem .. Since CSN == CI/CI = 1 = GSN = G~/G~, one readily
finds that Y SN == (Gs +iwCs)/G~ may be written as Y SN
==1 +iO ==p2.
It only remains to obtain useful expressions for YiN
J. Chern. Phys., Vol. 61, No. 10. 15 November 1974
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J. Ross MacDonald: I mpedance/frequency response for sol ids and I iqu ids
3980
and Y ZN ' Unfortunately, it is impractical to derive these
in general from the exact solution, both because of its
complexity and because the results would themselves be
too complex to be very useful. Nevertheless, the general solution and its simple exact results in certain specific situations can be of great value in determining the
general forms of YlN and YZN and limiting cases of these
quantities. Beyond this, exact computer results can
help one discover specific forms which jointly meet the
goal of maximum useful Simplicity married to adequate
accuracy.
Let (ZTN-ZSN)=Z12N={ZlN+ZZN). Thepreviouswork
then leads to the important quantity
Y1ZN =Zi~N =G 1ZN +mC 1ZN =p2r{JflR EN -0 +p2R~?N.YIN) •
(rp.rn ; 11'm' 11'z; 0, M)
M» 1
FIG. 3. General arcs and definition of quantities in the normalized impedance plane for (rp, rn; 7rm , 7r.; 0, M).
(2)
When the quantity YIN is taken from the exact solution
of the problem, 1 Y1ZN may be obtained exactly as well.
When Y1N or YZN is infinite, Y12N equals the noninfinite
member of the pair. Here RD and RE are frequency-independent elements in the exact equivalent circuit1,z; in
normalized form they are given by
RDN=GD~={€nrl+(2/rn)]-lHprl+(2/rp)]-1}"1.
(3)
and
REN=Gi"N={€nrl+(rn/2W1+€prl+(rp/2)]-1}-1.
In addition, €n={l+1T;,t}-l=f.J. n/(f.J. n +f.J.p), €p=(1+1T m t
= f.J.p/(f.J.n + f.J.p), and (€n + €p) = 1. Note also that (G EN
+G DN ) =1.
(4)
1
The normalized admittance YIN is the only frequency
dependent quantity in the exact theory and exact equivalent circuit, 1,2 two equivalent forms of which are shown
in Fig. 2. The impedance of the circuit of Fig. l{a) may
be made identical to that of Fig. 2{a) and (b) when some
Cg
II
II
•A
Go
I
I
Yj(w)
'v
(0)
Cg
(b)
FIG. 2. (a) Exact equivalent circuit involving Y j •
nate form of exact equivalent circuit.
IV. SOME APPROXIMATE RESULTS
A. Arcs
Let us first consider the simplest case: that where
all arcs present are well defined and the cusps between
them are sharp. Such behavior occurs when M» 1, so
that there are many Debye lengths contained between the
electrodes, and when 0lM «02M« 1, where these specific O's are defined in Fig. 3. Approximate expressions for the arcs and their underlying frequency response are then simplest and easiest to use to analyze
experimental data. Some exact and less approximate
results for various cases are discussed in the Appendix .
In general, three separate arcs can appear simultaneously, as illustrated in the diagrammatic normalized
impedance plane representation of Fig. 3. The encircled
numbers show the arc designations. In the present case,
arcs 2 and 3 are good semicircles with centers on the
real axis; arc 3 is fixed in size; and the sizes of arcs 1
and 2 may independently vary, making all sizes possible
relative to arc 3. Note that with the present normalization the point 1 on the real axis corresponds to Re(Z T)
=R~. The point 0 = 1 corresponds to w = TD1, the frequency where the capacitative reactance of C1 equals R~.
The general lengths: PaN, A1N , and 711N; points: P2N and
R DN; and normalized frequencies at arc maxima on the
imaginary axis: 02M and 0lM; are defined as illustrated
in Fig. 3. Note that A1N =-R DN - P2N .
GE
,A
I
I
frequency dependence is allowed for the Y1 and Yz conductive and capacitative elements of Fig. 1(a). The inverse of YIN may be written ZIN =RIN + (mCINt1, where
the series "interface" components RI and CI are both
frequency dependent in general. If we write for the total
normalized admittance of the system, Y TN =-z-A =-GpN
+iOC pN , then it follows that in the limit 0 - 0, which
will be designated hereafter by a subscript zero, GpNO
=-G DN and CPNo =-l +CINO=-G~N+G~NC12NO' We may also
write Z TNO =- G;}NO =R DN =Z 1NO + Z2NO + ZSNO = Gi10 + Gz10 + 1.
An exact expression for CINO is given in Appendix E.
(b) Alter-
Table I presents expressions for most of the above
quantities for specifiC (rp , rn) choices of interest and increasing complexity. Note that when a length is zero
the associated arc does not appear. The expressions
for RDN are exact here but those for the other quantities
J. Chern. Phys., Vol. 61, No. 10, 15 November 1974
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J. Ross MacDonald: I mpedance/frequency response for solids and I iqu ids
3981
TABLE I. Expressions for arc pOints and lengths for various
values of r, and rn'
Points
Lengths
z
~
YP,Yn
P2N
y,=rn==Ye
1+ (2/y.)
0,00
0,
1 + (2/EnYn)
Yn
P2N
1+(2/Ye)
y- I
e
0
cnl
0
,,-I
E;I[l + (2/Yn)]
(Enyn)-I
,,_I
(~)
Yp + 2En
0
7T;;.t
1 + (Y/2En)
RON
(EnYn+ EpYp)-1
(RON- P 2N)
YtuC()
r"Y n
1 + (EnYn:VJ
!::!
AIN
RON
....eI
m
m
are only good approximations when cusps are well defined. In cases where arc I or 2 but not both appear and
1T", =1T1I =1Te, exact expressions for YlN and Y2N may be
found (see Appendix B). Assessment of the accuracy of
the present approximations will be given later. Figures
4-6 show idealized arc situations for some of the specific cases of Table I. Approximate equivalent circuits
involving frequency-independent elements (except for Y l )
are also included in Figs. 4 and 5. A simple approximation for YlN is given later rEq. (17)] and more accurate ones in Appendixes D·and E.
Re (ZTN)
(b)
FIG. 5. Idealized arcs and approximate equivalent circuit for
(r" rn) =(re, re). (h) Idealized arcs and equivalent circuit for
(0,00).
B. Approximate components of Y 2
We have already seen that Y 3N =jJ =1 +m, a simple
Debye dispersion situation involving bulk properties
only. Richer possibilities appear in YiN and YlN' In the
present rn < 00, M» 1 case, we find that for OS 0 «Os
Y2N=G2N+mc2N~G2NO+mc2NO'
(5)
where the "0" subscript indicates the 0 - 0 limit as usual. The quantity Os is approximately 1T", for 1T",« 1. A
.more general definition is given in Sec. V. This approximation of taking G2N and C2N essentially frequency
independent is pertinent, of course, when arc 2 is a good
semicircle, when M» 1, and for frequencies where 0
«1 and thus jJ ~ 1. Exact results l for YiN' together
with Eq. (2), then lead to the expressions
G2NO ~ O. 5(f:n rn + f:, r,)
(6)
C2NO~r+(f:nrn+~r,) ,
(7)
=(f: rn + f:, r,t
l
must be of the order of unity or greater
for arc 2 to be comparable to or larger than arc 3, C2NO
may usually be well approximated,as
n
(8)
when M» 1. Equations (6) and (7) are exact when rn
=r, =re. Finally one finds, when arc 2 appears in the
region 0 <0 «Os,
z
~2M
~
N
and
where r
=(M)coth(M).
(rp,rn) =(00,00)
Since r ~ M here and P2N
Cg
(0,0)
~
Gm
z
~
z
~
N
~L::LJc:
o
I
0
e
7~:
o
--__
--------~~
Re (ZTN)
Re (ZTN)
(0)
(b)
FIG. 4. (a) Arc and equivalent circuit for (r" rn) = (00, 00). (h)
Idealized arcs and approximate equivalent circuit for (0,0).
~~
__
~
______-L_______
Re (ZTN)
(b)
FIG. 6. (a) Idealized arcs for (r" rn) = (0, rn) when rn «1 + 'IT",.
Here 'IT",»1, rn~1. (h) Idealized
arcs for (0, T)
when r n »1
2
n
+ 'IT",. Here 'IT". ~ 1, Tn:;:' 10 .
J. Chern. Phys., Vol. 61, No, 10, 15 November 1974
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3982
J. Ross MacDonald: I mpedance/frequency response for sol ids and I iqu ids
'l ZM ~ GZNo/CaNo =(2PzCZNO )-1 ~ (En rn + Ep rp )/2M •
(9)
Clearly we must have 'lZM «1.
Equation (8) leads to Czo '" CZNO Cg ~ MC g = E/81fL D ,
where C, '" E/41fl and E is the dielectric constant of the
underlying bulk material. But E/81fL D is essentially just
the expression for the unbiased double layer capacitance
of two completely blocking (rp = rn = 0) double layers in
series, one in the neighborhood of each electrode. 6 We
thus see that in the context of the present approximation
Z2 is made up of just the blocking double layer capacitance, Cd = Czo , in parallel with a resistance
R8 = G2~oR~ = 2R~/( En rn + Ep rp).
(10)
Since Re depends directly on the boundary parameters
rn and rp and is infinite when they are both zero, it may
be identified as an electrode reaction rate resistance,
here expressed for two identical electrodes in series.
Note that in unnormalized terms pz is just R8/2 and P z is
just (R~ +Re).
The above unnormalized results for arc 2 are formally
exactly those found in the conventional supported electrolyte situation for a perfectly irreversible redox reaction. 7 There, a semicircle of radius R8/2 begins at
(R~ +R8) on the real axis and ends on the same axis at
R~. (C, is ignored in the conventional treatment, thus
eliminating arc 3). But to what extent are the R/s appropriate in the two cases the same? Let us consider a
single working electrode and introduce the unsupportedsituation effective reaction rate constantsZ
(11)
and
(12)
where Dn and Dp are the diffusion coefficients of the mobile charges. Since ~n and ~p are intensive, rn and rp
are extensive.
For a single working electrode, deSignated with a
superscript (1), the unsupported Re thus becomes
(l)Re =(Rjl)/( En D~1 ~n + EpD;1 ~p)
= (2IfT/F Z)/(znnj +ZpPj)(Zn~n +Zp4)
(R T /FZ)/(z~ nj ~n +Z~ PI ~p) ,
The conventional supported Re is not usually given in
the present case of possible simultaneous parallel reactions involving both the positive and negative charges
(0 5 ~n5 00, 0 5 ~p5 00). Generalization to this case has
been given earlierz ; the supported result is
(l)R8=(RT/Fa)/(z~cnekn+z~cpekp),
(14)
which may be directly compared to Eq. (13). Here k n
and kp are the effective standard heterogeneous rate
constants in the supported case and cne and cpe are effective concentrations. If k n is associated with a redox reaction, for example, c ne is usually written as cgc}i'" ,
where a is the equilibrium transfer coefficient and Co
and c R are the concentrations of oxidizing and reducing
species involved in the reaction. If we take the fs and
the k's the same, (13) and (14) lead to the same resistance except for the apparent difference in the concentrations involved.
The quantities nj and Pj appearing in (13) are bulk
concentrations of charged species. On the other hand,
the supported-electrolyte concentrations cne and cpe usually involve the concentrations of both charged and uncharged species. But at the equilibrium potential, the
point at which the present unsupported and supported
treatments apply, co=cR"'c e and cgc~-"'=cne=ce. This
concentration is equivalent to nj, for example. Since
both the sand k's are standard rate constants and thus
account for both forward and reverse reactions, there is
evidently no significant difference between the present
unsupported Re and the conventional supported result at
the equilibrium potential. The present results apply
rigorously for very small ac perturbations around the
equilibrium potential, especially when it coincides with
the point of zero electrode charge in the electrochemical
case, but they may also apply approximately in other
regions where the current-voltage response is approximately linearz provided the rate constants are suitably
inte rpreted .
e
C. Experimental analysis of arc 2 results
= (RT/FZ)/(z~nl +Z~pj)(on~n + op~p)
=
= (0, rn) and the typical values T = 290 K, nj = 10-3M, zn
=1, and ~n=10-4 cm/sec, one finds (l)R 8 ",,2.6X103
ohm-cmz• For Dn = 10-5 cmz/sec and 1 = 0.1 cm, the rn
value corresponding to ~n = 10- 4 cm/sec is just unity.
In unnormalized form Eq. (9) becomes
(13)
where R~ '" (l/FHznMnnj +zpMp pd- 1 has been used along
with the electroneutrality condition Znnj =ZpPi; nj and Pj
are the equilibrium bulk intrinsic-conduction charge
concentrations; on'" (1 +1f;1)"1 =zi(Zn +zp); op '" (1 +1f~)-1
= Zp/(zn + zp); and on + op '" 1. In addition, the Einstein
relation D j = (R T/Fz j ) Mj has been employed here with
i =n and p. The Re of Eq. (13) is clearly intensive; it
differs somewhat from the corresponding Res resistance
given earlierZ because Re and Res appear in different
equivalent circuits. When there are no reactions, ~n
= 4 = 0 and Re = 00 in this blocking case. On the other
hand, when either ~n or ~p approaches infinity, R8 - 0 and
thus Zz-O as well. No double layer capacitance then appears in the present equivalent circuit. For (rp, rn)
W ZM
~ (Re Cd)"1
=(E" rn + Epr p)Gj2Cd •
(15)
Since R8 may be found from the P z intercept, Cd may
then be obtained from the experimental value of W ZM '
Although R~ may be found from the left real-axis intercept of arc 2, one must extend measurements to at least
'l::? 0.1 in order to determine C, adequately. Then C
"
Cd, and W ZM allow one to calculate M and L D'
Figures 5(a) and 6(a) show two principal cases where
only arcs 2 and 3 appear. While this is strictly true for
the rn = rp "'re case of Fig. 5(a), the condition rn« 1 +1fm
applying in the (0, rn) case of Fig. 6(a) only ensures that
arc 2 will always be far bigger than arc 1, not that arc
1 will be completely absent. Nevertheless, arc 1 may
then not be experimentally distinguishable. To what degree can we then distinguish between these two similar
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J. Ross MacDonald: I mpedance/frequency response for sol ids and I iqu ids
cases from experimental data? In the first case, Re
=2R",/re , while it is 2R"/€nrn in the second. Let us now
assume here that arc 1 is much smaller than arc 3 as
well as arc 2. Then1Tm »1, €n~1, andthusRe~2R.,/rn
in the case of Fig. 6(a). Unless vestiges of arc 1 can be
identified in this case, there is no way to distinguish it '
from the first case to the present order of approximation. One then will not know whether Re involves re or
rn and thus whether the experimental situation involves
equalrnandrporrp=O, O<rn<oo. Eventheresultsof
the exact theory will generally not allow distinction between the two cases except for data much more accurate
than usually achievable.
Suppose that a good arc 2 semicircle, or even part of
a semicircle, has been measured along with enough of
arc 3 to yield Cg • Arc 1 mayor may not be present.
Assume that l and the electrode area are known and that
both zn and zp are known as well, the usual experimental
situation. It is of interest to summarize just what significant material parameters can then be calculated using the four arc 2 and 3 results: C" R"" R e , and WZM '
Assume as usual that the circuit parameters are given
per unit area. We may then calculate the estimates:
€
(b)
Cd ~ (ReWzMtl
M~ Czo/C, ~ Cd/C,
L n =l/2M
nl
Incidentally, although very little least squares analysis
of complex data, such as impedance as a function of
frequency, temperature, and/or dc bias, seems to have
appeared in the literature, it is worth emphasizing that
recently described methods of generalized least squares
analysis 8 allow such data fitting to be carried out most
expeditiously. This approach should be particularly appropriate in obtaining best parameter estimates from
impedance or admittance data using one of the more accurate models, or impedance formulas, given in the Appendix.
D. Approximate components of Y,
Simplification of the approximate expressions for YlN
given in Eq. (A20) of Appendix D leads for r, - 0 to the
even more approximate result
YIN =G 1N +mclN ~ r1Tm(mbM Z)I/2 coth(mbM Z)I12]
x[1-sr.
(16)
where s =rp/rn' defined as unity when r, = rn = 0, and
b =6n6,/€n"-,. The (1- stZ term in (16) is needed to make
Y 1N - 00 when r, =rn =re , a situation where arc 1 should
not appear at all in the impedance plane plot.
It follows fromEq. (16) for the (0, rn) situation with
rn>O that G1NO~ 1Tm and C1NO~ b1T mM z/3 = 6n6,Mz/3"-~,
=41TlC,
(a)
3983
= foR T6 n/41T(Fz nL D )Z
PI =1T.nl
(c)
(J.l. n + J.l.p) =l/ FZn nl R",
(d)
(zn ~n + zp ~p) ~ (2R T / FZ)(zn nl R et 1
(J.l.nrn + J.l.p rp) ~2l/FZnnIRe
leading to extensive behavior for G10 and C10 • Equation
(16) holds most accurately when M2« 1. It leads to the
arc 1 shape of Fig. 5(b). To this order of approximation,
Y1N is independent of rn(> 0) when r, = 0; thus essentially
the same shape appears for arc 1 in Fig. 6(b), where
arc 2 is negligible. Note that when 1Tm = 1T., b = 1. Further when 1T z =1 and 1Tm« 1, b~1T;;/4 and C1No~M2/12.
For M» 1, C1NO is then much larger than CZNO~ M
~ CdN • The normalized admittance Y1N approaches its
low frequency limiting value when !l < 2. 5(bM Z 1, and
the maximum arc 1 height, T/1N ~ O. 4171T;;.1, is reached
at !l = !l1M ~ 2. 53( bM z 1 • Note that the conditions !l1M
« !lZM« 1 given earlier lead to the following conditions
on M when the present approximate expressions for !l1M
and !lZM are used: M» (2pz N 1 and M» 5b-1pZN' These
inequalities only apply when PZN < 00.
r
r
(€nrn + E:prp) ~ 2R"/Re =2R;1.
From the four independent inputs, we calculate the four
independent outputs €, n i and Ph (J.l. n + J.l.p), and
(zn~n +zp~p). It is clear that from these data alone we
are unable to obtain estimates of J.l.n , J.I." ~n' and ~,
separately.
r
Now when !l > 10(bM Zt 1 , coth(mbM 2)1/z"" 1 and for s = 0
one finds
(17)
To obtain J.l.n and J.I., we must know 1Tm. Further,
(zn~n +Z,~,) =zn~n [1 +1T;;.I(r,/rn)]
when zn~n *0. Thus to obtain ~n and ~, =(1T.!rrm)(rp/rnHn ,
we must also know (r,/rn). In most cases of interest,
this intensive quantity will be either unity or zero. If
no trace of arc 1 appears at very low frequency the r,
=rn =re assumption is indicated. Otherwise, one may
either see vestiges of arc 1 or have information suggesting that r, must be zero. Then in either of these cases
~n and ~p may be Obtained. Even though 1Tm is not needed
in the (rp/rn) = 0 case to obtain ~n' it is needed in the re
case. Although as we shall shortly see, 1Tm may be
readily obtained when arc 1 is significant, it can only be
estimated in the re situation through the use of a higherorder approximation than that employed in the present
section (see Appendix C).
simple Warburg behavior. If we write for a Warburg
impedance
(18)
and for
ZWN
(19)
then the present two-electrode results lead to
A
=(C,/R",)1/2Ao =(27f~bM2rl/2 =(,,-!/M)(2€'''-,6 n6,t 1/2
(20)
and
1
Ao = "-![(2,,t2R T/F 2) (Z.nl +Z,P1t ]
x {[(z"Dnr1 + (z,D,t 1] [Z~1 +Z;l
n
1 2
/
(21)
J. Chern. Phys., Vol. 61, No.1 0, 15 November 1974
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3984
J. Ross MacDonald: I mpedance/frequency response for sol ids and I iqu ids
Except for the presence of a factor of €: here and a
missing term involving rp and r n , these results for the
Warburg parameters A and Ao are exactly the same as
those found previously. Z The differences arise from the
different equivalent circuits in which the parameters appear. Note that for 1Tm «1 and 1T.. =1, A ~ (2/1r mMZ)1/Z.
The Ao result is quite different from that found in the
supported case, where Poisson's equation does not
couple positive and negative charge motion appreciably.
A detailed comparison has been given earlierz along with
much examination and discussion of approximate Warburg response. Some deviations from exact Warburg
behavior which follow from the present approach are
discussed in Appendixes D and E.
E. Further experimental analysis
In order that the good separation shown in Fig. 2 between arcs 1 and 2 be maintained, it is necessary that
0ZM»OlM' This condition leads to M»5€p/6n6prn for
rp=O, which becomes M»20/rn for 1T.. =1 and 1Tm«l as
well. Thus, in addition to the condition M» 1, rn may
not be too small if we wish to be able to distinguish arcs
1 and 2 adequately. Finally, comparison of Figs. 5(b)
and 6(b) shows that the cases rp = 0, rn = 00 and rp = 0,
rn» 1 + 1Tm cannot be well distinguished for the present
order of approximation. When PzN« 0. 5, arc 2 will be
indistinguishable, and it will not be possible to determine rn or ~n' For rp = and 1Tm «1, rn can thus only be
obtained from the present results when it falls in the
range 5/6n6p M« rn :S101T;.1. This is probably somewhat
too restrictive a condition on the low end, since even if
arcs 1 and 2 overlap appreciably, they may still be adequately separated analytically in many cases if overlap
is incomplete.
be increased by reducing 1 and thereby r n , leading to a
larger PzN' Such variation should not, of course, have
any effect on such intensive system parameters as €,
IJ. n , IJ.p, and ~n but it can help improve the accuracy with
which they may be obtained.
The above conclusions apply when the arcs are given
in normalized form. In unnormalized form, however,
the sizes of all arcs are proportional to Roo. Therefore,
arcs 1 and 3 will increase in size proportional to l. On
the other hand, Pz is intensive since rn and rp are themselves directly proportional to 1. Thus, ordinary unnormalized impedance plane plotting should lead to an
arc 2 size almost entirely independent of 1. As above,
it can be made larger relative to the sizes of arcs 1 and
3 by decreasing 1.
Although it is most convenient to determine 1Tm from
the arc 1 height, 'T/1N, 1Tm may still be estimated when
only the initial 45 straight-line portion of arc 1 is
available and measurements cannot be extended down to
H 1M • Let "1 be a value of
» 0lM) at a point in the
straight line region. Then in this region (Z TN)O ~ PZN
1
+ (ZlN)Ol' Therefore,
0
°(
1Y1N 101 ~ I[(Z TN)Ol - PZN ]-11 ~ 1TmM..;on1 ,
where the last equation follows from Eq. (16) when
flbM z .G'10.
°
Let us continue to consider only the (0, rn) case for
simplicity. If we now assume that PzN and 'T/1N must both
be 0.1 or greater to allow adequate observation of the
associated arcs, we find 1Tm$4 and rn $10€;1 or
10(1 +1T;.1). Even though one may sometimes be able to
measure with some accuracy impedance components
whose magnitudes are less than Roo/10, it is clear that
meaningful measurements in the arc 1 region will become very difficult to obtain when 1Tm» 10. Some other
aspects of this problem have been discussed earlier. Z
Although measurements frequently cannot be extended to
sufficiently low frequencies that all of arc 1 can be
delineated, it is sufficient that enough of the bend away
from Warburg behavior can be reached that extrapolation
based on the known approximate shape of arc 1 will allow
the height 'T/1N to be estimated. Then an estimate for
11'... is O. 41 711i~ = O. 417 Roo/[ - Im(ZT )maxd, where
- Im(ZT)max1 is the maximum height of arc 1 on the unnormalized impedance plane. Finally, this estimate of
1Tm may be used in the previous arc 2 equations along
with arc 2 measurements to yield estimates of IJ. n , IJ.p,
and ~n'
Since the sizes of arcs 1 and 2 depend on the separate
factors 1Tm and rn for (0, r n), one will have a certain
amount of control of the relative sizes in this case because rn( *0,00) is proportional to l. If arc 2, for example, is too small to measure accurately, its size may
(22)
Equation (22) readily leads to
2
1T 3/Z
lIZ""
)-l/ZI Y 1N 1 01
m + 1T m
- (M on 00
P 1
(23)
When Roo and C,. have been determined, one easily obtains 01 and I Y 1N I 01 from the experimental ZT results.
If it is assumed as usual that 1T.. is known, then 6nop may
be calculated. Finally, assume that M has been calculated from known quantities or estimated from an arc 2
experimental result such as a value of 0ZM' Then the
RHS of Eq. (23) is completely determined, and the cubic
in IT:';Z may be solved to yield an estimate of lTm' Final.,.
ly, previously presented expressions may be used to obtain IJ. n , IJ.p, and rn estimates. A typical case where the
above procedure is applicable is given by (0,80; lO-z,
1 ; 0,105 ) with 0lM« 0 1:5 (100Mt 1 • A somewhat more accurate approach is outlined in Appendix E. The situation
<rp <rn < 00 will usually be considerably more complicated1 and will not be discussed here.
°
It is worth noting that all formulas given in this paper
and earlier1•Z that involve only such 1Tm , 1T.. related quantities as En, Ep , On, and Op, and quantities such as b
derived from them, apply formally in the fully dissociated extrinsic conduction case as well as for intrinsic
conduction. 1 Even those results which explicitly contain
1Tm and/or 1T.. can readily be generalized for the dissociated extrinsic situation by carrying out the transformations 1Tm - 1Te and 1T.. -1Tf , where these quantities are defined in the full theory. 1 In the intrinsic case 1Tm 1Te and
1T1I =1Tf ; for dissociated extrinsic conditions 1Te and 1Tf
(and thus En, €p, On, and op) depend on the uncompensated impurity level and thus on the extrinsic parameter
X introduced in the earlier work. 1 The earlier work1 is
incorrect in implying that the theory holds for all extrinsic conditions. Its implicit assumption of homogeneous, immobile charged impurities limits its applica;'
=
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J. Ross MacDonald: I mpedance/frequency response for sol ids and I iqu ids
bility to tl).e fully dissociated situation. Further, although dissociation of intrinsic neutral centers to give
charged entities such as vacancies and interstitials determines the intrinsic charge level (nj and PI), the effect
of recombination has otherwise been neglected. 1 Although some justification for such neglect has been given
earlier, a it is hoped to include other effects of recombination exactly in future work.
.
V. TRANSITION REGION RESPONSE
Let the quantity a be defined as [(tl!/Ep)+ (O~/En)]' It
turns out that a = b when 1T& = 1. Further, when 1T~ = 1T""
a = b = 1. Finally, when 1T& = 1T;"\ b = 1 but a 1 unless
1T~ =1 as well. We may now define the quantity Os introduced in Sec. B as (a _1)-1. Thus, when (a -1)0« 1,
Eq. (5) holds quite well. When a =1, Eq. (5) holds well
over nearly the whole range of interest, 0:::0.$1.
Since 1T~ is limited by physical considerations to the
range 0.25:::1T&:::4, a and b cannot differ greatly, and
we may usually approximate Os by b- 1 when a» 1.
*
°
°
On taking s =b- 1 for simplicity, we find when GaM
«Os that for rn =rp =re
(24)
Thus, it is necessary in this case that (re/2M)« b- 1•
Note that symmetry ensures that ZTN is the same in the
(r e , re; 1T"" 1T.; 0, M) case for the two situations 1T",=B,
1T.=D and 1T",=F\ 1T&=D- 1• The quantity b=(2+1T",+1T;,.1)/
(2 + 1T. + 1T;1) is (2 + B +F1)/(2 +D +D- 1) for both these
choices. Clearly 1T", can be either too large or too small
here for (re/2M)« b- 1 to hold adequately when rand M
are fixed. It is this 1T", transition region where Eq. (5)
fails that we are particularly concerned with in this
section.
Let us now consider the rp=O, O<rn<oo situation.
Dominance of arc 2 as in Fig. 6(a) requires that rn
«1 + 1T"" which may be satisfied for small enough rn
when 1T", «1. Let us, however, ask for the conditions
that ensure that ZTN is essentially the same for the situations (re , re; B, D; 0, M), (re , re; B-1, D-1 ; 0, M), and
(O,rn ; B, D; 0, M), Figs. 5(a) and 6(a). It is thus necessary that I ZlN I « I ZZN I.
is much greater than that of the blocked carrier. Then,
arc 1 is negligible and arc 2 is essentially the same as
that for the (re , reI case. As we have already noted, it
may be virtually impossible to distinguish between ,the
two cases unless additional physical information on the
boundary processes is available. In the present 1T",
»1 + (r n /2) case, the dominant electrochemical step is
the heterogeneous electrode reaction, which involves the
most mobile charge carrier. On the other hand, when
rn» 1 + 1T"" and especially when 1T", «1, the dominant
process contributing to the total impedance is diffusion
to the electrode of the reacting charge carrier, which in
the 1T",« 1 case is much less mobile than the blocked
carrier [see Fig. 6(b)]. As usual, the slowest process
plays the major role in determining the impedance.
Now when Eq. (5) no longer holds well, Eq. (9) for
0ZM is no longer a good approximation either. Let us
see what happens under these conditions. Figure 7
shows some arc 2 results for various (1T"" 1T.) values.
Arc 3 has been omitted, as well as any vestiges of arc
1 for the two (0,1; 10\ 1T&; 0, 5X104 ) cases shown. Note
that although (re/2M) «b- 1 =1 in the (1,1 ; 1, 1 ; 0,5 X104)
case, this inequality fails for the 1T", = 104 cases, where
the (re , re) and (0, reI results are essentially indistinguishable.
The most interesting result of Fig. 7 is the appearance of arcs with depressed centers, first distinguishable when 1T", becomes appreciably larger than 100. Although the 1T", =1 curve is essentially a perfect semicircle with its center on the real axis, in agreement with
Eq. (5), the arcs for 1T", =104 are both depressed and
somewhat asymmetric. The broken line curves are
symmetric depressed semicircles; some deviation from
them occurs on the low-frequency side of the 1T", =104
arcs. In many cases, these deviations might, however,
be obscured by experimental error.
The listing in the figure shows the values of the ColeCole3 0, related to the included angle (01T/2) defined below the real axis. Here 0 values have been derived from
the best-fitting depressed semicircles, ignoring the lowfrequency deviations. Clearly, for constant 1T"" O! is
In the first two cases above, we have REN =1 + (re/2)
=ge, R DN =1+(2/re), and GaNO=re/2, all independent of
a
o
Band D. In the last case, on the other hand, one fin,ds
R EN =(l+B)gn/(B+gn) ,
RDN =
(1 +Fl)[l + (2/r n)] ,
0.05
(25)
1.0
(26)
~O.B
and
N
(27)
where gn =1 + (rn/2). It is clear that these results well
approximate those for the first two cases when rn ~ r e ,
1T", =B» 1, and 1T", =B»gn' These last two conditions
may be combined into 1T",» 1 + (rn/2). Arc 2 is still dominant, and it turns out that ZTN is very closely the same
for the (re , re) and (0, reI cases when 1T", »1 + (rn/2). The
results are quite different, a however, when 1T", « 1.
Note that the 1T",» 1 + (r n/2) condition in the (0, rn = re)
case ensures that the mobility of the discharging carrier
.5 0.6
I
I
L< I
0.0
1.0
!
I
1.2 ___ 1.4
!
I
1.6
!
1\
I.B
I
!---1----L----L----L---1---1---1....l.....l.-1.--.J
-------pi
0
2.2
2.4
2.6
2.B
3.0
Re (ZTN)
FIG. 7. Exact arc 2 results for (1,1; 71"",,71".; 0, 5X 104) and
(0,1; 71"",> 100, 71" •• 0, 5X 104) for various values of 71"", and 71" ..
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3986
J. Ross MacDonald: I mpedance/frequency response for sol ids and I iqu ids
larger than the 7T z =1 value when 7Tz <1, and it approaches
zero as 7Tz increases beyond unity. For 7Tz = 1, the maximum Q of about 0.054 occurs at 7Tm '= (7Tm )M, approximately 4 x 104 • As 7Tm increases beyond (7T m )M, Q approaches zero, reaching it by 7T m -107 •
Depressed semicircles of the present type are usually
described by the Cole-Cole dispersion formula, 3 which
implies a wide distribution of relaxation times when
ex >0. Here such curves, or at least quite good approximations to them, occur without such a distribution but
instead for a specific range of 7Tm values. For (0, rn;
7T m , 7T.; 0, M), this range clearly includes and is perhaps
centered near 7Tm - M. This is then the center of the
transition region, where (7T m )M - M, and where maximum
arc depression may be expected. Depressed arcs have
been observed for such widely different materials as dielectrics,3 biological substances,9 and glass electrodes. 10 In many cases, especially the biological and
glass areas just mentioned, it seems more likely that
the present mechanism is operative rather than that
there is a real distribution of relaxation times of the
Cole-Cole type present.
The quantity b(re /2M) is about 0.025 for the 7T m = 10\
= 1 curve of Fig. 7. Equation (5) no longer holds well
and usually the exact theoryl must be used to analyze results in detail in this depressed-arc region. Approximately, however, one may estimate 7Tm from the amount
of depression; but there is a chance of ambiguity in the
result if shape alone is considered since curves with 7Tm
both less and greater than (7T m )M have approximately the
same shape when 7Tz = 1. There is some significant difference in shape, however, if it isn't obscured by experimental errors. When 7Tz = 1, Fig. 7 shows that points
lie above the best-fitting depressed semicircle for 0
<02M when 7T m «7Tm )M' When 7Tm > (7T m )M , however, theoretical points lie above the depressed semicircle for
o >02M' The arc associated with (7T m)M is thus essentially
symmetric. It turns out, for example, that the arcs for
5
(7T m , 7Tz ) = 10\ 1 and 10 , 1 are almost mirror images of
each other in the plane through the line [Re(ZTN) =2,
- Im(ZTN) arbitrary] and perpendicular to the paper.
7Tz
To give some idea of frequency response, a few specific values are shown on the curves of Fig. 7. The 0
= 10-5 point corresponds to 0aM for 7Tm = 7T. = 1; it is clearly not 02M for the other values shown. Incidentally, the
curves obtained for 7T m = 1 and 0.25:S 7T.:S 4 cannot be distinguished from those with 7T m = 7T. = 1 when plotted as in
Fig. 7. This is not surprising since the RHS of Eq. (5)
does not depend on 7Tz at all. When 7T" = 1, the fa = 10-5
point moves progressively toward larger Re(ZTN) values
as 7Tm increases. By 7T m = 107 , it lies back on the (7T m , 7T,,)
= 1 semicircle but with the coordinates [Re(ZTN),
- Im(ZTN)]~2.33, 0.94 instead of 2,1. Alternatively,
when 7Tm = 107 and 7Tz varies, the point moves from 2.11,
0.99 at 7Tz =4 down to 2.66, 0.74 at 7T z = O. 25.
The change of curve shape with 7Tm variation when 7T.
< 1 is somewhat more complicated. Some typical results are shown in Fig. 8 for 7T" = 0.25. No depressed
semicircles have been drawn in here, and we see that
there may be very appreciable deviations and asymmetry for 7Tz =0.25. Clearly, for 7Tm >10\ the high fre-
(1,1; 1Tm,0.25; 0, 5x 10 4 )
1.2
(0,1;1Tm ,0.25;0,5xI0 4 )
1.0
;::0.8
N
~ 0.6
I
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
Re (lTN)
FIG. 8. Exact arc 2 results for (1, 1; 7r m, 0.25; 0, 5x 104) and
(0, 1; 7r m, 0.25; 0, 5x104) for various 7r m results with 7r m»1.
quency side of the arc approaches the limiting semicircle faster than the low frequency side. So much for the
transition region, although the present few curves
scarcely exhaust the different curve shape possibilities,
especially when extrinsic conduction behavior is present.
VI. PLATEAU REGION RESPONSE
In the plateau region,2 the smallest 0 value of interest
is appreciably larger than Os. Thus, the region to be
examined is roughly bounded by lOb- 1 :s 0:$ 0.1. It is
therefore only of consequence when b» 10 and either 7Tm
»1 or 7Tm « 1. It is here called the plateau region because within it CIN is virtually constant with a value CISN
less than ClNo' In the earlier work, 2 CISN was found to
be essentially g;2(M.fa; -0 for (r e , re) and g;2(M.fa; -1)
=(M
1) for (0, rn) when 7T m «1. Here gp '= 1 + (rp /2).
Arc 1 will then be very large. But the plateau region
extends over an appreciable 0 span when 7Tm » 1 as well.
In this situation, it turns out that C/ SN ~g;2(M.;r;; -1)
and g~2(MJ5" -1) for the two cases above. For 7Tm » 1,
arc 1 becomes negligible compared to arc 2, as we have
seen. As discussed earlier,2 the plateau region in CPN
is narrower than that which appears in CIN' Generally,
the range for the CPN plateau, where CPN = CPSN , is approximately 10b- 1 $O <M- 1 • Thus, there is no appreciable plateau unless a is sufficiently large that 7T!I» M and
conditions are thus beyond the 7Tm - M transition region.
Since the plateau region has already been considered2 for
the (0, rn) situation with 7T m «1, here main attention will
be given to the 7T m »gn case. Note that the shift from
.fr;; to IB,. in the above results arises because CISN essentially involves the real part of the eigenvalue 6 1 given in
Eqs. (A5) and (A 7) of the Appendix.
rr;;, -
The quantities M.fr;; = l.fr;;/2L D and M.;r;; = llB,./2L D
are of especial interest. Using the definition of the intrinsic Debye length, 1 one readily finds that
LD/.J7J; '=LDP '=(ERT/47TF2Z~PI)I/2
(28)
and
(29)
J. Chern. Phys., Vol. 61, No. 10, 15 November 1974
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J. Ross MacDonald: I mpedance/frequency response for sol ids and Iiqu ids
Here L Di> and L Dn are just the Debye lengths appropriate
for positive charges only mobile and for negative charges
only mobile, respectively. In the plateau frequency region, the mobility of the less mobile charge is so low
relative to the relatively high frequency in this region
that it can make no significant contribution2 to CI •
Figure 9(a), which follows directly from Fig. 2(a),
shows the exact equivalent circuit applicable in the plateau region, where C I ~ CIS' But it turns out that RI decreases rapidly as 0 increases in the plateau region.
By 0.c:25b- 1, for example, RIN is appreciably less than
unity. Thus, in most of the plateau region, RiN «R E ,
and Fig. 9(b) then applies, a circuit made up entirely of
frequency-independent elements. Finally, when 7Tm « 1,
RE~Roo andRD»Roo. ThenRD may be neglected and one
obtains2 Fig. 9(c).
Let us now see how an alternative series-parallel
equivalent circuit of the Fig. l(a) type can be obtained
for the plateau region. It is clear that when the Cg of
Fig. 2(b) is reconnected so that it is just in parallel with
Goo, one separates the total ZT into two series-parallel
circuits. The impedance of one of these circuits is just
Zs, and the other is, for 0«1, nearly Zl +Z2=Z12=ZT
- Zl' When Zl =0, as in the rp =rn =re situation, Y 2N is
just the Y 12N of Eq. (2). For this case, taking REN =ge,
one thus finds that for all 0
_
.
...z 2
iOCIN
\
2" HO
+ J> gel +iOCINR )
IN
_ 2(...zre
Y 2N - Y 12N - P
J>
.
(30)
The problem of obtaining a useful expression for Y 2N at
least for rp =rn =re would then be solved if a simple ex-
(bl
FIG. 10. (a) Approximate equivalent circuit for (r e , re; 1T m,
1T.; 0, M) and (0, re; 1Tm»1, 1T.; 0, M) when n«ns=ltz-l)-l.
(b) Approximate equivalent circuit for same situations when
n»ns·
preSSion for the frequency response of ZIN were available. Further development of this possibility is carried
out in the Appendix; here we need to consider only regions where CIN is essentially frequency independent.
First notice that for 0 - 0 and with
CINO = g;2(r -1),
Y 2N from Eq. (30) approaches
YaNO =[(re/2) + iO(r He)] ,
(31)
in agreement with Eqs. (5)-(7) for rp =rn =re' Figure
10(a) shows the equivalent circuit applicable when 0
«0 s, taking C2NO ~ r ~ M. The general relation C12NO
=R~l'NCINO + (2R EN -1), which applies here with CaNO
=C12NO , is implicit in Eq. (2) and consistent with the
term GiN Y l of Fig. 2(b). The corresponding G relation
is G 12NO =(REN - 1) =(R DN _1)-1 =R EN GDN'
Cg
II
RD
~Ir
(al
R·,IA
v
y
(al
Cg
II
II
RD
y
~I:
(bl
Cg
Now it turns out that for the plateau region, where
0» Os, (OC IN R IN )2« 1 and C2SN may be obtained to excellent apprOximation by replacing CINO in the above expression for C12NO by CISN ' Thus,
Y 2N ~ Y 2SN ~ jJ {jJ[(re/ 2) +iOM"ro:J)
(32)
where the second form applies when 0« 1 and M» reo
Here Ox is op for 7Tm «1 and on for 7Tm » 1. The Simple
equivalent circuit corresponding to the second result in
Eq. (32) is shown in Fig. 10(b) for 7Tm » 1. This circuit
will usually be easier to apply than one of the circuits of
Fig. 9.
Now in the (0, rn) case, we have seen that when 7Tm
»gn, ZlN- O and may be neglected compared to Z2N'
(e I
FIG. 9. (a) Exact equivalent circuit in plateau region (b)
Usual apprOximate equivalent circuit over most of plateau region. (c) ApprOximate equivalent circuit for the plateau region when 1Tm «1.
Thus, in this situation the plateau admittance is also well
approximated by Eq. (32) and Fig. 10(b) applies with re
replaced by f.nrn~rn' Then in both the rp=rn=re and
(0, rn) cases,
Y aN ~{o. 5(f.n rn + Ep rp)+iO[M"ro: + (f.n rn + f.p rp)]} . (33)
A somewhat more accurate expression is given in Eq.
J. Chern. Phys., Vol. 61, No. 10, 15 November 1974
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J. Ross MacDonald: I mpedance/frequency response for sol ids and I iqu ids
TABLE II. Some exact and approximate results for the (rp, rn;
12, and M=10'//2for rows 13-15.
rp
rn
7I"m
1; 0, M) situation. M= 5x 104 for rows 1 through
% Error
ZTN
Row
7I"m'
n
Re(ZTN)
-Im(ZTN)
2.9802
1. 99998
0.19802
1.00003
PR
% Error
ZTNI
PR
PI
10-14
10-14
ZTN2
PI
1
1
1
10- 6
10-5
2
0
2
10-6
10-'
2.9911
1. 9977
0.21452
1. 0023
-1.1x10- 1
-3.3x10-1
1.1
-2. 2x 10-1
-1.1 x 10-1
- 3. 4x 10-1
1.1
-2. 2x 10-1
3
0
1.5
2
10- 6
10-5
2.9846
1.9981
0.20629
0.99976
-7. 7x 10-2
-2. Ox 10-1
7.8 X 10-1
-2.2x10- 1
-7. 7x 10-2
- 2. Ox 10-1
7. 8x 10-1
- 2. 4x 10-1
4
0
3
0.5
10- 6
10-5
3.0024
1.9981
0.22807
1. 0061
-1. 5x 10-1
- 5. 2x 10-1
1.5
- 2. 3x 10-1
-1. 5x 10-1
- 5:1 x 10-1
1.5
- 2. 4x 10-1
5
1
1
10
10-6
10-'
2.9801
2.0026
0.19783
0.99746
-1. 4x 10-3
4. 3x 10-2
-3.2X10-2
- 8. 5x 10-2
-4.2x10-3
1.3x10-1
- 9. 6x 10-2
-2. 6x 10-1
6
0
1.1
10
10- 6
10-'
2.9798
2.0013
0.19934
0.99644
- 3. 6x 10-2
-3.1x 10-2
3.2X10-1
- 2.2 X 10-1
-3. 9x 10-2
5.3x10-2
2. 5x 10-1
-3.8X10-1
7
0
11
10-1
10- 6
10-'
3.0514
2.0012
0.28700
1. 0218
-3.2x10-1
-1. 2
2.7
- 3. 5x 10-1
- 3. 3x 10-1
-1.2
2.7
-3. 8x 10-1
1
10 2
10- 6
10-5
2.9797
2.0107
0.19723
0.98965
- 5. 7x 10-3
1. 8 x 10-1
-1. 3x 10-1
-3.4x10-1
-1. 7x 10-2
5.3x10-1
-4.0x10-1
-1. 0
1. 01
10 2
10-6
10- 5
2.9794
2.0103
0.19747
0.98922
-1.7x10-2
1.6x10-1
-2. 6x 10-2
-3.8x10- 1
-2.8x10-2
5.1x 10-1
- 2. 9x 10-1
-1.1
10-2
10- 6
10-5
3.2269
2.0093
0.50225
1. 0768
-9.7x10- 1
-3.9
4.8
-1.1
-1.0
-3.9
4.8
-1.1
8
-3xI0- 14
0
9.8x10- 6
-1. 9x 10-8
3. Ox 10- 5
1. 5x 10-3
9
0
10
0
11
2
0.5
2
10-6
10- 5
2.9861
2.0002
0.20518
1. 0019
-1.8
- 9. 4x 10-1
-25
-1.7
-1. 8
- 9. 4x 10-1
-25
-1. 7
12
0.5
2
0.5
10-6
10- 5
2.9861
2.0002
0.20518
1. 0019
-3. Ox 10-1
-2. 3x 10-1
-3.7
-2. 7x 10-1
-3.0
-2. 2x 10-1
-3.7
-2. 9x 10-1
-1. 3x 10-1
- 9. 8x 10-2
13
101
1
lOB
10- 6
10-5
2.9793
1.99996
0.19797
0.99955
4.6x10-1
-42
74
84
-6.1x10-2
-1. Ox 10-3
4.6x10- 1
-42
74
84
3.3
5x 10-2
-14
-94
89
47
14
0
1
lOB
10- 6
10- 5
2.9793
1.99996
0.19797
0.99955
15
0
lOB
10-8
10-6
10- 5
1.0997
1. 00097
19.9984
2.00001
-162
-99
_lx10 4 \
-200 !
-1.0
- 50
(A17). But arc 1 cannot generally be neglected when 1Tm
«1 for (0, rn). Therefore, in this case the full seriesparallel circuit should be used for the plateau region
with the usual YiN' arc 1, circuit added to that of Fig.
10(b), again with re replaced by En r n , which is here
essentially 1Tmrn' Alternatively, YiN in the plateau region can be approximated by the expression for Y 1SN
given in the Appendix, Eq. (A22).
both cases. Thus, discrimination will not be possible
without additional information. Finally when 1Tm is completely unknown, unless M and 1T. are known it will not
even be possible to distinguish between the arc 2 shape
associated with the 0 «Os condition and .that present in
the plateau region, 0» Os, when 1Tm» gn and all O's of
interest are much less than unity .
For rn =rp =re , the arc 2 radius PZN is r;i, independent
1Tm , and 0ZM in the plateau region is approximately
re/2M~. If 1T. is known, re and M may be readily obtained from PaN and 0ZM as in the 0 «0 S case. Consider
now the (0, rn) situation with the same PaN as that for
(re , re). It is then necessary that Enrn = r e , or rn
= (1 +1T;.l)re. To what degree can one distinguish between
the two cases? When 1Tm <10, even though the arc 2
sizes and shapes may be essentially identical, arc 1 will
be present in the (0, rn) case and not in that with (re , r e),
thus allowing immediate discrimination between the
cases. But when 1Tm» gn ,rn and re will be essentially
identical and no significant arc 1 will appear for (0, rn).
The quantity 0ZM will also be very nearly re/2M/6" for
VII. ACCURACY OF SOME APPROXIMATE RESULTS
of
Table II gives some numerical results for a variety of
(rp, rn; 1Tm, 1; 0, M) situations with PaN = 1 and 0ZM = 10-5
when these latter quantities are calculated from the pertinent formulas already given. We see that the actual
exact results for - Im(ZTN) at =OZM are very near the
expected value of unity for rows 1-12. Also shown are
values of the components of ZTN at the low-frequency
side of arc 2,
= 10-6. Very similar values for
4
- Im(ZTN) are found at the other side, 0= 10- •
°
°
The remaining columns in Table II show the percentage
errors ariSing in the various cases for the approximate
expreSSion for ZTN given in the Appendix, ZTNh Eq.
J. Chern. Phys., Vol. 61, No. 10, 15 November 1974
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J. Ross MacDonald: Impedance/frequency response for solids and liquids
(A27), and for the even more approximate expression
made up of the foregoing results:
ZTNZ
(34)
with Y 1NZ given by Eq. (16); Y ZNZ by Eqs. (5)-(7) for the
first twelve cases and Eq. (33) for the last three plateauregion cases; and Y 3N =tJ '" 1 +m. In terms of ZTNJ ,
j =1 or 2, P R and PI are
PR
'" 100 ([Re(ZTN) -
Re(ZTNJ)]/Re(ZTN)}
(35)
and
In row 1, the P's for ZTN1 arise solely from roundoff
since ZTN1 is exact for this situation. Although several
situations will be noted where the errors in ZTN1 are appreciably smaller than those in ZTNZ, many times they
are essentially equal. In such cases, the error observed arises principally from the way approximations
for ZlN and ZZN are combined in forming an approximation for ZTN' not from lesser inadequacies in the individual approximations.
The cases of rows 1-12 are just a few of the many
situations which can lead quite accurately to PaN = 1 and
02AI = 10-5. The exceedingly close similarity of most of
these results will make it exceptionally difficult to discriminate between them. Since in several cases the
numerical results at given are closer together than
the errors in Z'1'Nl> curve fitting using the exact expression for Z TN will be necessary as well as exceptionally
accurate experimental results in order to allow discrimination. Of course, outside the arc 2 region examined
here, discrimination will usually be possible between
such cases as rp = rn = 11m = 1 and rp = 0, rn = 2, 11m = 1 because of the appearance of some arc 1 behavior in the
rp = 0 case which is absent in the rp =rn =re case. Discrimination between such cases as rp =rn = 11m = 1 and rp
=rn = 1, 11m = 10 will be virtually impossible, however.
Since rp = rn is a more unlikely situation than rp"* rn or
rp = 0, r n "* 0, the need for such discrimination may not
often arise, however.
°
are far from small, however. In this case the contribution to ZTN from ZlN2 should have been negligible but was
not. In rn« 1 + 1Tm plateau cases with 1Tm» M, one must
clearly omit all ZlN contributions. Row 15 shows that
when 11m «M-t, arc 1 plays a large role compared to arc
2 and the desired values of PaN and 0ZM are not obtained.
VIII. SOME IMPORTANT CURVE SHAPES
In the original paper in this series,l numerous curves
were presented of the dependence of R 1NO and C1NO on the
parameters of the situation: M, r n , r p , 11m , 11., and
temperature. In the second paper, 2 many computer-calculated curves of CPN and GpN versus frequency and normalized impedance plane plots (such as Fig. 3) were included, again for various values of the input parameters.
The exact curves of the present paper help fill in response possibilities not adequately covered by the previous work. A few important further responses of this
kind are discussed in the present section.
Figure 11 shows some CPN frequency response curves.
Note that for 11;1» 1, three different dispersion regions
(associated with ZlN' Z2N, and Z3N) appear in CPN results when rn <00. From right to left, the first rise is
associated with the final bulk single-time-constant dispersion (Z3N) and leads to the plateau region (here appearing at 10-7 ;::; .s 10- 4). The second small rise, in the
region of 0-10- 8 to 10-9 , carries CPN up to approximately the ordinary double layer value of M, and the final
rise, whose onset clearly depends appreciably on the
magnitude of r n , . carries CPN to CpNO '" 1 + Cmo • For 11m
= 10-8 , only the rn = 00 curve shows an appreciable region
of slope - 0.5, Warburg frequency response. It is important to point out that for 2$rn $200, appreciable regions with a slope of - 1 appear. Such w- 1 frequency response has often been observed for various solids. The
important implications of the result that CPNO is asymptotically proportional to M2 for M» 1, not M, so that
CPNO may be very large indeed, have been discussed
earlier. 1,z
°
The impedance levels which appear for the
Rows 7 and 10 show that as 11m becomes small, even
though the desired values of PaN and 02M may be maintained by the proper choice of r n , ZTN will begin to be
appreciably different away from 0ZM. By symmetry, the
exact results of rows 11 and 12 must be identical as
shown. Note, however, that the ZTN1 results are appreciably more accurate for (rp , rn) = O. 5, 2 than for 2,
0.5. This happens because the expression for ZlN entering ZTN1 has been optimized for (rp , rn) with rp «rn
rather than with rp >rn. Modification of this ZlN approximation to make it show the symmetry of the exact solution would have made the expression given in the Appendix considerably more complicated. Such modification
is unnecessary since the symmetry relation may be used
to change any rp >rn case into a rp <rn case.
The final three rows of the Table show plateau cases.
As expected, the errors of ZTN1 are large here. On the
other hand, the ZTNZ expression appropriate for the plateau region has been used here, and the row 13 ZTN2 errors are small. The corresponding errors for row 14
3989
.....
11m
= 10-8
\
"'\
2
[Mffn-l]
\\...... g-n
.
+)
-:.--Mffp
--:..,----
10
- - (o,r n; 10- 8,4; 0,104 )
·········(0,2; 1, 1 ;0,10 4 )
- - - - (0, 2; 10 8,0.25; 0,10 4 )
FIG. 11. Curves for C pN
for several situations.
=0-
1
Im(YTN ) vsnormalizedfrequency
(1
J. Chern. Phys., Vol. 61, No.1 0, 15 November 1974
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J. Ross MacDonald: I mpedance/frequency response for sol ids and I iqu ids
TABLE III.
0, 104),
Values of impedance components for (0,2; 11m' 11,;
n ~10'!O
7rm, rr~
RDN
cPNO
Re(ZTN)
-Im(ZTN)
CpN
10.8, 4
2X 10 8
1. 337x 10 6
1,13x10 5
1,1lX10 6
8888.6
1, 1
4
2. 089x 10 6
:3,
lOB, 0.25
2
1. 341 X 10 6
99999
3. 34x 10-3.
8.93X10· 7
2.089X106
2232.9
curves are very high and hence relatively easy to observe. Thus for (0,2; 10-8, 4; 0,104 ) at n = 10-14 , ZTN
~(1.17-0.189i)X108. Atn=10- 8, ZTN~(3.31-19.1i)
x 103 • On the other hand, the situation is quite different
for the 7Tm = 108 curve. For (0,2; 108, 0.25; 0,10 4), ZTN
~ (2 -1. 89x10- 9i) at n =10- 14 , (2 - 5. 08XW"5i) at n =10- 8,
and (1.998-4.47XlO-2 i) at 10-5. Table III shows some
results for n = 10-10 . In these frequency ranges Re(Zr)
dominates, and it will usually be difficult to measure
Im(Z T) much below perhaps n = 10- 6 • Thus, although the
CPN curve is given for smaller n in Fig. 11, it will not
be measurable. This result is consistent with that of
Fig. 6(a) which shows the effect of ZlN to be negligible
for rn« 1 +7Tm •
The curves indicate that the lower limit of the plateau
region is between 10b-1 and b- 1 • Here b- 1 - 6 X10-8 for
both the two rn = 2 curves with 7Tm = 10- 8 and 108. Thus,
the lower frequency parts of the curves will appear at
lower and lower frequencies the larger b. For 7Tm - 0
and for 7Tm - 00, one will therefore again find the plateau
region extending all the way down to n = t:, with t: - O.
Under such conditions, only one species of charge is effectively mobile, and Cn cannot exceed CPSN '
The normalized impedance plane results of Fig. 12
are interesting because such shapes are frequently found
experimentallyll-14 and also because very nearly the
same sort of results appear in theoretical treatments of
the supported electrolyte situation. 7,15 Note that the
curves as shown do not extend far enough to the right to
include the low frequency limiting region of arc 1, associated with ZlN' It is interesting that the curve of Fig.
l(b) of Ref. 14 for the superionic solid electrolyte
A~RbI5 is very close in shape to that for M = 103 in Fig.
12. The authors of Ref. 14 were unable, however, to
obtain results in the high-frequency region of arc 3.
Although the shapes of theoretical impedance plane
curves for the supported electrolyte case7,15 are very
similar to those of Fig. 12, their dependence on M and
LD seems considerably different. Here the cusp between arcs 1 and 2 becomes sharper and sharper the
larger M, or the smaller L D • But Sluyters 15 gives supported curves which show sharper and sharper cusps as
the double layer capacitance C a decreases (L D increases). The intercept of the Warburg 45 line extended back to the real axis is also somewhat different
in the two cases. 7 These results suggest that it is quite
important to use theoretical results for the unsupported
situation1,2 rather than supported-electrolyte theory in
cases where the unsupported situation is likely. It is
quite probable that the unsupported assumption will be
applicable for many superionic conductors, fused salts;
0
and intrinsic-defect-conduction single crystals 16 at temperatures where electron-hole conduction is negligible.
The dependence of some of the present curves on l
variation is also of some interest. First note that nand
MCg are independent of l. When the Cn of Fig. 11 is
changed to C p ;: CPNCg , an increase in l (or in M arising
from the l increase) will make no significant change in
Cp in the plateau frequency region or above when M» 1.
Even in the Cp ""'-MCg region, when it appears, l variation
will have little or no effect on Cp • But whenever Cp
>MC g , an increase in l will tend to increase Cp , since
Cpo a: M 2 Cg a: l. These regions of the graph thus depend
strongly on l, and the results are extensive, not intensive there. Since the normalization of ZTN in Fig. 12 involves R., a: l, the scales of both axes change together
when l changes. It has already been mentioned that when
unnormalized impedance plots are considered, the sizes
of arcs 1 and 3 will be directly proportional to l, but the
arc 2 size is independent of l. Since arc 2 is associated
with electrode reactions, it should indeed be intensive.
It is, of course, in the intensive plateau region of Fig.
11 that arc 2 makes its contribution to CPN '
The region near the cusp between arc 3 and arc 2 (or
arc 1 if 2 is missing) is of particular interest. Frequently, frequency response measurements cannot be
readily extended to sufficiently low frequencies that all
of arcs 2 and 1 can be covered. Then, one must depend
on the limited measurement region available in order to
try to reach sensible conclusions about what happens in
the experiment. Separation of the effects of electrode
reactions and diffusion is particularly difficult when
these effects overlap in frequency, as they do somewhat
in Fig. 12 for M 5104 • But the ambiguity can be much
greater.
When the frequency region covered between the end of
arc 3 and the lowest frequency measurable is quite limited, it may be exceedingly difficult to tell whether the
situation being investigated involves (rp , rn) = (0,0),
(re , r e ), or (0, rn). Figure 13 shows some results for
(0,2 x 104 ; 7Tm , 1; 0,10 4). Although we have 0 <rn <00
6
5
(0,80; 10- 2,1; O,M)
2
<3> 10 M.o. = 1
B
/1;
10 M.o.=1
M.o.=1
.0. =1
P2N =1.2625
P2N = 3.525
RON =103.525
FIG. 12. Arcs in the normalized impedance plane for (0, 80;
10-2 , 1; 0, M) showing approximate Warburg-response lines at
the right. The figure does not extend to low enough frequencies
to show all of arc 1.
J. Chem. Phys., Vol. 61, No.1 0, 15 November 1974
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J. Ross MacDonald: I mpedance/frequency response for sol ids and I iqu ids
3991
ferent from unity. Again the low frequency curves are
not entirely vertical. These results apply to the situation where the capacitative reactance is far greater
than R.,; thus, arc 3 is too small to appear on this
scale. Here again curvature appears, even in a completely blocking situation when 1T ~1 »1. It is therefore
dangerous, even when complete blocking can in fact be
assured, to ascribe observed curvature only to electrode
roughness.
8
6
Z
lN
....e1
4
2
4
(O,2xl0 ; 1Tm,l; 0,10 4 )
'VI
4
Re (ZTN)
n=to- 5
6
FIG. 13. Normalized impedance plane results for (0, 2x 10 4 ;
1; 0, 104) and 1I'mSM-t. Arc 1 closure not shown.
1I'm'
here, the 1Tm« 1 curves shown are not good approximations over very much of their extent to the initial parts
of semicircles in the region Re(ZTN) 2:: 1. The effect of
rn here is to change the curve shape somewhat and slow
the eventual approach to Warburg behavior but not to
yield any distinguishable arc 2 semicircle. Here, in
fact, all curves eventually become asymptotic to the
dashed Warburg line through the point - Im(ZTN) =0,
Re(ZTN) =1 before finally approaching the real axis as
0-0.
Armstrong et al. have suggested17 that even in a completely blocking (0,0) situation, where one would ordinarily expect a vertical rise, as in Fig. 4(b), the rise
may be curved over, as in the curves of Fig. 13, because of the presence of electrode surface roughness. 18
Here, where we deal with smooth electrodes, such curvature can appear because blocking is not complete and
some electrode reaction occurs at the electrode.. One
may possibly be able to distinguish between the two
causes of curvature by varying the roughness of the electrode surface. But there will usually be no guarantee
that going from a very smooth surface to a rougher one
will not itself appreciably increase the probability of an
electrode reaction. A rough surface will have asperities
where the electric field will be far higher than that for
a smooth surface. Such a high field will 'itself most
probably enhance electron transfer at the electrode.
Therefore, in practice it may be difficult to distinguish
between a rough (0,0) surface and a rough (0, rn) surface.
The reason the· curves of Fig. 14 are not almost exactly vertical even in a completely blocking situation is
that the exact equivalent circuit of Fig. 2(a) leads to C,
in parallel with the series combination of CI ,R I , and
R.,. In this frequency range, where 0 <0.1, CIN ~ CINO
for complete blocking, virtually independent of frequency. But for 1T~I» 1, RjN is far greater than unity and is
frequency dependent. It is the presence of this "interface" resistance RIN in series with CINo =r-1 ~M~ CdN
that causes the departure from verticality. For the
present situation at 0=10-8 , RjN~2.98x102, 1.09x103 ,
and 3.31 x 103 for 1T~ =O. 25, 1, and 4, respectively. The
real part of ZTN is thus entirely dominated by R IN • Now
RjN increases from a very small value much less than
unity at n ~ 0.1 to R INO as n - O. In the frequency regions
covered by the curves of Fig. 14, 1 «R IN «R INO , but at
constant n RjN increases as R INO increases. Further, it
turns out that for 1T ~ » 1, so that a » 1, R INO is essentially equal to a. Thus, the larger the ratio of mobilities, the larger a, and the larger the departure from
verticality one may expect in a completely blocking situation. In this connection it is worth mentioning that the
solid electrolytes investigated by Armstrong et ai. 17 are
very likely to have a large mobility ratio.
Although the curves of Fig. 14 are shown as turning
Z
l-
N
....E1
2x 10
4
-8
4
(0,0; 10 ,1Tz ;0, 10 )
o MZ n = I
---- --- --
O~~--~~L-~~~~~-L
o
But Fig. 14 shows that the situation is even more uncertain. Here we see what happens for a theoretical
smooth (0,0) surface when the mobility ratio is far dif-
__
----------~:y
----4xl0
3
6xl0
3
SLOPE
__~~__~
4
10
Re (ZTN)
FIG. 14. Normalized impedance plane results for the completely blocking situation (0, 0; 10-8, 11'~; 0, 104).
J, Chern. Phys., Vol. 61, No. 10, 15 November 1974
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J. Ross MacDonald: Impedance/frequency response for solids and liquids
3992
TABLE IV. Components of ZTN at
for (O,Tn ; 10-8 , 1; 0, 104).
a = 10-8
Tn
Re(ZTN)
-Im(ZTN)
0
2 X 102
2 X 10 4
1.085X103
1. 124x103
3.151X103
3.629 X 103
1.273x104
1.276X104
1. 256 X 10 4
1.142 X10 4
00
(a - b) =(5 n - 5p )c =(5:1 €n)(11~ -1)(11~ - 11m) •
When 11m =11. , a=b=l, c=O, andd=2. Furtherwhen11.
l
= 1, a=b = (4€n€pt = (11m +2 +11;;.1)/4. This quantity will be
very large when 7Tm «lor 11m» 1.
Two approximate expansions of (AI) and (A2) are often
useful. When O(a -1)« 1, one finds
(A3)
and
over to the right, such behavior does not continue indefinitely in the completely blocking case as 0 gets smaller.
Clearly when 0 <2. 5 (bM 2tl, so that RiN2ERiNO, the
curves will again approach verticality, with Re(ZTN)
2E 1 +R iNo •
Finally, it is worth pointing out that the curves of Fig.
14 are primarily controlled by the small value of 11m
present, not by whether r. is zero or not. When 11m is
sufficiently small, the reaction rate of the negative
charge carriers is largely irrelevant because their mobility is so small relative to that of the positive carriers. This conclusion is quantified by the results of
Table IV taken at 0 = 10-8 • Here, all values of Re(ZTN)
given are essentially just RiN values. Not until rn
~2 x 104 does Re(ZTN) depart appreciably from its rn = 0
value. These results show that in the frequency range
covered by Fig. 14 it will again be virtually impossible
to distinguish between a (0,0) and a (0, rn) situation unless rn ~ M. Otherwise, good distinction will be possible
only if frequency response measurements can be carried
to low enough frequencies to show up the difference between (0,0) curves, which must eventually turn more
vertical, and (0. rn) curves, which must eventually curve
over to the real axis.
(M)
These results are exact for all 0 in the special c = 0 case
where 11m = 11. and a = b == 1.
Interesting things happen to 8~ in the 11m *1i~ case when
(OC)2 = 1 and when (OC)2» 1. When 11. = 1 as well, the
square root term disappears at the (OC)2 = 1 point and 8~
= O. 5 +iOa = O. 5 + o. 5i(l €p - €n I )"1. The branch point at
(OC)2 = 1 in this 11. = 1 case leads to possible ambiguity in
the assignment of 8! and 8~ to 8~ and 8~ for 0 ~ lei-I.
Such ambiguity is of no consequence in the exact solution
since it is indifferent to the assignment. Our present
inexact solutions may not be, however, and it is thus
important to choose that pairing that leads to simplest
and most accurate results in the region (OC)2 ~ 1.
When (OC)2» 1, one can readily expand the squareroot terms in the exact solution and obtain first-order
expansions for 8!. For this case, I have made the following ~, 8~ choices for the expansions:
2
(0c)2» 1
(A5)
81"", 5n +iO(5/€
n
n)"'}
(A6)
8~ 2! 5p +iO(5p /€n)'"
11m >11.
and
8~ 2! 5p +iO(5p /f.p)"· }
APPENDIX
(OC)2» 1
(A7)
8~ 2E
A. Eigenvalues
Two eigenvalues, 8~ and 8~, enter importantly into the
exact solution of the present problem. 1 Their use has
been avoided in the simplified formulas of the main text
but cannot be Sidestepped in the more complex expressions derived subsequently. Thus, before discussing
more accurate expressions for YlN and Y 2N , the eigenvalues need some examination.
In the general case, the full eigenvalue expressions
are 1
(AS)
5n +iO(5./€n)·· •
11m <7T•.
Note that the sign of c == (5 p/ €n)(11. - 11m) depends directly
on that of (11. - 7Tm ). The above choices ensure that under
all 11 m, 1i~ conditions Re(8~)>> Im(8i) for suffiCiently
small 0, as is also the case for 8iwhen (OC)2« 1. The pres·
ent expansion is of course invalid when 7T m =11. and c '" o.
In the exact theory, 1 several functions of 81 and 82 appear. Those that will be used in the admittance formulas of the Appendix are
(AS)
and
and
(AlO)
8~ =~ == 0.5 {I +md -
[1 +2m(a -
b) - (0c)2]1/2}.
Here
d==a + b
=(5
n/
f.,,) + (5p/€p);
a == (~/f.,,) + (5:/€p);
=
b 5n 5p / €nf.p ;
and
(A2)
where j = 1 and 2. When 7Tm = 11. =7Te, )'1 = (Mp)coth(Mp)
and )'2 == (iM20)l!2COth(iM 20)1!2.
B. Exact and approximate admittance results for
== 7re
First conSider the rp =rn=re situation. The full solutionl yields in this high symmetry case the Simple exact
result
7rm = 7rz
c == (5 n/f.,,) - (5p/f.p = (5p/€n)(11~ - 11m) •
It follows that
(All)
When this expreSSion
and'REN=ge
are substituted into
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Eq. (2) one obtains
(A12)
Y 12N =Y2N =tl'[p(re/2) +iOY1] ,
in agreement with an earlier result4 for re = O.
Notice that Eq. (A12), and hence ZTN' is wholly independent of 11e in this (re,r e ; 11e , 11e; O,M) case. It follows from (A12) that G2NO = (re/2) and C2NO = (r +re). The
parallel quantities GpNO and CPNO thus become GpNO
= [1 + (2/re)]-1 and CPNO = 1 +g;2(r -1), quite different
from the arc 2 quantities. When r» r e , the quantity
02M is closely given by the solution of the approximate
equationO(r+re )=0.5 (re -302 r). Then
02M ~ [1'e - (3r!/4r) ]/2(r +re) "'" re/2r •
It follows that
PzN~[re- (3r=/4r)r 1""'r;1.
The other situation where a simple exact result can be
derived is (rp ,00; 11e , 11e; 0, M). Earlier results 1,2,4 and
Eq. (2) yield
Y 12N = YlN
=p{ P[11eY2 + (rp/2€p)] +iOY1}'
3993
tain conditions that an approximate formula should satisfy, however, which may be helpful in synthesizing a
useful result. First, it should be invariant under the
transformation (11", ,'IT.) - (11;,.1,11;1). Second, it should lead
to the exact G2NO and C2NO results. Unfortunately, this
seeond condition is not particularly helpful here since it
turns out that G2NO and C2NO are the same for the general
(11",,11.) case as they are for 'IT", =11.='lTe• Some help
comes, however, from ensuring that terms in 0 2 are
correct as 0 - 0, especially when re = O.
The synthesis begins by adding plausible terms to
(A12) which nevertheless allow the results to still reduce
essentially to (A12) when 11",=11.=116 • Then by examining
the numerical differences between the approximate Y 2N
expression, here denoted as Y2Nl' and the exact Y2N as
a function of 0, modifications to reduce the differences
can be made. The best result obtained so far is
Y2N ~ Y2N1 =l[(p2/2)(E: nYn + E:prp) + in+ p2Q] ,
(A14)
where
(A13)
It follows that GlNO =[ 11e + (rp/2E: p)] and ClNO = 11e[ (~/3) +2]
+[r + (rp/ E:p)]. In addition, G PNO = G DN = E: n+ E:pr1 + (2/rp)r 1
and CPNo=1+gp2[(anapM2/3)+a~(r-1)], where here E: n
= an, E:p = ap. Notice that Eq. (16) does not reduce to
(A13) in the (0,00; 11e , 11e; 0, M) case because the difference between tl' and unity has been neglected in (16), as
has the term iOYlo valid when M» 102 •
It is natural to ask how well the present Y 1N and Y 2N
results, which are exact separately, work together in
the (O,r n ; 11e,11e; O,M) case. The ZTN calculated using
Eqs. (A12) and (A13) together is exact when rn=oo but
only valid when r n = 0 if Y 1N is then taken infinite. Some
intermediate rn results for M = 105 have been examined
over the 0 range 10-10 (10)1, i.e., at decade intervals in
O. For 'lTe = 1, it is found that for an rn range of
0.1(10)10 8 , IP R I is generally less than 0.1%, reaching
0.2% at rn=O.l and remaining below 10-6% for rn~108.
On the other hand, IPIl max may be appreciably larger.
For r n=10 8 , IPIl m"" is less than 2x10- 4%. It reaches
about 4% at r n=100 and is down to about 2% at rn=1.
Over most of the 0 range, IPI I is less than 0.1%, however, and the regions where it is considerably larger
coincide with those where - Im(ZTN) is near a relative
minimum. Such regions, where - Im(ZTN) <0.1, are of
small importance in the overall picture anyway and
probably are where experimental measurements for
- Im(ZTN) tend to be least accurate. Similar results to
the above are found for (0,104 ; 0.25, 0.25; 0,105 ), although IPIl max reaches about 5% for this case.
These general results are sufficiently favorable to
suggest that a useful plan of attack when 'IT", 'IT. is to find
adequate expressions for YlN in the (rp , 00) situation and
for Y 2N for (r e , re) and then combine them as above for
(rp , r n), or at least for the case of most interest, (0, rn).
This course of action is followed below.
*
C. Approximate admittance results for (re,re ; 7rm , 7rz ; 0, M)
In this more complicated situation, the exact theory
leads to no simple expreSSion for Y 2N • There are cer-
and
(A16)
Note that the full expressions for 61 and 62 must be used
here in calculating t1 and Y2' The ")'21 term in the denominator corrects for terms in Y2N of order OS/2.
This approximation for Y2N , which holds well down to
M -1 when it is a good apprOXimation for large M, could
be further improved, but only at the cost of conSiderable
additional complexity. I have written the expreSSion for
Y2N1 in terms of Yn and Yp in this rn=rp=re case so that
it may be used, to whatever degree it is applicable, in
the (0, Y n ) case and possibly even in some (r p' rn) cases
with rp« rn as well, situations where ZTN will involve
Z1N' ~N' and ZSN when r n, Yp"* 00. The expression for
R~No above is just the exact1 R INo in the rn = rp =re case
when gn is set equal to ge.
Equations (A14)-(A16) lead to Eqs. (5)-(7) in the text
for 0- O. Whenever the text equations are reasonably
good approximations for 0« Os, the present results
generally yield better approximations over the entire 0
range of interest: 0::; 0$ 1. Very considerable improvement generally occurs for rp=Yn=re cases, less
for others. Some comparisons have already been presented in Table II. None of the equations is really adequate in the transition region where 'IT~ ~ 100 and yet is
not large enough to lead to plateau conditions. A simple
equation for the plateau region itself has already been
discussed, and a somewhat more accurate result is
Y2N = Y2SN ~ [(E:nYn + E:p Y pHO. 5 + in) + (in -1. 5si)")'d ,
(A17)
where ")'1 here involves the 61 of Eqs. (A5) and (A7).
Thus, Eq. (A17) applies only when (0e)2 > 1, a condition
consistent with the plateau requirement 0» Os = (a _1)-1.
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J. Ross MacDonald: Impedance/frequency response for solids and liquids
3994
D. Approximate admittance results for (rp
,
~; 1rm , 1rz ;
0, M)
To derive an approximate relation for Y1N in this
case, one may first start with Eq. (A13) and generalize
it to yield the correct G1NO and ClNO results while still
requiring it to reduce to (A13) when 1Tm = 1T. == 1Te' The exact results 1 readily lead to
Let us continue to consider only the r p= 0, rn = 00 situation and investigate the approximate Warburg response
region following from Eq. (A20) when b-1 <M- 1 • Now
this region falls in the range 10(bM 2)"1 < '1 < b-1, which,
for the 1Tm« 1 case of interest, becomes 40 'IT", M-2 < '1
< 41Tm when 1T. = 1 as well. Then, for this region we may
take p2 "" 1 and rewrite (A20) symbolically as
(AlB)
(A23)
and
C1NO =[(2g /
t) - 1J+ E;2[(onOpM2 /3) + o!(r -1)],
(A19)
where [(2g/Ep) -1] may also be written as [(r/Ep)
+ 21Tm+ 1]= [(r/Ep) + 1Tm+ E;l]. The corresponding components of YTN are GPNo= GDN = 1- kp/g p)=g;l{En + (r/2)},
and C PNO = 1 + (oponM2/3g!) + [o!(r -1)/g!].
The above procedure leads to a result which is still
incorrect to order n S/2, though correct to order n. Numerical experimentation using the exact solution allows
one to establish the form of the needed n S/2 term. The
result is the still approximate expression
YtN
""
Y1NI = l{(lrp/2E~ + l1TmY2(1 + mCI)
+ WHo p/ Ep)2(Yl -1) + 1 - p2 1TmC1 ]},
(A20)
where
It follows that eo ""'IT",MfITj and el ""i[(op/E/(r -1)+ 1
-1T",Cl]""i(op/E~2(r-1). On taking (r-1)""M, one finds
that the ratio I eonl/21 /1 elnl becomes approximately
(EnEp/O~)(b/O)l/2. For O=b-t, this reduces to just lT z •
Thus, in the main part of the Warburg region where 0
«b-t, the ratio will be much greater than unity. Further, the third term on the right will also be much less
in magnitude than the first term.
Under the above conditions, one can immediately calculate ZlNl == Yi11 by series expanSion and obtain
ZlNl"" (e oOl /Z)"l[1 - (el/ eo)n1/2 - (ez/ eo)O·· • ]. (A24)
Now (e od/2 ).l = ZWN ==A(1 - i)//O, an exact normalized
Warburg impedance, with A defined in Eq. (20). On neglecting the ez term, one obtains
ZlNl "" ZWN - (el/ e~) == ZWN + RelN ,
(A21)
Although it would be possible to correct this expression
to order ft, the added complexity is unnecessary for
M ~ 100 since by the time incorrect '12 terms become
important, I ZlN I « IZSN I. The above approximation
does not apply very well in the plateau region where '1
» ns' An approximate expression for this region is
(A22)
where Yl again involves the (il of Eqs. (A5) and (A7).
Equation (A20) is exact for (r p , 00 ; 1Te , 1Te ; 0, M). Some
idea of its accuracy for rp= 0, rn "" 00 and 1Tm '" 1T. can be
formed from the following results. For M= 105 , 1T.= 1,
and 1 S 1Tm S 10 4, the maximum value of I PI I found is
about 3X10-\ with IPRlmaxmuch smaller. IPII does
not increase in the region 2 S 1Tm S 104 • On the other
hand, for 1T.=1 and 10-4 S1T m S.1, the error increases as
1Tm decreases. Again, it is prinCipally in PI and reaches
a magnitude of about 1% by 1Tm = 10-2 and 60% by 1Tm = 10-4.
It should be added that such maximum values of I PI I are
generally isolated and occur near '1- 10-2, where
- Im(ZTN) is very small. Over most of the range of '1,
I P RI and I PI I are usually well below O. 1%.
In the 1Tm = 1, 1T. ~ 1 Situation, results are somewhat
different. Again the maximum error shows up in I Prl •
Within the range 0.25 S 1T. S 4, IPrl max is less than lO-s%.
Even in the larger range 10-4 S 'IT. S 104 (important for
extrinsic conditions when 'IT. is re-interpreted) I PI I max
remains less than 3X 10-2%. When 1T", = rr;1, on the other
hand, I Prl max remains less than 10-~ for 1 S1T",S lOS.
As might be expected from the above results, it increases as 1T", decreases. It reaches about 1% by 'IT",
= 10-2 and 75% by 'IT", = lO-s • Again such a value as 75% is
an isolated peak; over most of the n range the formula
yields IPRI and IPII values less thanO.1%for'IT",=10- s .
(A25)
where
RelN == - (op/ Ep lT mM)2(r -l)/b "" - (1T m1T.. M)"l •
(A26)
Thus, to this order of apprOximation, Zl shows exact
Warburg response plus a negative, frequency-independent resistence R e1 •
E. Approximate admittance results for (rp.Tn; 11m , 11z ; 0, M)
Here, we shall primarily consider the rp=O situation
but with some investigation of the utility of the results
when rp<rn' The plan of approach is to combine the approximate results given in Eqs. (A14) and (A20), or
(A17) and (A22) for the plateau region. Since ZlN should
be zero when rp=rn and s ==rp/rn is unity, it turns out
that the Yl/s given in Eqs. (A20) and (A22) should be
multiplied by (1 - S)-2 in order to ensure that this is the
case. Although the (1 - S)"2 term will be shown to be
correct for s = 0 and 1, it is only an approximation other·
wise.
We may now write
ZTN "" ZTNl == ZINl + Z2Nl + ZSN ,
(A27)
where Z2Nl is formed from the Y2Nl of Eq. (A14) and
ZINl = Yi~l(1 _S)2, with YlN1 from Eq. (A20). The result
should be most applicable when either r p "" r n or 0 S r p
«rn'
Let us now investigate how ZTNI0 differs from ZTNO
= RDN . Analysis shows that
ZTNO==1 +[2/(Enrn+ Eprp)]+Hii1 ,
(A2B)
where
HO =={1Tm + (r p/2Ep) + s[2 + (r p/2Enh 'IT;s]) (1 - s)"2.
(A29)
But Eq. (A27) leads to
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J. Ross MacDonald: I mpedance/frequency response for sol ids and liquids
ZTNO= 1 + [2/(Enr n + Eprp)] +Hi
1
,
3995
(A35)
(A30)
and
where
(A31)
Comparison of (A29) and (A31) shows that they yield the
same results for s = 0 and 1 but that the s term in (A29)
has been omitted for simpliCity from (A31). Even when
rp - 0, the s term will only be negligible when s« 1T",.
The accuracy of Eq. (A27) in the general (rp , rn; 1T""
1T1l ; 0, M) case is illustrated by many of the results of
Table ll. There is some slight tendency for errors to
increase as rn decreases in the range rn« M. For rn
~M, errors are generally quite negligible. For M = 105 ,
rn= 10 3 , and Tf.= 1, I PII max increases with increase in
1T~. At Tf m = 10 3 and Tf m = 10-1 it is under 8%. For M
= 105 , rn = 0.1, and 1Tm = 0.1, it is under 12%. As usual,
the major percentage errors occur in PI rather than P R ,
usually because I Im(ZTN) I «Re(ZTN). All major errors
here arise not so much from errors in the individual
(r e , re) Z2N and (rp, 00) ZlN expressions but because of the
simple way they are combined in the present (rp , rn)
situation.
Finally, let us examine response in the Warburg region, restricting attention to the rp=O, O<r"soo situation. In the Warburg region, where 0« 1 and 0« OzM ,
Z3N ~ ZSNO = 1 andZ2N ~ Z2NO = (p2N -1). Then the approximate equivalent circuit of Fig. 15(a) reduces to that of
Fig. 15(b), where pure Warburg elements are enclosed
by dashes. On using these results and Eq. (A25) as
well, one may write
ZTN ~ZWN +RclN +P?N = (iOCwSNtl+RsN ,
(A32)
where
CWSN
=(.40 1/2 t 1 ,
(A33)
(A34)
RSN =RWSN +RRN'
(A36)
Thus in tp.is region ZTN is approximately made up of
an exact Warburg impedance ZWN =RWSN + (iOCwSN t 1 and
a frequency-independent resistance RRN. Such response
is frequently found experimentally. 2
The above results lead to
(A37)
Therefore, plotting - 1m (ZTN) versus [Re(ZTN) -R RN]in the
impedance plane should yield a straight line at a 45 ° slope.
We shall use this condition below as a test of Warburg response. Further, note that if RRN or RR is initially unknown, it may be determined from the data by selecting
the value subtracted from Re(ZTN) which yields the best
and straightest approximation to a straight line at 45 ° .
Then the present somewhat more accurate approach may
be used in the same way as Eq. (22) of the body of this
paper to estimate the value of 1Tm.
It is interesting that (RRN -1) or even RRN may be
zero for certain values of 1Tm, 1T., r n , and M. Note that
(RRN -1) corresponds to (RR -R,.), the usual subtraction
of the bulk resistance from the total frequency-independent reSistance. Further (RRN -1) is very nearly the
same under wide conditions as the normalized resistance
RCSN which appeared in the earlier work. 2 When r,,» 2,
it turns out that (RRN -1) and t:!RCSN are virtually identical. In the 1Tm« 1 case of principal interest, Ep~ 1.
The difference between the two quantities occurs because
they appear in different equivalent circuits. When 1Tm
«1, (RRN -1) is zero when r,,= 21T.M.
A more exact Warburg region condition may be derived
by requiring that
(A38)
be (45° ± 2°). It actually turns out that (45° -- 0°) and
(45 ° + 2°) are the limits of interest. Then on the basis
of numerous numerical results one finds that (3 reaches
",45 0 by 0~25(bM2tl and remains less than-47° until
0-11. Here 11 is the smallest of 1.= (10Mt 1, I b =(10bt1,
and l c =rn!(800bM). It is always necessary for 0$M""1
for Warburg response to occur. Thus, Warburg response, as defined by {3"" 45°, occurs when
.--------,
I
I
: Cws Rws :
0--7--1
:
I
I
IL _________ .JI
Zw
(a)
(A39)
When1T",«l, 1T.=1, r,,<80M, andr,,<80b, the conditions
become
r--------,
I
I
:
(A40)
I
C
ws
I
Rws:
RR
O--,-II~
I
I
I
I
IL _________ ...II
Zw
(b)
FIG. 15. (a) Approximate series-parallel equivalent circuit
in the Warburg response region. (b) Simplified approximate
equivalent circuit in the Warburg region. Ideal Warburg elements enclosed in dashed box.
Since OzM =E"r,.l2M ~ r,,1T ".12M for Tf",« 1, condition (A40)
requires that 0« 0zM for Warburg behavior to appear,
as initially stated. Finally, note that no such response
shows up when r"M$ 2x 104 • Under such conditions Z2N
rather than ZlN dominates ZTN.
Finally for completeness, since CPHO = 1 + CU/iO, it is
worthwhile to give the exact expreSSion for CiNO in the
present general case. It iS 1 •2
CiNO
= (g pgn)-2 [(O,,1i pM 2/3)(g J> - g,,)2 + (r - l)g!], (A41)
J. Chern. Phys., Vol. 61, No. 10, 15 November 1974
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3996
J. Ross MacDonald: Impedance/frequency response for solids and liquids
where g.=g,o"+g,,op, Some special results for R INO are
given elsewhere. 1
*Present address: Department of Physics and Astronomy,
University of North Carolina, Chapel Hill, NC 27514.
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J. Chern. Phys., Vol. 61, No. 10,15 November 1974
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