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ElecrrochimicaActa, Vol. 35, No. IO, pp. 1483-1492, 1990 Printed in Great Britain.

0013-4686/90 $3.00 + 0.00
0 1990. Pergamon Press plc.

IMPEDANCE SPECTROSCOPY: OLD PROBLEMS AND NEW DEVELOPMENTS
JAMES Ross MACDONALD Department of Physics and Astronomy, University of North Carolina, Chapel Hill,
NC 27599-3255, U.S.A.
(Received for publication 17 April 1990)
Abstract-The generality, scope, and limitations of Impedance Spectroscopy (IS) are discussed, with emphasis on unsupported conditions in ionic systems. For such conditions, the maximum reaction rate which can be determined from IS data is limited. The finite-length-Warburg diffusion frequency responses of unsupported and supported situations are simplified and compared, and similarities and differences emphasized. Two types of ambiguity possibly present in fitting IS data to equivalent circuits are discussed, one intrinsic and the other associated with distributed circuit elements. Powerful new features have been added to the author’s complex nonlinear least squares (CNLS) fitting program, and the results of a Monte Carlo simulation study of bias and statistical uncertainty in CNLS fitting of equivalent circuit data are discussed. The program now incorporates new variable weighting choices which can greatly minimize such bias. It also allows two unknown weighting parameters of the error variance model to be automatically estimated during the least squares fitting, thus best matching the weighting to the data and yielding most appropriate estimates of the parameters of the fitting model.
Key words: impedance spectroscopy, diffusion, reaction rate, complex nonlinear least squares fitting.

INTRODUCTION
Although a number of reviews exist of the burgeoning field of Impedance Spectroscopy (IS), many of them are primarily concerned with supported situations, those where a high concentration of indifferent electrolyte is present, rather than with unsupported ones[l-131. Although applicability only to supported situations is not always explicitly stated in these works, this restriction may often be identified by their concentration on liquid electrolytes and their assumption of electroneutrality, rather than their use of Poisson’s equation. But solid materials, and even some liquid electrolyte situations of interest, are not supported and their analysis requires satisfaction of the Poisson equation throughout the material.
It is convenient to partition IS into two subcategories, Electrochemical IS (EIS) and everything else. EIS deals with materials for which ionic conduction predominates, includes both supported and unsupported situations, and may involve either ionic motion and/or ion-vacancy motion. Besides liquid electrolytes, other ion-containing systems, such as superionic materials, non-stoichiometric ionically bonded single crystals, ionically conducting glasses and polymers, and fused salts, may also be included in this category. But it is worth emphasizing that IS, including its measurement and analysis methods, applies to other types of materials as well. In particular, it applies to materials exhibiting predominantly electronic conduction, such as single-crystal and amorphous semiconductors and polymers, and to solid and liquid dielectrics, whose electrical characteristics are associated with dipolar rotation. Now obviously most of these materials are unsupported, and in the whole area to which IS is applicable, there are

many more unsupported than supported situations. But, as mentioned above, the distinction between these possibilities is not always made clear, and one often finds equations and equivalent circuits which were derived and used for supported conditions also used without comment for the analysis of unsupported (usually solid) materials. One aim of the present work is to provide a brief history of theoretical analyses of unsupported situations and to compare some supported and unsupported equations and predictions.
In addition, the problem of two types of ambiguity in IS analysis will be discussed, and much attention given to complex nonlinear least squares (CNLS) analysis of small-signal UCdata. Important improvements in the author’s CNLS fitting program will be described, including a new and powerful method of automatic weighting choice, and representative bias in CNLS parameter estimates, derived from an extensive Monte Carlo CNLS simulation study, will be illustrated and discussed.
First, however, it is worthwhile to dispose of several minor points of usage. In recent years, it has become relatively common for writers in the IS field to refer to a “Nyquist diagram”, taken to mean a plot of the values of the real and imaginary parts of a complex quantity, such as impedance, in the complex plane. Such usage should be discouraged. A quantity such as impedance is basically derived from measurements of current and potential at a single (input) port. But Nyquist’s work[l4] dealt with two-port measurements of feedback in amplifiers and involved input and output voltage determinations. Thus, a complexplane Nyquist plot is intrinsically quite different from an impedance or admittance plot in this plane. Instead of “Nyquist diagram of impedance response,”

1484

J. R. MACDONALD

a better description would be, “complex-plane impedance plot.”
Further, there is little reason to refer to “UC impedance” rather than just “impedance.” Although one might define a higher harmonic impedance or even an indicial transient response impedance, these are uncommon in the IS area, and “impedance” may be taken, in its standard definition, to mean the quotient of vector voltage and vector current calculated from small-signal sinusoidal UCmeasurements. Any other type of impedance would require adjectival qualification. Also, impedance defined as above (see also Ref.[lZ], pp. 5, 134 and 169) needs no qualifier as “complex.” Impedance is complex by definition, and when it is only real it is called resistance. As usual, let “immittance” mean any of the four quantities of major use in IS: impedance, Z; admittance, Y; complex modulus, M; and complex dielectric constant, c or E*. It will be denoted by Z = Z’ + il”. Proper usage should distinguish between the meaning of a superscript asterisk as indicating a complex conjugate quantity (as hereafter) or just a complex quantity. Because of this ambiguity, its use for the latter purpose should be avoided when possible. It is desirable, however, when no asterisk is employed, to use the “complex” modifier for A4 and c to avoid ambiguity. Clearly, IS can, most properly, stand for “Immittance Spectroscopy”, rather than the less general “Impedance Spectroscopy.” Although it is customary in the IS field to plot -Z” on the imaginary axis USZ’ on the real axis and term the result an impedance plane (or impedance complex plane) plot, rigorous usage, which seems excessive, requires the designation “complex conjugate impedance plot” or Z* plot. Finally, note that presentation of IS data in Y, M, and c form, as well as Z, either as complex plane plots or, even better, as 3-D perspective plots, can often yield improved resolution (see Fig. 1) and/or highlight errors in the data not otherwise apparent ([121,pp. 174-179; [13], pp. 28-3 1).
ANALYSIS OF UNSUPPORTED AND SUPPORTED SITUATIONS
General background
Theoretical analysis of the small-signal UCresponse of unsupported materials essentially began with the work of Jaffe[l5] and Chang and Jaffe[ 161. Because of the complexity of the equations governing such response for a material with charges of both signs possibly mobile, able to recombine, and possibly partly or completely blocked at an electrode, the following short precis of work in the area shows that it progressed over a period of some 25 years through approaches which, until the last, involved various approximations, simplifications, and special cases for ease of calculation.
Chang and Jaffe’s failure to ensure full satisfaction of Poisson’s equation was corrected in Ref.[ 171,which dealt with completely blocking electrodes, with positive and negative charges of equal valence numbers, zi and zr, with arbitrary diffusion coefficients, D, and D,, and included dissociation-recombination possibilities. Soon thereafter, Friauf [ 181 presented a

similar treatment which involved partially blocking electrodes using Chang-Jaffe (CJ) boundary conditions, often appropriate for solid materials and even useful for many liquid material situations. This work and most of that discussed below thus best applies in the EIS area to materials with parent-ion or completely blocking electrodes, not directly to those with redox reactions. Several physical situations for which the assumption of parent-ion electrodes is pertinent are discussed by Buck[l9]. Beaumont and Jacobs[20] later investigated the response of a partly blocking system with charge of only a single sign mobile, results thus only strictly applicable to solids. Next came the first accurate treatment of the UCresponse of a fully dissociated, completely blocking situation with arbitrary z;s and DIs[21]. Reference [22] was the first to show finite length Warburg (FLW) diffusion response for an unsupported situation with equal Dis and equal zi)s and with charge of a single sign free to discharge at an electrode. General FLW response, at the impedance level, is of the form:

Z,(w) = Zw (0) [tanh[i/i]“.S/[i/i]o.~],

(1)

where n is given by (I/&)2. Here, 1 is the separation

between identical plane, parallel electrodes, and 1,

is the diffusion length, proportional to W-O.’ (see

below).

Further work on FLW and other response possi-

bilities appeared in Refs[23,24]. Reference[3] pointed

out that the main physical processes possibly impor-

tant in general IS response are, for either supported

or unsupported conditions, bulk, electrode reaction

(including discharge of ions at parent-ion electrodes),

specific adsorption, generation-recombination, and

diffusion effects. When these processes are loosely

coupled and thus appear in different frequency

regions, it was noted that they each lead to a separate

arc when overall impedance is plotted in the complex

plane. The first four arcs, but not that associated with

diffusion, were shown as semicircles with their centers

on the real axis, although their centers may be

displaced below this axis if there is a distribution of

relaxation times present for the process involved

([12], pp. 16-17, 34-36, 87-94; [24]). These con-

clusions turned out to be false in one respect. Gener-

ation-recombination effects do not, in fact, lead to

semicircular response in the Z plane. Recent work for

general unsupported conditions[25] shows, however,

that they do lead to all sorts of multiple arc shapes

when plotted in the complex dielectric constant (or

complex capacitance) plane for a completely blocking

situation.

Ambiguous circuit and element response
If one were able to analyze IS data with an appropriate mathematical model derived directly from discrete microscopic analysis, there would be little or no ambiguity present. But this is rarely if ever possible for unsupported-situation data, and fitting to an equivalent circuit, preferably by CNLS, is the remaining option. For such fitting there are two possible types of ambiguity. The first arises because there are situations where different geometric connections of the (ideal) circuit elements of a fitting equivalent circuit can yield, with appropriate element values, the same impedance over all frequencies. The

Impedance spectroscopy: problems and developments

1485

a

C,

0.6 ,
%;B

0.0 0.2 0.4 0.6 0.8 1.0
I’
Fig. 1. Four possible ambiguous circuits (D, A, C, J), a circuit containing a distributed element (E), and comparison of some of their complex plane response curves with Debye response for the parameter values listed in Table 1. The top Z and M curves apply for D, A, C, and J, and the bottom one for the E circuit
normalized to yield IA,, = 1.

matter is discussed in the literature ([12], pp. 95-99; [ 13,26,27]). All equivalent two-time-constant RC circuits of this type which allow some dc conduction are shown in Fig. 1 (designated D, A, C, and J), and they and the E circuit will be discussed later. Ambiguous circuiti involving inductance are also discussed in[26].
A considerable CNLS study of the theoretical response of an unsupported system was carried out in[26], and it was found that the circuit of Fig. 2, with Z, and R,, omitted and Z, taken as a FLW, was most appropriate. For this circuit, C, and R, account for bulk effects, C, and R, for electrode reaction effects, and C, and R, for adsorption-reaction processes. Now it is found both experimentally and theoretically that the RA-CA adsorption-reaction arc may involve either capacitative or (apparent) inductive effects: that is, it may appear above the real axis in a Z*-plane plot or below it. Although an inductive element has been used to model such below-axis response, it should not
Fig. 2. General, approximate equivalent circuit for unsupported conditions. The element Z,, when present, may account for diffusion of uncharged species in the electrodes, and Z, will be zero when charge of only one sign is mobile.

be interpreted as the usual real inductance, one which involves circulating currents and energy storage in magnetic fields, but instead as a pseudo-inductance: something which yields the needed phase shift.
Further, an inductive-type phase shift associated with a below-axis arc can also be produced by a negative differential resistance in parallel with a negative differential capacitance[26]. Thus, instead of using a pseudo-inductance of large value, such negative RC elements can yield exactly the same frequency response. Although it is entirely a matter of taste which approach to use, and either is as physically reasonable as the other, I find it preferable to make the latter choice in the interest of maintaining continuity. The R, and C, are often positive (arc above the axis), but as adsorption rates change the arc may move below the axis. By allowing negative as well as positive values for these differential RC elements, such response can be represented by R, and C, under all conditions, rather than requiring a change from R, > 0 and C, > 0 to the use of a positive resistance and a pseudo-inductance for below-axis response. In the theory of specific adsorption presented in[26], the values of R, and C, indeed change from positive to negative as adsorption conditions change. This paper also showed that expressions for R, and C, are independent of whether CJ or the more realistic Butler-Volmer (BV) boundary conditions, which take overpotential into account, were used. In addition, a useful transformation method was described which allows response theories using the simpler CJ conditions to be converted to ones appropriate for BV conditions. Applications of the method for treating unsupported diffuse and compact double layer effects appear in[27] and[28].
In a single set of IS frequency response measurements, there is no way to avoid intrinsic circuit ambiguity when it is present. Then why does it matter? It matters because one particular geometric

1486

J. R. MACDONALD

arrangement of the circuit elements is more likely to including many physically based ones which are

model better the actual connectivity of the physical associated with such response in solids. Thus, before

processes present than are the other possibilities. ascribing CPE behavior to fractal structures, one

Expressions for the circuit elements in terms of should first establish that this behavior is indeed

microscopic quantities will be simpler for this most associated with intensive electrode-interface regions.

appropriate arrangement and will generally show But even when this has been verified, the matter

simplest and most physically reasonable dependencies remains unproven. Bates et a/.[401 have carefully

on temperature, potential, and electrode spacing, 1. It compared measured CPE response for electrode-

is thus clear that if measurements are carried out for interface regions with well characterized electrode

a range of temperatures, potentials, and/or I values roughness profiles and found no correlation between

and are fitted using CNLS to the various equivalent the CPE fractional exponent n and the fractal dimen-

circuits, then this type of ambiguity may be resolved. sion of the rough interface. An approach possibly

It is frequently impossible, unfortunately, to obtain preferable to the fractal one which may still yield

a reasonable fit of IS data to an equivalent circuit CPE response may involve a detailed analysis of the which involves only a small number of R’s and C’s. effect of pore shapes and size-shape distributions,

The vast majority of such data involve at least one eg [38]. It seems likely that analysis of deviations from

appreciable frequency region where impedance or exact CPE response will help allow discrimination

admittance shows fractional frequency response and between the many various theories which lead, at

is thus proportional to w *‘, where 0 I n I 1. In the least approximately, to such response over a limited

absence of an exact response solution at the micro- frequency range.

scopic level, behavior of this kind is best handled by using an equivalent circuit containing one or more distributed circuit elements (DCE’s). Such elements

Some differences in unsupported and supported response

include, for example, FLW response, more general

The exact continuum treatment of small-signal IS

diffusion response, the constant phase element (CPE), response of unsupported situations without dc bias

Havriliak-Negami response, Williams-Watts (WW) culminated in the work of Ref.[41]. Mobile positive

response, and distribution of activation energies and negative charges were assumed present having

(DAE) response[ 12, 13,29, 301.The necessity of using aribitrary Q’s and z,‘s and general CJ boundary

equivalent circuits containing such elements leads to conditions were used. The charges were taken to arise

the second kind of ambiguity present in the fitting of from intrinsic dissociation and/or from extrinsic

IS data. It turns out, at least to first order, that any donor or acceptor centers. Arbitrary dissociation-

DCE which leads to symmetrical response in the recombination parameters appear for all three types

impedance plane, such as ColeCole (denoted ZC for of charge generation. The results are appropriate for

conductive systems), can be well fitted by any other either solid or liquid materials between plane, parallel

symmetrical DCE. Further, nearly any DCE which electrodes and apply for either ionic or electronic

leads to unsymmetrical response can be at least conduction. All five physical processes discussed

reasonably well fitted by any other such above in reference to[3] contributed to the overall

DCE[ 12, 13,29,3 l-351. Space limitations forbid illus- response. Because of the complexity of the situation

trations of these ambiguities herein. They are not and its solution, not all the implications of the latter

intrinsic, however. For a response region which is are even yet fully clear. But the exact response for

well separated from those of other processes and with completely blocking conditions has recently been

sufficiently accurate data and/or data which extend examined in detail[25]. A few further results are

well away from peak response in both frequency discussed below.

directions, CNLS fitting does allow adequate

The theoretical work of[41] showed that the circuit

discrimination to be made, provided appropriate of Fig. 2 with Zw , R,, , and Z, omitted was virtually

weighting is used. See especially the results described exactly applicable for the common fully dissociated

in Ref.[35]. Of the various DCEs available, the ones intrinsic situation with charge of only one sign

which can best fit the others and a wide range of mobile and free to discharge at the electrodes. This

experimental data involving thermal activation are hierarchical-response circuit has been widely used for

the DAEs[29,31-351. They further have the virtue both supported and unsupported situation data

of providing specific and physically meaningful fitting. Under the above conditions, no FLW

predictions for the temperature dependence of the response associated with the mobile charge appears,

fractional exponent n or its equivalent.

but such response may still arise from diffusion of

Recently, much effort has been devoted to develop- uncharged (discharged) entities in the material or the

ing fractal theories of rough electrode response, electrodes[42,43]. But later CNLS fitting of the exact

eg [36-391, to explain CPE-like behavior, Z a (iw)-", response for arbitrary Di values, but with charge of

for either supported or unsupported conditions. one sign completely blocked showed[44] that in the

Although there may indeed be instances where absence of specific adsorption (and presumably in its electrode-interface response is primarily associated presence as well) if R, is the usual reaction resistance,

with fractal structures, they may be rare. Consider the following. First, it seems somewhat unlikely that typical electrode roughness and pores could involve the necessary self-similarity over more than three to four levels, thus not leading to a good approximation

then an additional non-zero, non-reaction resistance,
R Rffi, is necessary, as well as a possible FLW response element for Z,, for unsupported situations. Thus RR, is a necessary element of the circuit even in the limit of infinite discharge rate, where RR = 0. Its

to full self-similar behavior. Second, there are numer- presence in the present ambipolar diffusion situation

ous other theories which lead to CPE response[29], arises from the drag of charges of one sign on

Impedance spectroscopy: problems and developments

1487

those of the other sign. Since RRm was found to be electrolyte ([ 11; 121, p. 106). Then, unsupported and

proportional to L,/l, the larger the mobile charge supported response may be quite different.

concentration and the larger the electrode separation

Thus far we have dealt with response for flatband

the smaller it will be. Here, L, is the Debye length. conditions: ones where, in the absence of any applied Nevertheless, for fixed conditions it sets an upper pd, the concentrations of mobile charged species are

limit for accurate estimation of the reaction rate uniform throughout the region between electrodes. constant from IS measurements on unsupported But even in the absence of an applied pd, a Frenkel

systems, a limit which decreases proportionally as the space charge layer can be produced near an electrode

mobility of the reacting charge decreases[44].

because of the difference of work function between

Another important difference between supported the electrode and conducting material. Further, an and unsupported IS response appears when diffusion external static pd, can also lead to nonuniform

effects are important. The exact solution in the concentration. For such situations the small-signal UC

unsupported case[24,26,41] shows that a diffusion continuum equations cannot be solved exactly. Yet

arc may appear when charges of both sign are mobile these situations are often of great experimental

and at least one or the other of them is partly or importance. Response has therefore been calculated

completely blocked at the electrodes. It is not present for a few situations by numerical methods. In

for completely ohmic electrodes or for complete addition to work on supported situations[48], Buck

blocking of charges of both signs[25]. The expression and Brumleve[49] and Franceschetti an’d the author

for the A of equation (1) is, in the notation of have applied such methods to the unsupported these papers, A = M2 bwR, C, , which becomes on case[50, 511. For a binary electrolyte, both blocking

expansion[41]:

and partly blocking electrode conditions were investi-

A = (01 ‘e/4kT)

1 he + zh
ZIPIP, + Z2P2%

x (h/P2)+2+(P2//4) [ (z,/z2)+2+(~2/z,)

gated using both CJ and BV boundary conditions

with charge of both signs or of only one sign mobile.

1 (2)Without specific adsorption effects included, it was found that the circuit of Fig. 2, with only C, , R,,

CR, and R, non-zero, was able to fit the results quite

’

well for an appreciable range of positive and negative

where pe and n, are equilibrium values of the mobile applied dc bias. The elements of the circuit depended

charge and p1 and p2 are their mobilities, respectively. on the bias in ways which were consistent with the

The quotient e/k may be replaced by F/R, the presence of accumulation and depletion layers near

Faraday and gas constants. Now the use of the bulk the electrode(s) [511.No such circuit fitting is available

electroneutrality condition, z,pe = z2n,, and the for comparison in the supported situation. Nernst-Einstein relation, Dj = (kT/e) (pi/z,), leads to:

A =(u*~/4)[D,;~~;,;,l

COMPLEX NONLINEAR LEAST SQUARES FITTING

(~,~,/~2~2)+~+(~2~2/~1~,) X

[

(z, /z2) + 2 + (z2 h )

On factoring this equation, one obtains

simplified symmetrical results:

A = (01z’4)

az,

+

[ (0, D,)(z,

&z2 +

1z2)

1. (3) the new

appropriate for use in equation (1) for sufficiently low
concentrations that one need not introduce activity coefficients.
For the supported case, FLW results which are consistent with equation (1) seem first to have been derived by Labes and Lullies[4547] in the biological membrane area. Because of the decoupling associated with the indifferent electrolyte which supplies the
support, one obtains the same result for a charged (parent-ion electrode) or uncharged diffusing entity ([12], p. 105; [19,41]). If its diffusion coefficient is D and the charged species is univalent, then A so12/4D, in agreement with equation (4) when D, = D, E D and z, = z2 = 1. When there are two charged species present in a supported situation, both free to discharge, each species leads to the presence of a separate FLW contribution of the form of equation (l), as in the redox situation, where there are both reduced and oxidized mobile species in the

Fitting of IS data to a model in order to obtain model parameter estimates is crucial to the identification, interpretation, and quantification of the physical processes leading to the data. The most powerful available fitting method, which also yields goodness-of-fit and parameter standard deviation (SD) estimates, is CNLS[52-561. Since the various CNLS approaches have been described and compared in[56], here only recent improvements to the author’s CNLS program will be discussed. The latest version of this program, LEVM, may be obtained from the author’s department for a nominal fee, and old versions will be updated free upon request.
Figure 3 shows some of the main fitting circuits available in LEVM. In these circuits, “DE” indicates a DCE, and “DAE” a DCE involving a general exponential (EDAE) or gaussian distribution (GDAE) of activation energies, or the exact smallsignal blocked-electrode model[25]. All the circuits allow the input of impedance of the measuring instrument to be included when appropriate. In addition, when desired, the actual fitting can be carried out for the combination of a given circuit and a known reference impedance, as in a frequency response analyser. A very important feature of these circuits is that only those elements which are taken non-zero in the CNLS fitting appear in a given circuit. These may be defined as fixed in value, free, or free and positive only. Because of this feature, the five circuits shown

1488

J. R. MACLIONALD

(bl (, cp

(d)
-II CP RP ,,Cn

R3

RAI

R4

R5

(cl

(e)

Fig. 3. Five general equivalent circuits for use in forming simpler circuits for CNLS fitting with the LEVM program. Here DE indicates one of many distributed circuit elements, and DAE denotes a full distribution
of activation energies distributed element.

actually encompass tens of thousands of circuit possibilities. In addition, there are ten different DCE choices available for any DE, each involving up to six parameters[56]. The input data may be in any of the four basic immittance forms, either rectangular or polar, and may be fitted in any other form desired. The program may also be used to fit WW, EDAE, and GDAE transient response.
A new feature of LEVM is optimization for complex data fits. When invoked, it automatically adjusts the weighting of the real and imaginary parts of the data in order to make them contribute optimally to the final fit. Another important addition is the choice of robust regression fitting rather than least squares fitting. It is particularly appropriate when errors are large. Even more important are the new fixed and automatic weighting options. First consider the form of the errors likely to be present in IS data. Let Z,(o) denote experimental IS data, Z,(w) the corresponding error-free data (generally unknown), and Zr(w, 6) denote a fitting model involving the parameter set 6. Further, take Zrc(w,0) as the correct fitting model and let 0, represent the set of exact parameter values. Then Zfc(w,6,) = Z,(w). Now the data, models, and errors, cj E 6; + icJ’ (not the complex dielectric constant), may be related, for j = 1,2,. . . , N, by:

Ze(Wj)z Z:(Wj) + iI:

=

Ifc(Wj,

00)

+

43

(5)

which is to be fitted, using CNLS, by the model Z,(o,, O), which may or may not be the usually unknown Z,(oj, 0). We now make the plausible assumption that the errors may involve independent resolution and power-law components ([57], pp. 5 and 57) of the form
6; = a,Prj(O, 1) + U,Pzj(O’ 1)
xsgn{Z&(oj,e,)}[IZ;,(wj, b)lc”l. (6)
and
c; = u,P,j(o, 1) + a,P,j(o, 1)
xsgn{Z~(wj,e,)}[lZ;,(oj, 60)lcoi. (7)
The P’s are independent probability distributions, assumed normal here, of zero mean and unity SD, and Pj is a random member of P. Thus ~1,and 6, are the SD’s of the additive resolution errors and of the power-law contributions, considered separately. For 6, = 0 and a, # 0 one thus has additive errors, while for u, # 0, x, = 0, and c,, = 1 the errors are of proportional form, For some situations, the choice P2 = P, is appropriate.
In CNLS one minimizes the sum of the weighted real-uart residuals and of the weighted imaninarypart- ones using weights, WI and WY, which are inverses of the approximate error vartances, 01 and

Impedance spectroscopy: problems and developments

1489

vy. Good parameter estimates usually require good

variance estimates. The proper variance model for

equations (5-7) is v; = tlf + 0: )Ii, (co,,0,) 1‘Co and

v; = txf + a:]l;(wj, e,)] 2CQE. ven when I, is known,

the true B0parameter values are not known exactly

since it is the main object of CNLS fitting to obtain

good estimates of them. Therefore, the best one can

usually do is to replace the Z,(w,, 0,) in vi and v; by

Ir(o,, 0). Further, the true values of tl,, or, and &,are

also virtually always unknown, so one must use

estimates of them in the variance model. Since a good

estimate of or is unavailable until fitting convergence,

when or # 0 we use a scaled variance model, that

above with &, replaced by its estimate t, ar replaced

by unity, and u, by the variance parameter U, an

estimate of (cx,/a,). Then as fitting progresses toward

convergence, proper weighting will be approached if

Zrequals or well approximates If,; if I/ and 5 are good

choices, and if the final set of 0 values well approxi-

mates the 0, set. Then, for example, Sr, the SD of the

overall fit, will be a good estimate of Q, for the c(,= 0

situation.

If the errors are believed to be additive, one would

set u, = 0 and use any constant value of U, leading to

unity weighting (UWT). On the other hand, when Q,

and 5 are non-zero, consider first two possible choices

for the If, components in the variance model: com-

ponents of If or of I,. We shall identify the first

choice, where the weighting involves function values,

and thus varies with each iteration as the B’s vary, by

adding the letter “F” to the weight designation.

Further, for either choice, when 1II, rather than the

components of 1, is used in both vi and v;‘, yielding

modulus weighting, an “M” will be added to the

designation. Thus in the common and important

situation where t(, is negligible and 5 is set to 1, the

constant-coefficient-of-variation

model, one may

define and use PWT, MWT, FPWT, or FMWT.

When 5 is non-zero and free and U is zero or free, we

may, analogously, define the general weights, GWT,

GMWT, FGWT, and FGMWT. Thus far, only

UWT, PWT, or MWT have been used in most CNLS

fitting, but Monte Carlo CNLS fitting with the other

weightings suggests that they may often be more

appropriate. All the above weighting possibilities

are incorporated in the new version of the CNLS

program.

Modulus-type weighting is primarily appropriate when 1I: I and 11;I remain comparable in size for the

full data set, often not the case. Although the UWT,

PWT, and MWT weights involve I, and so do not

vary during CNLS iteration, fitting with varying

weights is essentially no more difficult than with

constant ones. More important, PWT and MWT

generally lead to much more bias in parameter

estimates than do FPWT and FMWT, even when

If = If, (see below). Only when one is quite uncertain

about how well the fitting model 1, approximates 1,

is PWT more reasonable than FPWT. Thus it is often

appropriate to use PWT in preliminary fitting; then

if a seemingly good fit is obtained, FPWT should be

used for final fitting.

Monte Carlo (MC) simulation CNLS fitting of the

circuits of Fig. 1 has been carried out to quantify the

above bias differences. Exact values of the parameters

(in ohms and farads) of these circuits are listed in

Table 1. Those for the ambiguous circuits D, A, C,
and J have been selected to yield exactly the same
response for all the N = 33 frequencies used, extend-
ing over the range 10m4I w I 104, for each of them. The impedance of the ZC DCE in Fig. 1 is R,/[l + (iwr)*]. Each individual k out of the total K simulation Z-level fits involves data with normal
random errors satisfying equations (6) and (7), and, initially, with the fixed values c(,= 0, U = 0, and &,= 5 = 1. Let p denote the total number of free parameters, m = 1,2, . . . ,p, and define the relative error of the mth parameter in the kth fit, e,, the estimate relative bias of the mth parameter, b,, and the MC estimated relative SD of the m’* parameter, sml as

emk = iemk - brnk ]ieomk )

(8)

b,,,= K-’ f e,,,k,
k=l

s..=U/K)k$,ledl,

(10)

wheref- 1.483 for a normal distribution and 2/J3
for a uniform one. The first value has been used
for the results below. It is, however, appropriate only
for negligible bias and depends somewhat on bias
when this is not negligible. Simulation also yields a
linearized estimate of parameter relative SD’s sr,,,,
which may not depend so directly on the parameter error distribution[56]. Now although b,,,/u, depends strongly on 6, for the model parameters, the ratios ram=- sam/a, and rr,,,= ~,/a, are independent of it for the D, A, C, and J circuits. To obtain well-defined estimates of s,, or s, one needs a K of 0( 104), while good b, estimates often require a K of 0(106) when lb,,,1 is <<I.
The r values listed in Table 1 for each em value provide a measure of how well a given parameter is
likely to be determined in a single CNLS fit: 0,,,[1 + Q,,,]. They are thus concerned with potential precision of estimates, while a bias is a measure of the
limits of ultimate accuracy of estimation (unless it is
known and can be taken account of). Note that

Table 1. Exact (0 to 2 decimal places) and approximate parameter values and corresponding values of the r ratios
for FPWT fitting. T and $ apply only for E

en,

Circuits

rm

D

J

C

A

E

R,

0.3

1

I

0.530 106

rd

0.47

0.13

0.13

0.22

0.20

rr

0.43

0.16

0.16

0.22

0.21

C,

I

1

1.49

2.530

10-12

ra

0.15

0.15

0.35

0.49

0.23

rr

0.17

0.17

0.32

0.44

0.25

R2

0.7

0.429

0.0463 0.947 2 x 106

ra

0.26

0.67

1.62

1.66

0.20

rr

0.25

0.59

1.39

1.42

0.22

Cz.(T) 1

0.49

3.041

1.654

1

ra

1.17

1.03

0.75

0.85

1.42

‘r

1.00

0.88

0.64

0.73

1.20

*

-

-

-

-

0.3

ra

-

-

-

-

0.29

rr

-

-

-

-

0.25

CIRCUIT D BIAS

J. R, MACDONALD

3r-l -’ 10”

-
’

Rz
3 4 1 ‘7yD_,

P ,* Sf

CIRCUIT A BIAS

CIRCUIT E BIAS

1

- R;

10 10

Fig. 4. Dependence of the bias b, estimated from simulation fitting on the mean standard deviation of K fits +, for the five circuits of Fig. 1 and for PWT and FPWT. Dotted lines indicate b < 0.

parameters which are equal in value and play the same role in two circuits, such as J-R, and C-R,, necessarily have equal r values. It is found that the r values for FMWT are appreciably larger than those for FPWT (eg, for D-R,, the FMWT values of r, and r, are 1.93 and 1.49, respectively), and those of UWT are very much larger. Thus in a fit of a single data set, FMWT parameter estimates will actually be appreciably more uncertain than FPWT ones for data of the present type.
Bias results us St for the five circuits are shown in Fig. 4 for the variance model parameter values given above. There are three possible sources of bias: wrong fitting model, wrong variance model, and the intrinsic nonlinearity of the fitting model. The first source is absent here since we take Ir(w, 0) = Ifc(w, 0). The

figure shows that wrong weighting, here PWT, usually leads to bias 10-100 times or more larger than the residual nonlinearity bias of the correct FPWT. The results of Fig. 4 and Table 1 show that for 0,~ 0.1, for the D or J circuit the PWT C, bias even exceeds its statistical uncertainty, s, or s,. Further, UWT yields appreciably greater bias even than PWT. Although the large PWT bias results are nearly the same for the four ambiguous circuits, appreciable differences appear for FPWT. Note also that the FPWT bias results are so small that generally (b, + Srr,,,) N S,r, for even large values of S,, and such bias is thus negligible compared to normal statistical uncertainty.
All previous CNLS work has involved constant values of U and <, most often with either UWT or

Impedance spectroscopy: problems and developments

1491

PWT. For real situations, however, appropriate values of U and 5 are usually unknown. Therefore, it is desirable to allow them to be free parameters of the least squares fit as well as the 0’s. By modifying and extending an approach discussed in Refs.[57,58], it has been possible to do so, and the results, which usually yield U and 5 estimates with high resolution, are incorporated in LEVM. Most appropriately, the U and 5 estimates from such a fit may then be used as fixed values in a final fitting. For a situation where LY=, 0, to = 1, and 5 is free, one finds for the four equivalent circuits of Fig. 4 that FPWT yields 6; = 2.7 x 10m3,essentially independent of Q,, ra5‘5 2.63, and rrS 2: 1.52, both quite small enough that 5 estimates will be useful in most cases of interest. Results for the E circuit are comparable. These relative results are all independent of the data scaling as expected. Present space limitations, prohibit a detailed discussion of the important situation where both tl, and u, are nonzero and fitting involves both U and 5 free. MC results show, for example, that for the D, A, C, and J circuits U N GI,/Q,as expected, and s,” u 0.16, independent of Q, and Q, values. One finds that for these four circuits with CI,= 10e3, &,= 1, and tr, = 0.1, then U N 0.01, b, N 0.03, s,( = 0.08, and s,; N 0.07. No convergence problems appear for any of the circuits until IX, becomes comparable with o, ( 1Z,(max)io.Then, ordinary UWT is appropriate.
Acknowledgemenfs--I much appreciate the preliminary Monte Carlo work of Larry D. Potter, Jr. The support of the Army Research Office is gratefully acknowledged.
REFERENCES
1. M. Sluyters-Rehbach and J. H. Sluyters, in Electrounalyrical Chemistry, Vol. 4, (Edited by A. J. Bard), pp. I-127. Marcel Dekker, New York (1970).
2. R. Parsons, Adv. Electrochem. electrochem. Engng 7, 177 (1970).
3. J. R. Macdonald, in Superionic Conductors, (Edited by G. D. Mahan and W. L. Roth), pp. 81-97, Plenum Press, New York (1976).
4. J. R. Macdonald, in Electrode Processes in Solid State Ionics, (Edited by M. Kleitz and J. Dupuy), pp. 1499183, Reidel, Dordrecht, Holland (1976).
5. R. D. Armstrong, M. F. Bell and A. A. Metcalfe, Electrochem. them. Sot. Spec. Rep. 6, 98 (1978).
6. W. I. Archer and R. D. Armstrong, Electrochem. Chem. Sot. Spec. Rep. 7, 157 (1980).
7. J. R. Macdonald, IEEE Trans. Electrical Insulation El-15, 65 (1981).
8. F. Mansfield, Corrosion 37, 301 (1981). 9. F. Mansfield, M. W. Kendig and S. Tsai, Corrosion 38,
570 (1982).
*The I in equations (33) (35) and (36) on pp. 105-106 should be replaced by l/2.
-iErrata published in J. them. Phys. 56, 681 (1972). In addition the words “intrinsic” and “extrinsic” are improperly used in this work (in place of “intensive” and “extensive”.
$The term (1 - i - n) in equation (8) of this paper should be (1 + i-n).
§The exp( - N,, E term in equation (17) should be replaced by exp( - n,, E) and the “ + ” sign in equation (24) should be “=“,

10. R. P. Buck, Ion-Selective Electrode Rev. 4, 3 (1982). 11. M. Sluyters-Rehbach and J. H. Sluyters, in Comprehen-
sive Treatise of Electrochemistry, Vol. 9, (Edited by E.
Yeager, J. O’M. Bockris, B. E. Conway and S. Saranga-
pani), pp. 177-292. Plenum Press, New York (1984). 12. J. R. Macdonald (Editor), Impedance Spectroscopy-
Emphasizing Solid Materials and Systems. Wiley-lnter-
science, New York (1987)‘. 13. J. R. Macdonald, J. electrounal. Chem. 223, 25 (1987). 14. H. Nyquist, Bell Syst. tech. J. 11, 126 (1932). 15. G. Jaffe, Phys. Rev. 85, 354 (1952). 16. H. Chang and G. Jaffe, J. them. Phys. 20, 1071 (1952). 17. J. R. Macdonald, Phys. Rev. 92, 4 (1953). 18. R. J. Friauf, J. them. Phys. 22, 1329 (1954). 19. R. P. Buck, J. electroanal. Chem. 210, 1 (1986). 20. J. H. Beaumont and P. W. M. Jacobs. J. Phvs. Chem.
Solids 28, 657 (1967). 21. J. R. Macdonald, Trans. Faraday Sot. 66, 943 (1970). 22. J. R. Macdonald, J. them. Phys. 54, 2026 (197l)t. 23. J. R. Macdonald, J. electroanal. Chem. 32, 317 (1971);
47, 182 (1973); 53, 1 (1974). 24. J. R. Macdonald. J. them. Phv_s. 58. 4982 (\1973,),: 61.,
I
3977 (1974). 25. J. R. Macdonald, J. elecrrochem. Sot. 135,2274 (1988). 26. D. R. Franceschetti and J. R. Macdonald, J. elec-
troanal. Chem. 82, 271 (1977). 27. D. R. Franceschetti and J. R. Macdonald, J. elec-
troanal. Chem. 87, 419 (1978)$. 28. J. R. Macdonald and D. R. Franceschetti. J. elec-
troanal. Chem. 99, 283 (1979). 29. J. R. Macdonald, J. appl. Phys. 62, R51 (1987). 30. R. L. Hurt and J. R. Macdonald, Solid St. Ionics 20, 111
(1986). 31. J. R. Macdonald, Bull. Am. phys. Sot. 30, 587 (1985). 32. J. R. Macdonald, J. appl. Phys. 58, 1955 (1985). 33. J. R. Macdonald, J. uppl. Phys. 58, 1971 (1985N. 34. J. R. Macdonald, J. uppl. Phys. 61, 700 (1987). 35. J. R. Macdonald, J. uppl. Phys. 65, 4845 (1989). 36. A. Le Mehaute and G. Crepy, Solid St. Ionics 9/10, 17
(1983). 37. S. H. Liu, Phys. Rev. Lett. 55, 529 (1985). 38. J. C. Wang, in Sixth International Conference on Solid
State Ionics, Garmisch-Partenkirschen (1987); Solid St.
Ionics 28/30, 1436 (1988). 39. B. Sapoval, J.-N. Chazalviel and J. Peyriere, Phys. Rev.
A38, 5867 (1988). 40. J. B. Bates, Y. T. Chu and W. T. Stribling, Phys. Rev.
Lett. 60, 627 (1988). 41. J. R. Macdonald and D. R. Franceschetti, J. them.
Phys. 68, 1614 (1978). 42. D. R. Franceschetti and J. R. Macdonald, J. elec-
troanal. Chem. 101, 307 (1979). 43. D. R. Franceschetti and J. R. Macdonald, J. elec-
trochem. Sot. 129, 1754 (1982). 44. J. R. Macdonald and C. A. Hull, J. electroanal. Chem.
165, 9 (1984).
45. R. Labes and H. Lullies, Pllugers Arch. ges. Physiol.
230, 738 (1932). 46. H. Lullies, Biol. Rev. 12, 338 (1937). 47. K. S. Cole, Membranes, Ions, and Impulses, pp. 167-203.
University of California Press, Berkeley, CA (1972). 48. S. W. Feldberg, J. electroanal. Chem. 3, 199 (1969). 49. T. R. Brumleve, J. electrounal. Chem. 90, 1 (1978). 50. D. R. Franceschetti and J. R. Macdonald. J. elec-
trounal. Chem. 100, 583 (1979).
51. D. R. Franceschetti and J. R. Macdonald, Proceedings
of the 3rd Symposium on Electrode Processes (Edited
by S. Bruckenstein, J. D. E. McIntyre, B. Miller and E. Yeager). Proc. electrochem. Sot. 80-3,94-l 14 (1980). 52. J. R. Macdonald and J. A. Garber, J. electrochem. Sot.
124, 1022 (1977).
53. J. R. Macdonald, J. Schoonman and A. P. Lehnen, J. electroanal. Chem. 131, 77 (1982).

EA35,1-

1492

J. R. MACDONALD

54. Y.-T. Tsai and D. H. Whitmore, Solid St. Ionics 7, 129 (1982).
55. B. A. Boukamp, Solid St. Ionics 20, 31 (1986). 56. J. R. Macdonald and L. D. Potter Jr, Solid St. Ionics 23,
61 (1987).

57. R. J. Carroll and D. Ruppert, Transformation and Weighting in Regression. Chapman and Hall, New York (1988).
58. S. L. Beal and L. B. Sheiner, Technometrics 30, 327 (1988).