ELECTROCHEMICAL IMPEDANCE SPECTROSCOPYA Nyquist plot allows one to identify the elements present in the circuit from the shape of the proﬁle. The basic elements will appear on Nyquist

Characterization of Materials, edited by Elton N. Kaufmann.Copyright 2012 John Wiley & Sons, Inc.

ELECTROCHEMICAL IMPEDANCESPECTROSCOPY

YEVGEN BARSUKOV1

AND J. ROSS MACDONALD2

1Texas Instruments, Inc., Dallas, TX, USA2Department of Physics and Astronomy, University of NorthCarolina, Chapel Hill, NC, USA

INTRODUCTION

Electrochemical impedance spectroscopy allows accessto the complete set of kinetic characteristics of electro-chemical systems, such as rate constants, diffusioncoefficients, and so on, in a single variable-load exper-iment. It is restricted to characteristics that describesystem behavior in the linear range of electrical excita-tion, for example, when it can be approximated by ordi-nary differential equations. It can be contrasted withother methods where explicitly nonlinear properties areinvestigated, such as cyclic voltammetry.

A requirement for linear behavior is a voltage excita-tion below 25mV for experiments at room temperatureor less thankBT/e, in general, where theV(I) dependencyof charge-transfer reactions can be approximated aslinear. Some relevant parameters remain constant overa wide range of conditions, however, and, once foundunder linear conditions, can then be applied to modelmuch wider range response. Example of such para-meters would be the ohmic resistance of electrolytesor the thickness of passivating films on an electrode.

While impedance spectroscopy shares the variable-load experimental method with many other linear-exci-tation electrochemical techniques, it also involves thetransformation of time-domain signals and response tothe frequency domain and the calculation of the relevantimpedance, a complex quotient of voltage divided bycurrent. It thus involves impedance calculation from theresults of time-domain excitation of the system at fixedfrequencies. Although “impedance” is often character-ized as “complex impedance,” this is unnecessary sinceit is an intrinsically complex quantity. Its behavior over arange of frequencies forms an impedance spectrum,leading to impedance spectroscopy.

Model creation and visualization by approximationwith discrete circuit elements when possible can greatlysimplify treating otherwise intractable complex systemsinvolving multiple different processes. Further analysisis then aimed at deriving system parameters from anexperimental impedance spectrum, typically by devel-oping a model function connecting the impedance spec-trumwith appropriate system parameters. Suchmodelsare also usually much simpler in the frequency domainthan they would be in the time domain for the samesystem, which can lead to parameter optimization evenfor quite complex systems that is easily achievable withexisting computing systems.

Impedance spectroscopy is an extremely wide fieldand its scope is described in detail in Barsukov andMacdonald (2005), Stoynov et al. (1991), and Orazem

and Tribollet (2008). While another article in this workdiscusses impedance spectroscopy of dielectrics andelectronic conductors, IMPEDANCE SPECTROSCOPY OF DIELEC-

TRICS AND ELECTRONIC CONDUCTORS, here we primarily dealwith its application to materials containing mobile ions,althoughmuch of the present discussion is independentof those specific elements in the material that lead todispersive behavior. The aim of this work is to provide abasic introduction to the technique, such as its princi-ples, mathematical approaches for developing modelfunctions for most common systems, analysis of theexperimental data to obtain system parameters, and thebasics of experimental implementation.

PRINCIPLES OF THE METHOD

Basic Concept of Electrical Impedance

The simplest relationship between voltage and currentfor electric elements is the Ohm’s law, I¼V/R, whereelement resistance R is dependent on neither I nor V andcan be found by simply applying constant current andmeasuring resulting voltage across the resistor. How-ever, nature contains not only energy dissipative ele-ments but also energy storage elements. Current orvoltage dependence of such elements as capacitors andinductors cannot be directly expressed by Ohm’s law,because of time dependence (I¼C dV/dt, V¼L dI/dt)and the relationship between voltage and current thatrequires a differential equation. Finding parametervalues (in this caseCandL) require observing the systemunder variable voltage and current conditions and over aperiod of time, which makes system response analysiscomplex, especially if multiple components are present.

Fortunately, there is an indirect way to apply Ohm’slaw-like treatment to time-dependent systems, becauseall linear differential equations can be transformed intothe Laplace-domain where they become ordinary equa-tions but in terms of “complex frequency” variables,s¼Reþ io, where i ¼ ffiffiffiffiffiffiffi1

p(sometimes also denoted as

“j”) and o is the circular frequency, related to usualfrequency f as o¼2pf. For example, on taking theLaplace transform L of i(t)¼C dn(t)/dt gives

IðsÞ ¼ Cðnð0ÞsLðnðtÞ; t; sÞÞ ð1Þ

Under the condition where there is no energy stored inthe system before the test, V(0)¼0, and writing thevoltage in the Laplace domain, L(n(t), t, s), as V(s), we get

IðsÞ ¼ CsV ðsÞ ð2Þ

This is equivalent to Ohm’s law, which becomes obviousif we make a definition, Z(s)¼1/Cs that turns the equa-tion into

IðsÞ ¼ V ðsÞZðsÞ ð3Þ

Here, I(s) andV(s) are complexcurrent andvoltage,Z(s) isthe complexequivalent of resistance called “impedance,”

and s is the complex frequency. This simplification forsolving electric circuits and the definition of impedancewas first given by Oliver Heaviside in 1880.

Conveniently, having the expression for impedance ofsimple element (ZR(s)¼R, ZL(s)¼ sL, and ZC¼1/sC), wecan derive the impedance of any complicated circuit,remembering that the combination of impedances in acircuit follows the same rules as combination of resis-tors. So instead of first making a very complex differen-tial equation and then applying the Laplace transform toit, we could define equations directly in the Laplacedomain, by following a simple rule that for serial ele-ments the impedances add up and for parallel elementsthe admittances Y (defined as Y(s)¼1/Z(s)) add up. Letus use an example of a circuit shown in Figure 1.

Firstwefind impedance of sectionR1 andC1,whicharein series. The impedance of this section will be Z1(s)¼R1

þ1/C1s.Theimpedanceofaninductiveelementparalleltoit isZ2(s)¼L1s. Nowwecan turnbothof these impedancesinto admittances so thatwe canuse the rule about addingparallel admittances: Y1¼1/Z1¼1/(R1þ1/C1s) andY2

¼1/L1s. Now we get the admittance of the total circuit asthe sum of the parallel section admittances as Y¼Y1þY2

¼1/L1sþ1/(R1þ1/C1s). Now converting it back toimpedance as Z¼1/Y, we get the impedance for thewholecircuit:

ZðsÞ ¼ 11

L1sþ 1

R1 þ 1

C1s

ð4Þ

Measuring Impedance Values

In many practical cases, there is a need to solve aninverse problem.We have an actual electrical circuit butthe values of theparameters of the circuit are not known.Since we know the relationship between the impedancefunction Z(s) and the parameters, we could find theparameters if the impedance function is also known.How can we find it experimentally?

Thefirst thought is that sincewehave the relationshipZ(s)¼V(s)/I(s) we could experimentallymeasure I(s) andV(s), whichwould give us the desired function. Indeed, itis possible to collect a set of time domainmeasurementsof i(t) and n(t), and then use a Laplace transform toconvert the data to Laplace domain.However, this trans-formation requires large data amounts and computingpower and is prone to high noise sensitivity and integra-tion problems; so this direct method is only practical forsimpler systems.

Historically, an approach that employs periodicallyrepeating excitation n(t) has been used instead, whichdoes not require data collection at multiple time-pointsand was, in fact, used long before computers existed.This method benefits from the fact that after periodicexcitation has been applied for a time much more thanthe time constant of the system under test, the expo-nential components of the response function i(t) declineand become negligible. This means that if input n(t) is asinusoid, the response will also be a sinusoid, althoughchanged inmagnitude and shifted inphase. For a single-frequency sine wave applied as an excitation to a circuit

nðtÞ ¼ VmsinðotÞ ð5Þ

where the circular frequencyo 2pf is defined from thebase frequency f, and after the stabilization time theresponse current will be observed as

iðtÞ ¼ Imsinðot þ yÞ ð6Þ

Here y is the phase difference between the voltage andthe current in radians (from 0 to 2p). Since both magni-tude and phase can be measured directly using analogequipment, this approach already allowed impedancemeasurements in the eighteenth century.

Toconnect thephaseand themagnitudeofa sinewavewith the impedance function we discussed in the previ-ous section, we can take advantage of the periodicity ofbothsignalsanduseFourier transformation to transformboth input n(t) and response i (t) into the frequencydomain, I(io)¼F (i (t)), whereFdenotes the Fourier trans-formation. Since Fourier transformation deals with peri-odic functions, it does not contain a real part in itstransform variable and so complex frequency will be justio. It is the same frequency as used inEquations 4 and 5.ApplyingFourier transformation to these equations givesV (io) and I (io) as Vmp and Imp exp(iy), respectively. SinceFourier transformationhas the sameproperty asLaplacetransform of transferring differential equations into lin-ear equations (only for periodic signals), it alsomaintainsthe Ohm’s law-like relationship between excitation andresponse, I(io)¼V(io)/Z (io).

Substituting the values for complex current and volt-age for a single sine wave into this equation, we obtainthe desired equation for the impedance Z(io) from themagnitude of sine waves Vm, Im, and the phase shift y,measured using analog means

ZðioÞ ¼ V ðioÞ=IðioÞ ¼ Vm ImexpðiyÞ ð7Þ

Tohelpunderstand the notationsused in the impedanceliterature, we should mention that complex numbers,historically, have also been represented as phase angleand modulus. When a complex number is graphicallyrepresentedasapoint in theXYcoordinate systemwhereX¼Re(Z) and Y¼ Im(Z), the phase y will be the anglebetween the X-axis and the vector, and modulus |Z|will be the length from the center to the pointwith coordinates X and Y. This defines the relationshipsy¼ tan1(Im(Z)/Re(Z)) and |Z|¼ [Re(Z)2þ Im(Z)

2]1/2.

R1C1

L1

Figure 1. Example of a serial/parallel circuit.

2 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY

Impedance Spectrum

What can be expected from the values of Z(io) obtainedfrom the experiment, in particular how will they dependon the circular frequency o? This can be easily checkedusing the equations for Z(s) that we derived in the firstsection, now substituting io instead of s. For the baseelements, ZR(s)¼R, ZL(s)¼ sL, and ZC¼1/sC becomesZL(io)¼ ioL and ZC¼1/ioC.

The samesubstitutionapplies formore complex func-tions. For example, the function for the circuit used inFigure 1 becomes

Z ðioÞ ¼ 11

L1ðioÞ þ1

R1 þ 1

C1ðioÞ

ð8Þ

To analyze the properties of an unknown system, it isuseful to graphically represent impedance functions atmultiple frequencies. This is known as an impedancespectrum. Since we are dealing with complex numbersZ (io)¼Reþ i Im, we need to visualize not only the realpart but also the imaginary part. The most commonlyused plots for this purpose are Bode plots, where Re(o)and Im(o) are plotted separately versus ln(o) or log(o), orthe so-called Nyquist plot (actually a misnomer; seeBarsukov and Macdonald, 2005), where Im(o) versusRe(o) is plotted, while frequency is implied (higher fre-quency is on the left). The scales of the X- and Y-axesshould be the same to avoid distortion of the elementsshape.

A Nyquist plot allows one to identify the elementspresent in the circuit from the shape of the profile. Thebasic elements will appear on Nyquist plot as follows:resistor as a shift on the X-axis, capacitor as a verticalline in a direction of increasing of Im, inductor as avertical line in a negative direction (increasing Im ), andresistor in parallel with a capacitor as a semicircle.Inductive effects are rarely observed in electrochemicalsystems below frequencies of 10kHz. An example circuitthat exemplifies all of the elements common in electro-chemistry is given in Figure 2.

Using the rule for adding serial impedances, we getZ1(s)¼Rserþ1/Csers for Rser and Cser.

For the parallel combination R1 and C1, we usethe rule about adding the admittances to find its admit-tance Y2(s)¼1/R1þC1s. Converting back to impedanceZ2(s)¼1/Y2(s)¼1/(1/R1þC1s).

Finally, adding Z1 and Z2, which are in series witheach other, we get the total impedance as given in

Equation 9. To calculate the impedance values at differ-ent frequencies, we will assume periodic excitation andsubstitute s¼ io.

ZðoÞ¼Z1ðoÞþZ2ðoÞ¼Rserþ1=Cserioþ1=ð1=R1þC1ioÞð9Þ

For a numerical example, we can use values of para-meters Rser¼0.05O, R1¼0.1O, Cser¼10 F, and C1

¼0.01 F. The range of frequencies f is from 0.1Hz to100kHz, where o¼2pf.

Usually, a logarithmic distribution of frequencies isused to give enough points to cover different processesthat might be far apart in terms of the direct frequencyrange.

It can be seen that Rser is causing the high frequencyportion of the spectrum (on the left) to intercept withX-axis at 0.05O, where imaginary portion becomes zero.Parallel R1 and C1 are causing the semicircle, whoseright side again approaches zero imaginary part at lowerfrequencies with real value close toRserþR1. Finally, thecurve goes vertical due to the serial capacitanceCser thatis very large and, therefore, shows a noticeable effectonly at low frequencies (right of the graph). Note thatsuch simple considerations allow to estimate the valuesof circuit elements from just looking at the Nyquist plot.However, that is not always possible for more complexcircuits where time constants of different elements canoverlap. This topic is discussed in the data analysissection.

Applications of Impedance Spectroscopy to ElectrochemicalSystems

Electrochemical processes are in general nonlinear,which means they cannot be described by linear

S1

R1

S1

C1

RserCser

Figure 2. Example of a serial/parallel circuit.

0 0.05 0.1 0.15 0.20

0.05

0.1

0.15

0.2

Re(Z )/Ω

–Im

(Z)/

Ω

Figure 3. Impedance spectrum of the circuit in Figure 2 in thefrequency range from 100kHz to 0.1Hz.

ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY 3

differential equations or expressed as electric elementslike resistors, capacitors, and inductors. Nonlinear sys-tems cannot be solved by Laplace transformation andthe concept of impedance is generally notwell defined forthem.

In spite of this limitation, all merits of the above-mentioned formalism can still be applied to electrochem-ical system provided that voltage changes during elec-trochemical processes are small. Analysis of theButler–Volmerequation that is central inelectrochemicalkinetics shows that at voltage changes below the thermalvoltage value kBT/e (about 25mV at room temperature),the relation between current and voltage is linear. There-fore, analysis of electrochemical processes at small volt-age changes can be replaced by analysis of equivalentelectric elements. In particular, the relationship betweenvoltage and current for simple electrochemical charge-transfer reaction becomes similar to Ohm’s law, wherethe resistive element that corresponds to charge transferis known as “charge-transfer resistance.”

Other processes important in electrochemistrysuch as concentration polarization in adsorption anddiffusion processes can be approximated throughcombinations of capacitors and resistors. Finally, elec-trochemical systems can exhibit actual physical capac-itance across some films deposited at the electrodesand resistances (such as electronic and ionic resis-tance of porous electrodes). An overview of most of theimportant electrochemical processes that can be ana-lyzed by impedance spectroscopy is given in subse-quent sections. Furthermore, the Macdonald website(Macdonald, 2011) contains an extensive list of refer-ences dealing with complex nonlinear least squares(CNLS) analyses of a wide variety of solid and liquidmaterials; see the downloadable guide to electrochem-ically oriented publications listed there.

In addition to linearity conditions assured by smallvoltage excitation, several other requirements areneeded to be preserved during an experiment in orderto satisfy the assumptions of impedance spectroscopy.They include the following.

1. “Steady-state” requirement, whichmeans that thesystem should not change during the measure-ment. For example if the system under test is abattery, its state of charge should not changeduring test, a result that can only be achieved ifthere is no bias current flowing between the elec-trodes that is not caused by a small excitation.Also, there should not be any other changes thatcan affect system response, such as change oftemperature, pressure, and so on.

2. “Causality” means that the response should bereflecting only the excitation and no other effect,such asmemory effects, fromsomepriormeasure-ments.Satisfying all the conditions can be checked by

the absence of any additional frequencies ofresponse sine waves except those of the excitationsine waves. This will be discussed in themeasure-ment section.

PRACTICAL ASPECTS OF THE METHOD AND METHODAUTOMATION

Basics of Measurement Apparatus

Impedance spectrum can be measured by modulatingthe voltage or the current signal and measuring voltageor current response, correspondingly. The device usedfor modulating voltage signal is called a “potentiostat”and for modulating current signal is called a“galvanostat.” In most cases, electrochemical systemshave their own potential between the electrodes, so it isvery rare that voltage excitation is overlaid over a zerovoltage difference.

Potential between electrodes reflects the state ofelectrochemical system; therefore, it is convenient toenforce certain potential that activates a process ofinterest. To isolate effects on just one electrode, typi-cally a three-electrode configuration is used, wherepotential between electrode of interest (working elec-trode) and reference electrode is measured. Referenceelectrode potential difference from standard HþH2

electrode is known beforehand, which allows to com-pare the potential of working electrode with the stan-dard half-cell potentials available in referenceliterature for various electrochemical redox couples.

For example, to observe process of metal dissolution,potential needs to be set in the range close to the equi-librium potential Eeq of Me/Menþ couple, otherwise pro-cess would be too slow to be noticeable (impedance willbe close to infinite). Equilibrium potential Eeq can befound from standard half-cell oxidation potential E0

given known concentrations of active species as definedby Nernst equation (Equation 18).

Typically, potentiostats are capable to provide boththe constant “bias” potential (e.g., 1.6 V) between theworking and reference electrodes and variable voltageexcitation of25mV. Ideally, the current caused by biaspotential should be low so that the system does notchanges fast enough to have different parameters overthe duration of impedance spectra measurement. Forexample, if there is an active corrosion current, surfacearea could change enough to cause different effectivecharge-transfer resistance. To achieve small currents,bias potential can be set at Eeq and then slowly rampedup until noticeable current starts. In galvanostaticallycontrolled experiment, bias current can be set, whicheliminates the need of searching for optimal potential.However, there is an additional caveat, since excitationis done by variable current, voltage response might turnout to be outside of linearity rage of less than 25mV, sovariable excitation level needs to be adjusted.

Overview of Available Measurement Systems

Although we have seen in the section “Principles of theMethod” how to obtain impedance values from compar-ing magnitude and phase of excitation versus responsesine waves, currently it is more common for electroche-mists to use automated measurement systems. Themost common system includes a signal generator to


generate a sine-wave excitation of a required frequency,a potentiostat or a galvanostat, that amplifies the signaland forces the required voltage or current across a mea-surement system, and an analyzer that obtains phaseand magnitude signal of a resulting response.

Such systems are widely commercially available. Thefield is dominated by Solartron Analytical with their“frequency response analyzer” (FRA) combined with apotentiostat. Control software allows one tomake a scanofmultiple frequencies to obtain a spectrum. The detailsof their system can be seen at http://www.solartrona-nalytical.com/Pages/1260AFRAPage.htm (SolartronAnalytical). Other competing systems providers includeNovocontrol, Hewlett Packard/Agilent, Gamry Instru-ments, Ametek Princeton Applied Research, Autolab,and ZAHNER-Elektrik.

FRA-based spectrometers provide high-qualityimpedance spectra but share common disadvantages,such as the requirement for a complicated signal gen-erator or phase-sensitive detector and long measure-ment times for exploring a wide frequency range. Thelatter occurs because the impedance at each frequencyis measured sequentially and the excitation for eachfrequency should be applied at least for two periods toprevent transient effects.

Accordingly, the time increase for the sequentialmea-surement is excessively large if low-frequency data needto be measured. A method using a perturbation signalconsisting of multiple sine waves and analysis of theresponse by fast Fourier transformation (FFT), whichremoves the necessity for a phase sensitive detector andallows a fastermeasurement, has been proposed in Pop-kirovandSchindler (1992).Withthismethod, impedancedataatmanydifferentfrequenciescanbeobtainedsimul-taneously. Therefore, the total measurement time isequal to the time required for the lowest frequency mea-surement. This approach also allows one to check for theabsenceof “additional” frequencies in the response spec-trum, whose presence would indicate nonlinearity ornonstationary behavior of the system under test.

Another approach suitable for “self-made” systemsdue to its simplicity, that is based on Laplace transfor-mation and does not require a frequency generatorbecause a simple pulse excitation can be used, isdescribed in Barsukov et al. (2002).

Impedance Spectra Analysis Systems

It is important to understand that there is no fullyautomaticway to obtain systemparameters from imped-ance spectrum, with the exception of some most simplesystems such as electric conductivity measurement of ablock of conducting material. Even when using a com-mercially available impedance spectrum analysis soft-ware, it is necessary to understand the basic principlesof impedance spectraanalysis that is outlined in thenextsection. Given such familiarity, it is possible to shortenthe development time by using commercial or publicdomain programs rather than developing your own.Most commercial programs are based on the open-source code in LEVM (Macdonald and Potter, 1987).

Popular programs include Scribner’s ZView, EChemSoftware ZSimWin, Dr. Boukamp’s Equivalent Circuit,and Kumho’s MEISP (MEISP, 2002).

DATA ANALYSIS AND INITIAL INTERPRETATION

Obtaining Model Parameters from the Impedance Spectrum

Let us consider the simplest case where the structure ofthe circuit is known but the values of the elements areunknown. How can we find the values given the imped-ance spectrum measured at the circuit terminals?Before attempting to find parameters, we have to ensurethat the impedance spectrum was taken in a frequencyrange thatmakes all the circuit elements “visible” on thespectrum, for example, their impedances are notnegligible.

This is a very important requirement since largecapacitors, for example, would not appear in the high-frequency spectrum because their impedance valueswould fall below the noise level. Consider Figure 3. Ifwe would have measured the impedance spectrum onlyuntil the frequency before the vertical line starts, wewould not even know that capacitor Cser existed, andcertainlywewouldnotbeable to obtain its value fromthefit. If we measured at frequencies above 100kHz, wewould not see any circuit elements except Rser, becausethe impedance would show only a real part and look likea “dot” on the X-axis of the Nyquist plot at 0.05O. Forparallel RC elements, it is useful to know approximatelywhat time constant, t¼RC, is expected for it, and makesure that the 1/t is within the frequency range of theexperiment. In the case of physical systems, it is usuallyknown from prior art in what frequency range a processis to be expected to appear; for example, a charge-trans-fer reaction or electrochemical double-layer RC elementis typically between 1kHz and 0.1Hz and diffusioneffects often appear below 0.1Hz. Examples of typicalfrequency ranges are given in Appendix II discussingdistributed elements.

Once it is assured that thespectrumactually containsinformation about the parameters, this problem can beconsidered as a parameter optimization problem in theform

Z ¼ fZ ðo;PÞ ð10Þ

where Z is a vector of complex impedance values thatcorresponds to circular frequency values in vectoro, fZ isthe complex function of circular frequency (e.g., thefunction Z(io) detailed in Equation 8), and P is a vectorof function parameters to be optimized. Typically, suchoptimization problems can be solved using a suitablenonlinear least-square fit algorithm.Most popular is theLevenberg–Marquardt algorithm. Care should be takenwhile using a version of the algorithm that supportscomplex function values, the CNLS approach. Manymathematical packages such as Mathcad or Mathema-tica support such complex optimization. There are alsostand-alone programs specially developed for


immittance (either impedanceoradmittance) datafittingsuch as LEVM (Macdonald and Potter, 1987).

Unfortunately, nonlinear fitting is, in general, an ill-posed problem and is not guaranteed to converge to theglobal minimum or to converge at all. Convergence islargely dependent on the closeness of the parameterinitial guesses to final values. Some other optimizationchoices such asweighing of outliers and different criteriafor optimization also affect convergence. For details onsuchchoices,refer toMacdonaldetal. (1982).Meanwhile,let us consider some ways to find a good initial guess.

Manually, such a search would be similar to param-eter estimation fromNyquist plots, as demonstrated, forexample, in Figure 3. Serial resistance can be estimatedfrom the real part of the “left,” high-frequency edge of thespectrum, while “width” of each semicircle gives anestimate for resistor values associated with that partic-ular RC couple. The sum of all resistors in the circuitusually adds up to the real part of the low-frequencyedge. Once resistor values are estimated, capacitorvalues can be found by taking the points in the middleof corresponding semicircles and finding which fre-quency fmid corresponds to these points. The time con-stant t of the RC couple will be t¼1/fmid, and from it,C can be estimated as C¼ t/R.

For simpler circuits that consist of just multiple RCcouples connected in series, this finding of initialguesses can be automated. After substitution of C¼ t/Rinto the impedance equation for an RC couple, we get anequation that is linear with respect to R:

ZðoÞ ¼ 1=ð1=R þ ioCÞ ¼ Rð1=ð1þ iotÞÞ ð11Þ

Optimization of a sumof any number of linear equationsis a linear regression problem that is guaranteed toconverge. This allows us to assign fixed values of t[index]that are logarithmically distributed along the frequencyrange of the spectrum, and find an estimate ofR[index] foreach RC couple by linear regression. After that, a non-linear fit, where both R and t are free, may be performedto find final values.

An automatic analysis that attempts to find the num-ber and values of RC elements that can represent par-ticular impedance spectrum is useful if a circuit for thesystem under test is unknown. In this case, we couldstart, for example, with 10 RC elements and keep reduc-ing them until the sum of least square errors startsincreasing. The final number of RC elements and theirvalues gives an indication of how many distinct pro-cesses are responsible for this impedance spectrum.Such “distribution of relaxation times” analysis can bedone automatically in LEVM. An alternative method ofpreliminary analysis of a spectrum of unknown systemknown as “differential impedance analysis” is describedin Vladikova (2004).

A variation of the RC element-finding method can beperformed to find raw initial guesses not only for cir-cuits that do not consist of multiple RC elements butalso for elements that can be associated with timeconstants. Resulting values of RC time constants canbe assigned to time constants of actual circuit elements

that give a good initial guess for the nonlinear fit of theactual system.

Such anapproach also allows reducing the ambiguityof the results of the fit. For example, it is known that adiffusion time constant should be lower than that of acharge-transfer reaction. Assigning an initial guess fromthe “low-frequency” RC element to a diffusion elementand that from high-frequency RC element to a charge-transfer resistance or double-layer couple will force thechoices to be in the right frequency ranges. Note that anonlinear fit itself is oblivious to the physical meaning ofparticular elements of the circuit and thus it might notproduce parameters that would make diffusion with asmaller time constant than charge transfer, unless“pushed” in right direction by assigning initial guesses.Automatic initial guess finding with time constant orderassignment is implemented in MEISP (MEISP, 2002).

After optimization is performed, it is important toverify that a fit reflects all features of the spectrum, forexample, no semicircle visible in the spectrum is“ignored” by the fit line or two semicircles are notdescribed with just one by the fit. Other features suchas a 45 line or depressed semicircle that we will discusslater should also be correctly represented. A plot ofresiduals (differences between fit and actual data)should look “randomly distributed” if fit is correctlydescribing all features of the spectrum, and not havebiases such as systematic errors in either direction. Thisis, unfortunately, an ideal not actually fully achieved inmost fitting of models to data sets. There is usually atrade-off between model complexity and the degree ofreduction of systematic errors.

It is also important to verify that time constants for allthe processes have their expected position with respectto each other, as discussed previously.

Discrete Elements

Many common electrochemical processes can be repre-sented as discrete “equivalent” elements with analyticalsolutions for their impedance versus frequency equa-tions. Refer to Appendix I for a collection of frequentlyused discrete elements. All equations are given for a unitarea of electrode surface except where noted otherwise,so electric parameter units would be i¼A/cm2, R¼Ocm2, and C¼F/cm2.

Distributed Elements

Many electrochemical processes do not correspond tosimple elements such as resistors and capacitorsbecause they are described by differential equationinvolving partial derivatives (e.g., diffusion or distribu-tion of activation energies in the solid). However, they dosatisfy all the linearity conditions required to apply theimpedance spectroscopy approach, and impedanceequations canbederived for these processes by applyingthe Laplace transform to the governing equations (see,e.g., Barsukov and Macdonald, 2005, Chapter 2.1.3).

Such processes can be still used as elements of anequivalent circuit along with resistors and capacitors,since their impedance follows the same “additive” rule


for serial connection. Some of these processes can berepresented by simple electrical circuits, but those ofmore complex nature, such as diffusion, cannot berepresented with finite number of discrete elements(although it can be represented as an infinitely repeatedchain-connected resistor or capacitor network, in elec-tronics, known as a “transmission line”). They are com-monly called “distributed” circuit elements. Appendix IIincludes a collection of commonly used distributed ele-ments. Some others are included for use as parts offitting models in the LEVM computer program andare listed in its extensive manual (Macdonald andPotter, 1987; Macdonald website, 2011).

Some Common Models for Fitting Impedance SpectroscopyData

Although nearly all impedance spectroscopy data setsinvolve some constant phase element (CPE) power-lawbehavior at low, high, or both frequency regions, suchdata generally includemore than one dispersive processand several time constants. Analysis and fitting modelsmore complex than a CPE function or a single transmis-sion line are thus frequently required. Since Maxwell’sequations show that it is impossible to distinguishbetween displacement and conduction currents by elec-trical measurements alone (Macdonald et al., 1999;Macdonald, 2008), one must usually invoke otherknowledge about the system when attempting to choosean appropriate model and apply it at either the imped-ance or the dielectric immittance level. Therefore, whenmodels may be readily expressed in algebraic form, herewe shall express them in terms of a general I(o) immit-tance function that can be applied at either level. Theresult can thus describe either conductive-system ordielectric-system relaxation/dispersion.

1. An iconic, widely used empirical model is the Hav-riliac–Negami, which, along with its simplifica-tions, is described, e.g., in Macdonald (2010). Itmay be written as

IkðoÞIkð0Þ

¼ 1

½1þ ðiotÞag ð12Þ

Here t is a characteristic relaxation time, and aandg usually fall in the range from zero to unity. Forimpedance response, the subscript symbol k¼0and I0ð0Þ Z 0ð0Þ R0, while for dielectric-levelresponse k¼D; therefore, it is customary to writeIDðoÞ eDðoÞ ¼ e‘ þ De=½1þ ðiotÞag, whereDe e0e‘ and e‘ is the high-frequency-limitingdielectric constant. The Havriliak–Negami modelthus potentially involves four or five free para-meters.Note that when the values of a and g are both

fixed at unity, Equation 11 reduces to single-time-constant Debye response at either the impedanceor the dielectric permittivity level. Alternatively,when g¼1, it reduces to Cole–Cole response,widely used for analysis of liquid electrolyte data.

Unfortunately, the k¼0 responses of both thegeneral Havriliak–Negami model and that of theCole–Cole response do not reduce to a physicallycorrect low-frequency-limiting power law with anexponent of two for the real part of the admittancein this limit. This defect isnot present, however, forthe Davidson–Cole model defined for a¼1. Fur-thermore, the Davidson–Cole model is less empir-ical than the other two and may even be derivedfrom fractal considerations. Even though the Hav-riliak–Negamimodel with its extra free parametersmay frequently be found to fit data better than theDavidson–Cole model, the latter is more realisticand should be preferred for both fitting and inter-preting data, especially when an impedancemodelis appropriate.

2. Althoughmanymodels are onlyavailable for fittingin numerical form, an early physically plausiblemodel that can be expressed algebraically is thePoisson–Nernst–Planck (PNP) effectivemedium forordinary diffusion of positive and negative mobilecharges of arbitrary mobility in the material of afinite-length cell with fully blocking electrodes(Macdonald, 1953; Macdonald and Franceschetti,1978): its anomalous-diffusion version, the PNPA(Macdonald, 2010; Macdonald et al., 2011), andthe partially blocking PNP/PNPA model (Macdo-nald, 2011). For these models Poisson’s equationis satisfied throughout. The fully blocking PNPAexpression (Macdonald et al., 2011) (Equation 6)may be written for full or small charge dissocia-tion as

Za ¼ R‘

ðiotÞg þ tanhðMqaÞMqa

ðiotÞgð1þ iotÞ þ iotðiotÞg½ tanhðMqaÞMqa

whereR‘ 1=G‘,G‘is the high–frequency-limitingconductance, t R‘C‘, C‘ is the high-frequency-limiting bulk capacitance,M L=2LD, the numberof Debye lengths in half the celllength,qa ¼ ½1þ ðiotÞg1=2, and 0 < g 1. Wheng ¼ 1, the result is PNP response that is found tobe closely equal to dielectric-level dipolar Debyebehavior when M 1, even though the conductionprocess here involves mobile charges (Macdo-nald, 2010).When g < 1, however, the PNPAanom-alous diffusion case, this model has been shown towell fit experimental data sets of several disparateionically and electronically conducting materials(Macdonald, 2011; Macdonald et al., 2011) andinvolves CPE-like behavior at low frequencies. Fur-thermore, in its LEVM instantiation (Macdonaldwebsite, 2011) it includesarbitrarymobilities of thetwo charge types, the possibility of their generationrecombination from neutral centers, and canaccount for full or partial blocking of charges at theelectrodes, as well as possible specific adsorption(Macdonald and Franceschetti, 1978). In addition,it can lead to estimates of both the original


neutral-center concentration as well as those of thedissociated charges, particularly important for theusual case in solid materials of small dissociation(Macdonald, 2011).

3. Several interesting conductive-system imped-ance-spectroscopy data analysis models, refer-enced in the Macdonald website Guidementioned in the section “Applications of Imped-ance Spectroscopy to Electrochemical Systems,”are only available in numerical form for fittingdata. A particularly valuable one, instantiated inthe LEVM computer program, is the KohlrauschCK1 model, whose strengths and relevant refer-ences are summarized in a section of the Macdo-nald website. Its K1 part dates since 1973 and hasbeen derived from both macro and microscopicanalyses with the latter showing that it is a con-tinuous-time, random-walk hopping model.

In the time domain, it has frequently been found, sincethe work of Kohlrausch in the 19th century, that thedischarge current of a fully charged material containingmobile charges decays as a stretched exponential of theform fðtÞ / exp½ðt=t0Þb with 0 < b 1, stretched whenb < 1 In the stretched case, the one-sided Fourier trans-formto the frequencydomainmustbecarriedoutnumer-ically that directly leads to what has been called the K0model, whose high-frequency capacitance approacheszero, thus requiring the addition of a parallel capacitancerepresenting e‘ and leading to the CK0 model.

Interestingly, the K1 model, also derived fromstretched exponential temporal response and expressedat the electricmodulus level (Macdonald, 2009), leads toa constant capacitance in the high-frequency region,wrongly identified in the earlier work as e‘ Thus, the fullfitting model must also include a parallel capacitanceand is termed the CK1. It has been found to well-fitconductive-system data sets for many different materi-als with a value of b very close to one third and indepen-dent of the temperature and ionic concentration! In thiscase, it is named the CUN, with UN standing for“universal” (probably an exaggeration), and it theninvolves only three free parameters. Nevertheless, whenfitting impedance spectroscopy data in either the time orthe frequency domain, it is important to fit with severaldifferent models to find the most appropriate one.

Example of Impedance Spectrum Analysis for BithiopheneElectropolymerization

This section gives an example of impedance spectra andits analysis. Data used for this section were recordedduring electropolymerization of bithiophene resulting inthe formation of a thin electrically conductive polybithio-phene layer. The test setup is the same as described inthe section “Sample Preparation.” A Nyquist plot of thedata is shown in Figure 4.

This data has been published earlier in Barsukov(1996, Chapter 4.3, Fig. 32 (0.1 s)). However, the anal-ysis given here uses a simplified model that focuses onthe most pronounced electrochemical processes.

The analysis of impedance spectroscopy mostly fol-lows five steps.

1. Understanding the System: In this case, thesystem includes twoprocesses. The first process isdue to a highly conductive polybithiophene filmdeposited on the electrode surface. Polybithio-phene undergoes an oxidation/reduction reactionif the applied voltage is varied, with subsequentdiffusion of the counter-ions that compensate theinjected charge into the bulk of the film. Sincethe film is thin (polymerization is at early stages),the diffusion is limited by the thickness of the film.Conductivity of the film is very high so its resis-tance can be neglected.The second process is a polymerization of bithio-

phene. It is limited by its charge-transfer resis-tance. Bithiophene also diffuses from the bulk ofthe electrolyte but due to the small electrode sizeand excess of bithiophene monomer, this is not alimiting step.

2. Devising a Model: Charge-transfer resistancedue to oxidation/reduction of the polymer wouldbe in serieswith the finite-length diffusion elementand both of these components would be in parallelwith the electrochemical double layer. Polymeri-zation reaction is not limited by diffusion in thepolymer, andwill therefore appear as independentcharge-transfer resistance in parallel to all thesecomponents. There is also a serial resistance thatis always present due to ohmic resistance of theelectrodes and wires as well as small portion ofelectrolyte between the tip of the Luggin capillarytube that connects reference electrode and the celland the working electrode. Such model can berepresented as an equivalent circuit in Figure 5.

Figure 4. Nyquist plot of impedance spectrum of a thin poly-bithiophene film during electropolymerization of bithiophene.Circles are raw data and the line is the model fit.


3. Deriving the Impedance Equations for theModel: The impedance equation for the modelin Figure 5 can be derived using the simple rulethat serially connected impedances add up andparallelly connected admittances (an inverse ofimpedance) add. This way we start with Rser thatgives usZ(o)¼RserþZcir1, where Zcir1 includes every-

thing else, and o is the circular frequency, relatedto the usual frequency f¼o/2p. Note that mostcommercial impedance spectrometerswould storeusual frequency “f” unless specified, so you wouldneed to convert it to o to use the above equations.Now we have three components that are in par-

allel in the remaining circuit: Rpol, Cdl, and Rct

serially to X1. First, let us calculate impedancesof these components. They are Rpol, 1/(Cdlio), andRct þ ZX1ðoÞ. Now to use the admittance additionrule, we turn these values into admittances asY¼1/Z. This gives us 1/Rpol, Cdlio, and1=½Rct þ ZX1ðoÞ.The admittance of the total circuit will be the

sum of parallel element admittances, for example,

Ycir1 ¼ 1

Rpolþ Cdlioþ 1

½Rct þ ZX1ðoÞ

Nowtoget the impedanceof thecircuit,weuseZ¼1/Yto get

Zcir1 ¼ 1

f1=Rpol þ Cdlioþ 1=½Rct þ ZX1ðoÞg

Substituting it into equation for the total impedanceZ(o) we get

ZðoÞ ¼ Rser þ 1

f1=Rpol þ Cdlioþ 1=½Rct þ ZX1ðoÞg

Onemissing impedance function is ZX1ðoÞ—reflectivefinite-length Warburg impedance. This impedance

function was first used in electrochemistry by Hoet al. (1980), and it is described in more detail inAppendix II, Equation 21. Substituting this functioninto Equation 21 and substituting s¼ io we get

ZðoÞ¼Rserþ 11

RpolþCdlioþ 1

RctþffiffiffiffiffiffiffiffiffiffiffiRd

Cdio

rcothð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

RdCdioÞp

ð13Þ

4. Fitting the Data to Optimize theModel Para-meters:To simplify this example, we can use the trialversion of MEISP impedance analysis software(MEISP, 2002) (which is a front-end to LEVM) tofit this data by optimizing parameters Rser, Rpol,Cdl, Rct, Rd, and Cd. It allows one to make a circuit,as in Figure 5, using a built-in circuit editor and touse a ready-made function for ZX1ðoÞ (reflectivefinite-length Warburg impedance). Normally, youwouldalsoneed to comeupwithan initial guess forthe values, but MEISP has a “pre-fit” option whereit finds initial guess values by roughly mappingtime constants of the analyzed circuit to that of aseries of RC elements.Time-constant order has to be specified accord-

ing to the prior knowledge of the system. Rser isalways thehighest frequency (soorder0),whereas,otherwise, the typically highest frequency RC ele-ment corresponds to Rct and Cdl (so order 1). Thencome the polymerization reaction (order 2) anddiffusion components Rd and Cd involving lowestfrequency response (order 3).The resulting fit curve is shown as a continuous

line in Figure 4 and the parameters and theirestimatedstandarddeviationsare given inTable1.It is important to evaluate the fit accuracy and

parameter confidence levels. In this case, the fit isvisually good (more detailed analysis wouldinclude looking at the distribution of residuals).Rser was not determined with good confidencebecause of its extremely low value (value 1 forrelative standard deviation means “singularmatrix”). Confidence intervals obtained for otherparameters are below0.2,whichmeans that phys-ical processes are sufficiently prominently

ref Rser

RctX1

difopen

F

Cdl

gmd

Rpol

Figure 5. Equivalent circuit of polybithiophene/electrolyte sys-tem in thepresenceof apolymerizationreaction.Here,Rser is theserial resistance and Rct is the charge-transfer resistance ofpolybithiophene redox reaction, X1 is a distributed elementrepresenting limited length diffusionwith a blocking boundary,Cdl is the double layer capacitance, and Rpol is the charge-transfer resistance of the bithiophene polymerization reaction.

Table 1. Optimized Parameter Values and Relative StandardDeviations Obtained During Fitting the Spectrum of Figure 4with the Function in Equation 13

Parameter ValueRelative Standard

Deviation

Rser 3.1210e004O 1Rct 2.7379eþ003O 6.7e-03Cdl 4.3673e008F 2.5e-03Rd 3.9333eþ002O 1.0e-01Cd 6.7774e006F 1.3e-02Rpol 7.4008eþ003O 1.4e-02


represented by this impedance spectrum. Theworst value here is for Rd (0.1); it is larger thanvalues for other parameters because of the smallvalue of Rd. By plotting the values of all resistivecomponents as in Figure 6, it can be seen thatRd ismuch smaller than other values.

5. Using the Parameters to Find Physical Prop-erties:The relative reaction rates of redox reaction andpolymerizationmaybecomparedby just lookingatthe values of Rct and Rpol. Clearly, the polymeriza-tion reactionhasasmaller rate constant since theyboth take place on the same surface. To comparewith other experiments, charge-transfer resis-tances are usually expressed in area-independentform Rct¼Rct/S, where S is the electrode area. Inthis case, it is known fromthediameterd¼0.5mmof the platinum disc used as electrode (see Section“Sample Preparation” for details). This gives usS¼ pd2/4¼1.967e-07 m2 and relative charge-transfer resistances of Rct¼1.39e010O/m2 andRpol¼3.769e010O/m2.In this case, quantitative values of the rate con-

stant can also be found. The relationship betweenthe charge-transfer resistance and the rate con-stant is given in Equation 19 in Appendix I. Itrequires, however, some additional values notusually available from impedance experiments,such as the standard half-cell potential of theoxidation reaction, E0, and the concentration ofthe reduced and oxidized species ctot. E0 can befoundusing cyclic voltammetrymethod (seearticleCYCLIC VOLTAMMETRY).The voltage effect of concentrationchange, dE/dc,

may be found fromCd, as given in the description ofthe “reflective finite-length Warburg” element inAppendix II, if the diffusion length “d” (in this casethickness of the polymer layer) is known:

dE

dc¼ FndS

Cd

Here n is number of electrons participating in reac-tion, in this case 1.From polymer density and amount of polymer

considerations, d has been estimated for this casein Barsukov (1996) as 80nm.On substituting the values ofCd andd, we get dE/

dc¼2.236e-04 V m3/mol.ThediffusionconstantDcanbe foundfromRdand

Cd values, given a value of the diffusion length d, as

D ¼ d2

CdRd

On substituting the values from Table 1, we get

D ¼ 2:4e012m2=s:

SAMPLE PREPARATION

While studying the electrochemistry of liquids, it is com-mon to investigate the behavior of an electrochemicalreaction on just one of the electrodes by subtracting theeffect of other components of the cell. This is done bycomparing the potential at the “working” electrode undertest with the electrochemical potential of a passive“reference” electrode close to its surfacewithapotentiom-eterwithhigh-ohmicinput.Sinceonlynegligiblecurrentisflowing between theworking and the reference electrodesdue to the voltage measurement, the impedance of thereference electrode itself does not matter.

At the same time, the excitation signal is appliedbetween the “working” and the other much larger“counter” electrode located in the bulk of the cell andsized to minimize its impedance and to provide a homo-geneous field around the working electrode. Since mea-surements are made only between the reference and theworking electrode, the impedance of the “counter” elec-trode, as well as that of the electrolyte between theworking and the counter electrodes, is also excluded.Here, the electrolyte should be highly ionically conduc-tive to avoid any local potential gradients, so typically asalt that is passive in the potential window under inves-tigation, such as LiClO4, is added to the solvent in largeamounts (typically, 1M or more).

An example of such a three-electrode cell is shown inthe Figure 7.

The working electrode used for investigation can bemade by inserting a platinum wire into a glass capil-lary, melting the capillary so that the wire is imbeddedin the glass, cutting the left-over wire, and polishingthe glass/platinum composite perpendicular to the cutuntil the platinum surface is coplanar with the flatglass surface. The platinum disk is exposed to theelectrolyte and constitutes a mini electrode whosediameter is precisely controlled by the diameter of theoriginal platinum wire. In the experiment described inthe section “Example of Impedance Spectrum Analysisfor Bithiophene Electropolymerization,” a 0.5-mmdiameter wire was used.

Figure 6. Relative contributions to total Re(Z) by differentprocesses.


Thiscell issuitable formeasurementof the impedanceof theprocessesontheworkingelectrode.However, someunusual considerations apply to a reference electrode tobe used for impedance measurement as compared totraditionally used references. Because distortions in thecaseofACmeasurementcanbebothresistiveandcapac-itive, it is important that theelectrode itself, aswell as theLuggincapillary tubefilledwithelectrolyte,doesnothavean impedance too large so that the impedance remainsless than that of the electrometer. This ismore difficult toachieve for organic electrolytes, required for many sys-tems commonly investigated. A good reference electrodefor impedance measurement in organic electrolyte is asilverwire inasolutionof its ownsalt.Details ona typicalsetup of electrochemical impedance measurement canbe found in Barsukov (1996).

PROBLEMS

Most problems are related to attempts that apply themethod to systems that donot completely satisfy some ofthe impedance concept requirements, in particular, lin-earity, steady state, and causality.

The linearity requirement would be violated if theprocess in question cannot be described by an ordinarydifferential equation. Inelectrochemistry, itwouldbe, forexample, a current/voltage relationship whose voltagedeviation ismuch larger than25mV. In this case, the fullButler–Volmer equation (Equation 14) should be used todescribe the relationship rather than the linearized formthat is similar to Ohm’s law. The impedance concept nolonger applies to suchsystems. If youapply a single sine-wave excitation andmakeanFFT from the response, youwill observe multiple additional frequencies instead of a

single-response frequency. Such “frequency splitting” isa sign that some of the requirements of impedance spec-troscopy are violated, although it requires a complexadditional analysis to find which one.

The simple approach is to try eliminating one of thethreepossible suspects and retest until frequency split-ting disappears. For example, reducing the magnitudeof the excitation current can reduce voltage response tobelow 25mV, thus eliminating nonlinearity. Makingsure that all chemical (e.g., change of state of charge)or physical processes caused by temperature changeare completed prior to measurement by adding relax-ation time can ensure the steady-state condition. Elim-inating additional factors during the experiment canassure the causality requirement. Note that for all ofthis analysis, it is beneficial to have the ability to ana-lyze the response signal in greater detail (FFT) thanmany of the commercial FRA allow. The ability of mul-tifrequency analysis could thus be one of the criteria inchoosing or designing an impedance measurementapparatus.

Another common problem especially for low-fre-quency impedance measurement is that the sine wavemight not be applied for sufficient intervals before themeasurement is done. First, sine-wave responseincludes a transient component due to the load onset,while sine-wave analysis assumes that sine waves wereapplied long enough so that the initial transients arecompletely dissipated. This effect also causes a multi-frequency response that is seen as multiple frequenciesin the FFT spectrum. It can be simply fixed by adding atleast one period delay after the sine-wave onset beforethe measurement, which unfortunately means a longertest time (e.g., for 1 mHz measurement, you will need toadd 1000 s additional measurement time). Laplacetransform measurement does not suffer from this prob-lembecausenoassumptionof absenceof transient effectis made.

Amost commonproblem in impedanceanalysis is thatthe same spectrum can fit perfectly with a variety of dif-ferentequivalentcircuits.Forexample,seriallyconnectedparallel RC elements and parallelly connected R–C serialchainswouldproducethesameimpedancespectrumifallR and C values were optimized. Thismakes it clear that agood fit of impedance spectrum by a particular circuit (orequation) does not prove that this circuit or equation is agood representation of the physical processes in the sys-tem. If amodel isnotphysically relevant,agoodfitwill stillproduce completely meaningless parameter values. Forthis reason, it is critical to follow the steps from physicalunderstanding of the system to model creation and onlythen followed by fitting the data, as exemplified in thesection “Example of Impedance Spectrum Analysis forBithiophene Electropolymerization.”

LITERATURE CITED

Barsukov, E. 1996. Investigations of the Redox Kinetics ofConducting Polymers. PhD thesis, Kiel University, Kiel,Germany.

Luggin capillary tube

Reference Working Counter

Figure 7. Three-electrode cell used for electrochemicalmeasurements.


Barsukov, E., Ryu, S. H., and Lee, H. 2002. A novel impedancespectrometer based on carrier function Laplace transform ofthe response to arbitrary excitation. J. Electroanal. Chem.

536:109–122.

Barsukov, E. andMacdonald, J. R. 2005. Impedance Spectros-copy, 2nd ed. Wiley-Interscience, New York.

Conway, B. E. 1999. Electrochemical Supercapacitors, KluwerAcademic/Plenum, New York.

Ho, C., Raistrick, I. D., and Huggins, R. A. 1980. Applicationof AC techniques to the study of lithium diffusion intungsten trioxide thin-films. J. Electrochem. Soc., 127:343–345.

Macdonald, J. R. 1953. Theory of AC space-charge polarizationeffects in photoconductors, semiconductors, and electro-lytes. Phys. Rev. 92:4–17. 11

Macdonald, J. R. and Brachman, M. K. 1956. Linear-systemintegral transform relations. Rev. Mod. Phys. 28:393–422.37

Macdonald, J.R. andFranceschetti,D.R.1978.Theoryof small-signal AC response of solids and liquids with recombiningmobile charge. J. Chem. Phys. 68: 1614–1637. 124

Macdonald, J. R., Schoonman, J., and Lehnen, A. P. 1982. Theapplicability and power of complex non-linear least-squaresfor the analysis of impedance and admittance data. J. Elec-troanal. Chem. 131:77–95. 149

Macdonald, J. R. 1984. Note on the parameterization of theconstant-phase admittance element. Solid State Ionics

13:147–149. 162

Macdonald, J. R. and Potter, L. D. 1987. A flexible procedure foranalyzing impedance spectroscopy results: Description andillustrations. Solid State Ionics 24:61–79. 179

Macdonald, J. R. 1999. Dispersed electrical-relaxationresponse: Discrimination between conductive and dielectricrelaxation processes. Braz. J. Phys. 29:332–346. 218

Macdonald, J. R. 2000. Comparison of parametric and non-parametric methods for the analysis and inversion of immit-tance data: Critique of earlier work. J. Computat. Phys.

157:280–301. 220

Macdonald, J. R. 2008. Analysis of dielectric and conductivedispersion above T-g in glass-forming molecular liquids.J. Phys. Chem. B 112:13684–13694. 246

Macdonald, J. R. 2009. Comments on the electric modulusformalismmodel and superior alternatives to it for the anal-ysis of the frequency response of ionic conductors. J. Phys.Chem. Solids 70:546–554. 248

Macdonald, J. R. 2010. Utility of continuum diffusion modelsfor analyzing mobile-ion immittance data: Electrode polar-ization, bulk, and generation-recombination effects. J.

Phys.: Condens. Matter 22:495101. 252

Macdonald, J. R. 2011. Effects of different boundary conditionson the response of Poisson-Nernst-Planck impedance spec-troscopy analysis models and comparison with a continu-ous-time, random-walk model. J. Phys. Chem. A

115:13370–13380. 254

Macdonald, J. R.website. 2011. All of the J. Ross Macdonaldpapers citedhereinwith xx referencenumbers are availablein PDF format for downloading from its address: http://jrossmacdonald.com.

Macdonald, J. R., Evangelista, L. R., Lenzi, E. K., and Barbero,G. 2011. Comparison of impedance spectroscopy expres-sionsandresponsesof alternateanomalousPoisson-Nernst-Planck diffusion equations for finite-length situations.J. Phys. Chem. C 115:7648–7655. 253

MEISP v3.0. 2002. Multiple impedance spectra parameteriza-tion software. Kumho Chemical Laboratories. Trial versionavailable at: http://impedance0.tripod.com/#3.

Orazem,M. and Tribollet, B. 2008. Electrochemical ImpedanceSpectroscopy (The ECS Series of Texts and Monographs)Wiley-Interscience, New York.

Popkirov, G. S. and Schindler, R. N. 1992. A new impedancespectrometer for the investigation of electrochemical sys-tems. Rev. Sci. Instrum. 63:5366–5372.

Solartron Analytical Frequency Response Analyzer (FRA).Available at http://www.solartronanalytical.com/Pages/1260AFRAPage.htm.

Stoynov, Z., Grafov, B., Savova-Stoynova, B., and Elkin, V.1991. Electrochemical Impedance. Publishing House Sci-ence, Moscow.

Vladikova, D. 2004. The Technique of the Differential Imped-ance Analysis, Part II. Differential Impedance Analysis. InProceedings of the International Workshop on AdvancedTechniques for Energy Sources Investigation and Testing,Sep. 4-9, Sofia, Bulgaria.

KEY REFERENCES

Barsukov and Macdonald, 2005. See above.The standardbook on the subject that covers theory, experiment,

and applications of impedance spectroscopy and includes

contributions from the leading experts in different aspects

of the field.

Macdonald, J. R., Schoonman, J., and Lehnen, A. P. 1982. Seeabove.

An early justification and demonstration of this approach before

the full development of LEVM.

Macdonald and Potter, 1987. See above.This publication describes and illustrates in detail the use of

LEVM near its introduction in 1987 and in 2000. See the

Macdonald website for up-to-date information on both LEVM

and its WINDOWS version, LEVMW.

Macdonald, 2011. See above.Summarizes PNP and PNPA current work with special emphasis

on boundary conditionpossibilities andmodel fitting of exper-

imental data.

Macdonald, J. R. website.It includes a link (http://jrossmacdonald.com) to download the

free newest WINDOWS version, LEVMW, of the comprehen-

sive LEVM fitting and inversion program. The program

includes an extensive manual and executable and full

source code. Commercial programs derived from this open

code include ZView and MEISP. All of the J. R. Macdonald

papers cited herein are available in PDF format for down-

loading from the above link. Their numbers in the temporally

ordered list in the website are shown as xx in the present

citation list.

APPENDIX I: COMMONLY USED DISCRETE ELEMENTS

1. Current due to activated electron transfer whendiffusion limitations are negligible.

(a) Governing Equation (Butler–Volmer):

I ¼ Si0

e

ð1aÞnFRT

ðEEeqÞe

anFRT

ðEEeqÞ ð14Þ


Here and below, “R” in thermodynamical equa-tions, such as Equation 14, where it is multipliedby temperature T, denotes gas constant,8.3144621 J/K/mol, and “F” denotes Faradayconstant 96,485.3365C/mol. In all “electrical”equations, R and F denote resistance (Ohm) andcapacitance (Farad), respectively.

(b) Electrochemical Parameters:If n¼number of electrons,a (usually around 0.5)¼ transfer coefficient,E¼present potential at the electrode surface, andS¼ surface area, then i0 exchange current isgiven as

i0 ¼ nFk0coxe

anFRT

ðEeqE0Þ

ð15Þ

where k0 is the rate constant of the redox reaction.

(c) Small Voltage Equivalent Equation: I¼DE/Rct (Ohm’s law).DE¼EEeq is overpotential and is the differencebetween the present and equilibrium potentials.

(d) Equivalent Electric Element:Charge-transfer resistance is given as

Rct ¼ RT

nFi0ð16Þ

It can also be expressed through concentrationand rate constant by substituting the equationof exchange current i0 given in Equation 15 as

Rct ¼RTe

FanðE0EeqÞRT

F2k0n2coxð17Þ

Here cox is the concentration of oxidized species,k0

is the rate constant of the redox reaction, and E0 isthe standard half-cell oxidation potential.The ratio between concentrations of oxidized

and reduced forms of active species, cox and cred,depends on the equilibrium potential Eeq asdefined by Nernst equation:

Eeq ¼ E0 þ RT

nFln

coxcred

ð18Þ

Sinceboth coxandEeq inEquation17dependontheamount of chargepassed through the electrochem-ical system before equilibrium was reached, theyare both “state variables.” To make just one statevariable, cox inEquation17canbeexpressedbyEeq

and ctot (the sum of cox and cred) using the Nernstequation. Since the sum of the concentrations ctotremains constant and is a permanent systemprop-erty,we are then left with just one state variableEeq

defining Rct shown in Equation 19.

Rct ¼RTe

Fnða1Þ E0Eeq

RT

eFn E0Eeq

RT þ 1

CtotF2n2k0

ð19Þ

2. Double-layer capacitance.

(a) Governing Equation:

I ¼ CdldE=dt ð20Þ

(b) Small-Voltage Approximation:Same.

3. Reaction of monolayer of active species adsorbedat smooth surface.

(a) Governing Equation:

i ¼ dE

dt

Fn

dE=dcð21Þ

Here, dE is the voltage deviation from steady-statevoltage and dc is the surface concentration devia-tion from the equilibrium value.dE/dc can be further determined if a process is

exactly described by the Nernst equation (typi-cally, for diluted solutions and low coverage levelsof adsorbed species) as

dE

dc¼ RT

Fnctot

e

nFðEeqE0ÞRT

e

nFðEeqE0ÞRT þ 1

24

352

ð22Þ

Note that in cases of higher concentrations,Frumkin isotherm has to be used instead ofNernst equation. See Conway (1999) for detailedtreatment of pseudocapacitance for varioussystems.

(b) Electrochemical Parameters:E0¼ equilibrium potential of redox reaction,E¼applied electrochemical potential, andctot¼ coxþ cred is the bulk concentration of activespecies (sum of oxidized and reduced-formconcentrations).

(c) Small-Voltage Equivalent:

I ¼ Cps dE=dt ð23Þ

(d) Equivalent Electric Element:Pseudocapacitance of redox reaction of an activespecies adsorbed on the surface of an electrode inF/cm2

Cps ¼ Fn

dE=dc


In the case of Nernstian redox reactions, dE/dccan be substituted from Equation 22 as

Cps ¼2F2n2 cosh

FnðEeqE0ÞRT

þ 1

ctot

RTð24Þ

APPENDIX II: COMMONLY USED DISTRIBUTED ELEMENTS

1. Reaction of Mobile Active Species Distrib-uted in Infinite Layer:Linear diffusion from a medium whose length canbe approximated as infinite results in an imped-ance analogous to that of an infinite length trans-mission line composed of capacitors and resistors

Z ðsÞ ¼ dE

dc

1

nFffiffiffiffiffiffiffisD

p ð25Þ

Here D is the diffusion coefficient of the reactionrate-limiting mobile species in the medium.Governing equation in the Laplace domain: (infi-

nite-length Warburg impedance)

IðsÞ ¼ EðsÞ=ZðsÞ ð26Þ

dE/dc can be further determined for Nernstianreactions as in Equation 22.. Typical shape in complex presentation is shownin Figure 8.

. Impedance function in terms of electric para-meters:

ZðsÞ ¼ffiffiffiffiffiffiffiffiffiRd

sCd

rð27Þ

. Schematic presentation of the circuit illustratedin Figure 9.

. Fit parameters.Cd in F/cmRd inO/cm

. Conversion into electrochemical parameters.Here S is the geometric electrode area in cm2,n is the number of electrons participating in thereaction, and c is the volume concentration inmol/cm3.

dE

dc¼ FnS

Cd

dE/dc in V cm3

D in s1 cm2

D ¼ 1

CdRd

2. Finite Length Diffusion for a System with aReflective Boundary (Reflective Finite War-burg Impedance):In the case of a reaction of a mobile active speciesdistributed ina layer of finite length, terminatedbyan impermeable boundary, the impedance is anal-ogous to that of an open–circuit-terminated trans-mission line. For thin homogenous layers ofintercalation materials, this type of impedancewas observed by Ho et al. (1980).

. Typical shape in complex presentation is shownin Figure 10.

. Impedance function in terms of electric para-meters

Z ðsÞ ¼ffiffiffiffiffiffiffiffiffiRd

Cds

rcothð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRdCds

pÞ ð28Þ

. Schematic presentation of the circuit is shown inFigure 11.

. Fit parameters.Cd in FRd inO

. Conversion into electrochemical parameters.HereS is the geometric electrode area in cm2,n

is the number of electrons participating in the

0

0

1

2

3

4

5

6

7

8

9

2 4 6 8

Re(Z )/Ω

–Im

(Z)/Ω

Figure 8. Impedance spectrumof diffusionwith infinite bound-ary condition,Rd¼1O/cm,Cd¼1 F/cm, from10kHz to 1mHz.

ref Rd Rd

CdCd Cd

N = inf

Rd

gmd

Figure 9. Equivalent circuit of an infinite boundary transmis-sion line.


reaction, and c is the volume concentration inmol/cm3.

dE

dc¼ FnS

Cd

dE/dc in V cm3

D in s1 cm2

D ¼ 1

CdRd

3. Finite Length Diffusion with a ConductingBoundary (Transmissive Finite WarburgImpedance):In the case of a reaction of a mobile activespecies distributed in a layer with finite length,terminated by a permeable boundary, the imped-ance is analogous to that of open–circuit-terminated transmission line. This case is realizedif the species diffuses through a semipermeablemembrane before reaching the electrode inter-face, or in the case of impedance measurementswith a rotating disc electrode, where diffusionlength is constant and determined by the rota-tion speed.

. Typical shape in complex presentation is shownin Figure 12.

. Impedance function in terms of electric para-meters

ZðsÞ ¼ffiffiffiffiffiffiffiffiffiRd

Cds

rcothð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRdCds

pÞ ð29Þ

. Schematic presentation of the circuit is shown inFigure 13.

. Fit parameters.Cd in FRd inO

. Conversion into electrochemical parameters.HereS is the geometric electrode area in cm2,n

is the number of electrons participating in thereaction, and c is the volume concentration inmol/cm3.

dE

dc¼ FnS

Cd

dE/dc in V cm3

D in s1 cm2

D ¼ 1

CdRd

ref

Cd Cd Cd Cd

N = inf

Rd Rd Rd Rd Rd

gmd

Figure 11. Equivalent circuit of open–circuit-terminated trans-mission line.

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0 0.2 0.4 0.6 0.8

Re(Z )/Ω

–Im

(Z)/

Ω

Figure 10. Nyquist plot of an impedance spectrum of open–circuit-terminated transmission line,Rd¼1O,Cd¼300 F, from10kHz to 1 mHz.

0.9

0.8

0.7

0.6

0.5

0.4

0.6

0.2

0.1

0

–Im

(Z)/

Ω

0 0.5

Re(Z )/Ω

1

Figure 12. Nyquist plot of an impedance spectrum of finitelength diffusion with conductive boundary, Rd¼1O,Cd¼10 F, from 10kHz to 1 mHz.

ref

Cd Cd Cd Cd

N = inf

Rd Rd Rd Rd Rd

gmd

Figure13. Equivalentcircuit ofafinite-length transmission linewith short-circuit termination.


4. CPE:Impedance dependence on frequency that isdescribed by the CPE occurs, for example, inmany cases of inhomogeneous porous layersand in solid-state conductors. This endemic

response function is described below (Macdo-nald, 1984).No physical process can result in ideal CPE

frequency dependence over the entire range fromzero to infinite frequency (except for specialvalues of f such as 0 or 1), although its spec-trum is compatible with the Kramer–Kronigtransformations (Macdonald and Brachman,1956). However, in a restricted frequency region,CPE response can be strictly valid. The use of aCPE for analysis is recommended only if there isno way to obtain a more physically relevantmodel of the process because its f exponentdoes not usually have a physical meaning. Luck-ily, however, physically plausible analysis mod-els that include CPE-like behavior over restrictedfrequency ranges exist and some are discussedin the following section.

. Typical shape in complex presentation.

. Impedance function.

ZðsÞ ¼ 1

Csfð30Þ

. Electrochemical parameters conversion.No direct conversion is possible. In the case

where f>0.5, the meaning of C is near to that ofa capacitor, while if f<0.5, it is nearer to aresistor. For f¼1, C is an inductance.

Figure 14. Nyquist plot of a CPE element, R¼1O, C¼1 F,f¼0.4 . . . 1.


Documents

ELECTROCHEMICAL IMPEDANCE SPECTROSCOPYA Nyquist plot allows one to identify the elements present in the circuit from the shape of the proﬁle. The basic elements will appear on Nyquist