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Title: An analysis of drifts and nonlinearities in electrochemical impedance spectra

Author Names: Niket Kaisare a , Vimala Ramani

b , Karthik Pushpavanam

c, and S.

Ramanathan a *

Affiliations:

a Department of Chemical Engineering, Indian Institute of Technology-Madras, Chennai

600036, India

b Department of Mathematics, Anna University, Chennai 600025, India

c Department of Chemical Engineering, Sri Venkateswara College of Engineering

(SVCE), Anna University, Chennai 602105, India

* Corresponding author

MSB 239, Department of Chemical Engineering

Indian Institute of Technology-Madras

Chennai 600036

India

Ph : +91 44 2257 4171

Fax: +91 44 2257 0509

Email: srinivar@iitm.ac.in

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Abstract:

The response of an electrochemical reacting system to potential perturbations during

electrochemical impedance spectrum measurement is investigated using numerical

simulation. Electrochemical metal dissolution via an adsorbed intermediate species is

analyzed and it is shown that applying the potential perturbation causes the average

surface coverage to drift. For high frequency perturbations, the final value of the average

surface coverage depends mainly on the kinetic parameters and the amplitude of the

applied perturbation. Acquiring the data during the first few cycles of perturbations leads

to an incorrect calculation of the impedance, particularly for large amplitude

perturbations. Repeating the experiments will not identify this drift, while Kramers-

Kronig transform (KKT) can successfully detect this problem. The correct experimental

methodology to overcome this effect and obtain the impedance spectra is also described.

Another reaction with two adsorbed intermediates is also investigated and it is shown that

in certain cases, the violations of linearity criteria can also be detected by KKT. The

results illustrate the importance of validating the impedance data with KKT before further

analysis.

Key words: EIS; Non-linear; Adsorbed Intermediates; Surface Coverage; Kramers

Kronig Transform

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1. Introduction

Electrochemical systems are inherently nonlinear, but small amplitude electrochemical

impedance spectroscopy (EIS) is frequently used to characterize them with the

assumption that the system can be approximated by a linear system [1,2]. In a typical EIS

experiment, the frequency of the perturbation is varied, and the system response is

recorded [3-6]. Once the measurement at one frequency is completed, an excitation at the

next frequency is commenced. Gabrielli et al. analysed the effect of this change in

perturbation frequency [7] . It was shown that the fast swept sinusoidal excitation causes

a systematic error. They also showed that the error can be reduced by “keeping the initial

phase of the input signal zero or by delaying every measurement for several times the

longest system time constant” [7]. The electrochemical system may be examined in quasi

potentiostatic or quasi galvanostatic mode. Many electrochemical reactions are reported

to occur via one or more adsorbed intermediates [8-11]. In the absence of any passivating

film, diffusion limited processes and significant solution resistance, the electrochemical

system can be modeled by a double layer capacitance in parallel with the Faradaic

process [12]. The high-frequency impedance is determined mainly by the double layer

capacitance and the low-frequency impedance is determined mainly by the Faradaic

process, while both double layer and Faradaic process play a role in determining the mid-

frequency impedance. The equations of impedance for the Faradaic processes involving

the adsorbed intermediates can be obtained analytically if it is assumed that the system

response is linear [8,9,13].

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One of the assumptions used in the analysis is that the system is stable, “in the sense that

it returns to its original state after the perturbation is removed” [14]. There are limited

investigations of EIS under nonlinear conditions [13, 15-20] or with stability issues [21-

25]. Recently we reported the results of the numerical solution of the nonlinear

impedance equations for a reaction with an adsorbed intermediate, under stable and

unstable conditions [25]. In this work, we extend it to analyze the effect of applying

sinusoidal potential perturbation to the system and show that the surface coverage

changes with time due to the perturbation itself. The case of reactions with one or two

adsorbed intermediates is analyzed. We also evaluate the implications of this drift on

different methods of acquiring EIS data. The ability of the Kramers Kronig Transform

(KKT) to identify the drifts as well as violations of linearity is analyzed. While EIS may

be acquired under quasi potentiostatic or quasi galvanostatic mode, in this work EIS

acquisition in the quasi potentiostatic mode is considered with the electrode at 500 mV

anodic with respect to the open circuit potential.

2. Theory

It is assumed that the kinetic parameters depend exponentially on the voltage and that

Langmuir isotherm model is applicable. Consider the dissolution of a metal via an

adsorbed intermediate given by the following equation

1

2

+ -

ads

+ +

ads sol

M M +e

M M

k

k

→

→ [1]

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Here, M represents the bare metal site, +

adsM the adsorbed intermediate species and +

solM

the dissolved metal ion. In addition, k1 and k2 are the reaction rate constants which

depend on the overpotential as 0ibV

i ik k e= . Here ki0 is the rate constant pre-exponent and

V is the overpotential. The exponent bi is given by the formula ii

Fb

RT

α= where αi is the

transfer coefficient with a value between 0 and 1, F is the Faraday constant, R is the

universal gas constant and T is the temperature. The current corresponding to this

reaction is given by

( )1 1M

i Fk θ += − [2]

where M

θ + is the surface coverage of ads

M+ and the surface coverage of bare metal sites

can be denoted by 1M M

θ θ += − . The mass balance equation is given by

( )1 21M

M M

dk k

dt

θτ θ θ

+

+ += − − [3]

where τ corresponds to the total number of sites at monolayer coverage and t is the time.

In the absence of any perturbation, the steady state surface coverage is given by

21

2

1 2

1

1

1

DCSS DC

M

DC DC

iDCi

kk

k kk

θ +

=

= =

+

∑ [4]

6

Here iDC

k refers to the rate constant at the overpotential Vdc, given by 0i dcbV

iDC ik k e= .

During impedance measurement, an ac potential of angular frequency ω and amplitude

Vac0 is superimposed on the dc overpotential Vdc and the total potential applied is given

by 0 sin( )dc ac

V V V tω= + . The frequency f is related to ω by the equation ω = 2πf. It is

assumed that before the beginning of the EIS measurement, the system is in steady state.

The surface coverage θ at any time is obtained by solving equation [3] numerically with

the initial condition given by equation [4]. For the second example, the following

reaction with two adsorbed intermediate species is considered.

1

2

3

2

2 2

k

ads

k

ads ads

k

ads sol

M M e

M M e

M M

+ −

+ + −

+ +

→ +

→ +

→

[5]

where the rate constants ki’s are defined as before. Here, 2+

adsM denotes the adsorbed

intermediate species with two charges, distinct from the bare metal site M and the

adsorbed intermediate with a single charge +

adsM . The corresponding mass balance

equations are

( )2

2

2

1 2

2 3

1M

M M M

M

M M

dk k

dt

dk k

dt

θτ θ θ θ

θτ θ θ

+

+ + +

+

+ +

= − − −

= −

[6]

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where the surface coverage of the two species ads

M+ and 2

adsM

+ are M

θ + and 2M

θ +

respectively. The surface coverage of the bare metal sites can be denoted by

21M M M

θ θ θ+ += − −

The steady state surface coverage values of these two species are given by

2

2

3

1

3

3

1

1

1

1

1

SS DC

M

iDCi

SS DC

M

iDCi

k

k

k

k

θ

θ

+

+

=

=

=

=

∑

∑

[7]

while the current is given by

( )( )21 21M M M

i F k kθ θ θ+ + += − − + [8]

The current component at the fundamental and other harmonics can be extracted by

applying fast Fourier transform [25]. The methodology can be employed for analyzing

other reactions such as reversible reactions and for other adsorption isotherms such as

Frumkin isotherm model. The source code, written in Matlab® programming language,

can be obtained by contacting the corresponding author.

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3. Results and Discussion

3.1 Reaction with one adsorbed intermediate

3.1.1. Surface coverage drift

Fig. 1 shows the average surface coverage ( av

Mθ + ) and transient surface coverage (

Mθ + ) as

a function of time, for 10 mV perturbation amplitude and four different frequencies. The

starting point (at t=0) corresponds to the surface coverage at steady state conditions

( SS

Mθ + ) given by equation [4]. The av

Mθ + is calculated by averaging the transient surface

coverage over one complete cycle and the average is assigned to the time corresponding

to the middle of the cycle. The transient surface coverage values for the frequencies 10

Hz and 100 Hz are shown in the inset. For 100 mHz and 1 Hz perturbations, M

θ + and the

av

Mθ + values are presented in the same plot. The relevant kinetic parameters are given in

the figure caption. For 1 Hz or higher frequencies, av

Mθ + clearly moves away from the

initial value of approximately 0.5975 and settles at a final value of approximately 0.5986.

The final av

Mθ + is the same for all the frequencies, as long as the frequency is above 1 Hz.

At 100 mHz, the final value of av

Mθ + is approximately 0.5983, which is slightly lower

compared to that observed for high frequencies. The duration needed for av

Mθ + to settle

about the final value does not depend on the frequency employed. Simulations with

different starting values for the surface coverage show that the final av

Mθ + value is always

the same for a given set of parameter values (results not shown). Fig. 1 also shows that a

wait time of 20 s is sufficient for av

Mθ + to stabilize around its final value for this system.

Fig.2 shows the final av

Mθ + values for Vac0= 10 mV and 100 mV for various frequencies. It

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is clear that as long as the frequency is 1Hz or more, the final value is independent of the

frequency. It is also strongly dependent on the Vac0 employed.

We have not been able to get the exact analytical solution to equation [3] when the ac

potential is superimposed on the dc potentials. However, the solution in limiting cases

can be calculated. Using the Fourier series expansion of ( )sin t

eα ω

[26]

( ) ( ) ( ) ( ) ( ){ } ( ) ( ) { }sin

0 2 1 2

0 1

2 1 sin 2 1 2 1 cos 2m mt

m m

m m

e I I m t I m tα ω

α α ω α ω∞ ∞

+

= =

= + − + + −∑ ∑ [9]

where In is the nth

order modified Bessel function of the first kind, it can be shown that

the surface coverage can be written in the form of the following integral

( )1 0 sin

1

acb V t

X X init XDC

M M

k ee e dt e

ω

θ θτ

+ +

− −= +∫ [10]

where

( )( ) ( )

( )( ){ }

( ) ( )( ){ }

( )( ) ( )

( )( ){ }

( ) ( )( ){ }

1

2 1 1 0

0 1 0

01

2 1 0

1

1

2 1 2 0

0 2 0

02

2 2 0

1

112 cos 2 1

2 1

112 sin 2

2

112 cos 2 1

2 1

112 sin 2

2

m

m ac

ac

mDC

m

m ac

m

m

m ac

ac

mDC

m

m ac

m

I bVI bV t m t

mkX

I bVm t

m

I b VI b V t m t

mk

I b Vm t

m

ωω

τω

ω

ωω

τω

ω

+∞

+

=

∞

=

+∞

+

=

∞

=

−+ +

+ =

− +

−+ +

+ +

− +

∑

∑

∑

∑

[11]

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In eqn.[10] init

Mθ + is the value of

Mθ + at t = 0. At the limit of infinite time and for high

frequencies (ω >>1) , it can be shown (appendix) that M

θ + will oscillate around a final

value f

Mθ + given by

( )( )

( )( )

2 0 2 0

2

0 01

1

1

DC acf

M

iDC i aci

k I b V

k I bV

θ +

=

=

∑ [12]

regardless of the value of init

Mθ + . i.e. the pre exponent ki0 values have to be multiplied by a

correction factor ( )0 0i acI bV and the av

Mθ + for high frequency and long time can be

estimated. For small values of biVac0, equation [12] reduces to equation [4]. The results of

equation [12] are plotted as continuous lines in Fig.2 and they match the numerical

results very well. Since we do not have an analytical expression for low frequency limit,

a comparison could not be made for the data at low frequencies. However, the match at

high frequency indicates that the numerical results are real and that the average surface

coverage does drift during the application of ac perturbation. The dependence of the final

value of av

Mθ + on the kinetic parameters bi and the amplitude of the perturbation Vac0 is

also mapped out clearly when ω >> 1. The drift in the average surface coverage occurs

even for the cases where Langmuir isotherm model is not applicable. For example, if

Frumkin isotherm model is employed, numerical results indicate that a similar drift in the

average surface coverage occurs (results not shown). Thus, the Langmuir isotherm

assumption is not a necessary condition for the occurrence of the surface coverage drift

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3.1. 2 Effect on Impedance Spectra

Usually in EIS measurement, a sequence of perturbations of decreasing frequencies is

employed. In many cases, the electrode is subjected to a wait time or a number of wait

cycles before the measurement starts. This is because the electrode must be at steady state

for the EIS measurement to be valid [3, 23, 24]. If the ac perturbation is applied during

the wait time and a sufficiently long waiting time is employed, then the surface coverage

will be oscillating around the final value f

Mθ + and the impedance measurements would be

accurate. On the other hand, if between the measurements, the electrode is maintained at

the dc overpotential for a long time, then the surface coverage would return to SS

Mθ +

When many cycles are employed and a sufficiently long time is allowed for the system to

come to final state before the measurements are made, the measured impedance would

not be affected by the drift in av

Mθ + . The simulated impedance for data acquired after a

wait time of 20 s are presented in Fig. 3 as stable indicating that the data are acquired

after the av

Mθ + is close to f

Mθ + . On the other hand, if the system is allowed to come to the

initial steady state by applying only the dc overpotential for a sufficiently long duration,

between each measurement, the drift during the measurement would affect the impedance

spectra. The results of such data acquisition methodology are presented in fig. 3 as

drifting indicating that the data are acquired when the average surface coverage was

changing. Note that for given amplitude, the major difference between the two spectra is

in the mid-frequency regime. In the high-frequency regime, the total change in av

Mθ + is

significant. However, in the high frequency regime most of the current would pass

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through the double layer structure and hence the total impedance is the same for either of

the two data acquisition methods. In the low-frequency regime, the total change in av

Mθ + is

relatively small. Besides each measurement cycle takes a significant time and even if the

surface coverage begins at SS

Mθ + , the av

Mθ + will come close to the final surface coverage

f

Mθ + within one cycle. Thus the Faradaic impedance, and hence the total impedance, will

not depend strongly on the data acquisition methodology. Only in the mid-frequency

regime, where the Faradaic impedance is different for these two measurement strategies

and where it contributes significantly to the total impedance, the differences between

these two measurement strategies are clear. The differences are small for 10 mV

amplitude and large for 100 mV amplitude perturbations.

A comparison of the results for both perturbation amplitudes in fig. 3 shows the if data

are acquired after applying the perturbation cycles for a sufficiently long time (under

stable conditions), then both 10 mV and 100 mV perturbations yield the same spectra. i.e.

the nonlinearity effects are negligible for this reaction with the given set of parameter

values. On the other hand, if the data are acquired under drifting conditions, the spectra

would be different for these two perturbation amplitudes. It is worth noting that if any of

the experiments simulated in fig. 3 is repeated, then the results of the second run would

be identical to those of the first run, as long as other conditions remain the same. This is

because while there are changes to the system during the experiments, the changes are

created by the measurement process and the changes disappear a while after the

measurement process is stopped. If sufficient time is allowed between the repeat runs,

then the system will come back to the original state. However, during data acquisition,

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the system is not under stable conditions since av

Mθ + drifts during the measurement, even

for a small amplitude perturbation. If the data are acquired under drifting conditions,

clear differences would arise between the results of 10 mV perturbation and that of 100

mV perturbation, as seen in fig. 3. However, the differences might be incorrectly

assigned to non-linearity effects, rather than the larger drift caused by the 100 mV

perturbation.

3.1. 3 Kramers-Kronig Transform

KKT can be used to validate the EIS data [22-25]. If the system under investigation is

linear, causal and stable and if the amplitude of the transfer function is finite at all

frequencies including the zero and infinite frequency limits, then the data would be KKT

compliant. However, the converse is not always true. i.e. if a set of data is KKT

compliant it may not indicate that the system response was linear, causal and stable or

that the values would be finite at all frequencies. Macdonald and Urquidi-Macdonald [27]

analyzed the impedance of a CPE which is given by ( )0

1CPE n

ZY jω

= , where Y0 and n are

empirical constants (0 ≤ n ≤ 1). Even though the impedance of a CPE is infinite at zero

frequency, a circuit with CPE was shown to be KKT compliant as long as the real

component of ZCPE is positive. Experimental and simulated data indicate that violations

of stability criterion are usually detected well by KKT [21-25].

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The impedance spectra in fig.3 were subjected to KKT and the results are presented in

fig. 4. For the data acquired under drifting conditions, the original data and transformed

do not match well, especially for the 100 mV perturbation, as can be seen in fig. 4(c). A

comparison of fig. 4(a) and fig. 4(b) shows that when the perturbation amplitude is 10

mV, the transformed data deviates slightly from the original data. In this case, the

maximum drift in surface coverage is also small and correspondingly the deviations are

small. On the other hand, fig. 4(c) shows that for 100 mV amplitude and drifting

conditions, the transformed data deviates significantly from the original data. For the

data acquired during stable conditions, the transformed data and the original data match

reasonably well, as seen from fig. 4(b) and fig. 4(d). The system corresponding to fig.

4(a) or fig. 4(c) may not be considered as unstable in the traditional sense since it will

revert back to the steady state some time after the perturbation is removed. However,

during impedance measurement the system drifts due to the application of the ac

perturbation and in that stage it should be considered as unstable. A mere comparison of

repeat run data of fig. 4(c) will not show any sign of drift, but the KKT is successful in

detecting the problem.

3.2. Reaction with two adsorbed intermediates

3.2.1. Surface coverage drift

In case of the reaction with two adsorbed intermediates given by equation [5], we could

not analytically solve the corresponding mass balance equation[6]. The numerical

solutions of av

Mθ + , 2

av

Mθ + and av

Mθ for 100 mV perturbation amplitudes are presented in

fig. 5. The parameters employed are given in the figure caption. The parameters were

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chosen such that the initial surface coverage values are in the range of 0.3 to 0.33 for all

the three species. Here again, the values av

Mθ + , 2

av

Mθ + and av

Mθ drift during the application of

ac perturbation and they are not always a monotonic function of time. Based on the

results of fig.5, it is estimated that the average surface coverage stabilizes in 25 s or less

for this system. For each frequency the perturbations are applied and after waiting for 25

s, the impedance was calculated.

While analytical solutions could not be calculated for equation [6] even at the high

frequency limit, a comparison was made between the final avθ values for each species

from the numerical solutions and the following heuristic equations:

( )( )

( )( )

( )( )

( )( )

2

2

2 0 2 0

3

0 01

3 0 3 0

3

0 01

1

1

1

1

1

DC acf

M

iDC i aci

DC acf

M

iDC i aci

f f f

M M M

k I b V

k I bV

k I b V

k I bV

θ

θ

θ θ θ

+

+

+ +

=

=

=

=

= − −

∑

∑ [14]

Essentially, each one of the kinetic rate constants used in the calculation of steady state

surface coverage values was multiplied by a correction factor ( )0 0i acI bV . The heuristic

results are plotted as points in fig.5 and it is seen that the numerical and heuristic results

match very well. Since equation [14] was not derived analytically, it is not clear if the

same heuristic approach can be used for other mechanisms.

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Another set of parameters were employed such that the surface coverage values of at least

one species was close to the extreme value of 0 or 1. The numerical solutions of

av

Mθ + , 2

av

Mθ + and av

Mθ values for 100 mV amplitude perturbations for this system are given in

fig. 6. The final average surface coverage of ads

M+ species ( av

Mθ + ) is close to zero, while

that of 2

adsM

+ species ( 2

av

Mθ + ) is close to 1. Since the range of av

Mθ + , 2

av

Mθ + and av

Mθ are very

different, they are presented in separate sub figures. Although the average surface

coverage values stabilize within 2 s for Vac0 = 100 mV, they take longer time to stabilize

for Vac0 = 10 mV and hence a wait time of 10 s was employed for each measurement

before the impedance data was calculated.

3.2.3. Impedance Spectra

The impedance spectra corresponding to the system in fig. 5, for various amplitudes of

perturbations, acquired after av

Mθ + , 2

av

Mθ + and av

Mθ have stabilized, are presented as complex

plane plots in fig. 7. For 1 mV and 10 mV perturbations, the spectra are practically

identical and the plots completely overlap. Thus, the system response is linear at least up

to Vac0 = 10 mV. On the other hand, the impedance values at Vac0 = 100 mV and 200 mV

are clearly much less than the impedance values at Vac0 = 10 mV or 1 mV. The charge

transfer resistance is also less at larger amplitudes of perturbation. If the system is linear,

then the measured impedance will not be a function of the perturbation amplitude. From

fig. 7 it is clear that this system exhibits nonlinearity at 100 mV, and that the nonlinear

response manifests particularly at low frequencies. In contrast, for the mechanism given

by equation [1] and for the parameters employed in fig.1, nonlinear effects did not

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manifest at Vac0 = 100 mV, as seen from the complex plane plots in fig.2, acquired under

stable conditions. This shows that the exact potential where the nonlinear effects are

dominant depends on the reaction mechanism and the parameter values and that this can

be even larger than Vac0 = 100 mV.

The impedance spectra corresponding to the system in fig. 6, for various amplitudes of

perturbations, acquired after av

Mθ + , 2

av

Mθ + and av

Mθ have stabilized are presented as complex

plane plots in fig. 8. Again, the impedance values for Vac0 = 1 mV and 10 mV are

identical. Apart from the high frequency loop corresponding to the double layer

capacitance, two other loops at the middle and low frequencies are seen. It is known that

a linear electrochemical system with two adsorbed intermediates can give rise to a

maximum of three loops, including one for the double layer capacitance [13, 28-30]. The

occurrence of positive values of Zim (i.e. negative –Zim values) at the intermediate

frequency range, as seen in Fig.8 can be described as inductance or pseudo inductance or

negative capacitance. While the exact term to be used is debated in the literature [31,32

and references therein], the validity of the data (either experimental or simulated) has

been generally accepted. For reactions with a single adsorbed intermediates, the

conditions under which such loops can occur have been mapped out [13, 28]. While a

similar mapping is not reported in literature for the mechanism in equation [5], the values

of the parameters employed suggest that the case of b1 > b2 > b3 allows for the possibility

of negative –Zim values. The impedance values depends on the applied perturbation when

Vac0 = 100 mV or more, indicating that the nonlinear effects dominate at these conditions.

While the value of the impedance changes with the perturbation, the pattern remains the

18

same for all the amplitudes investigated. However, here the charge transfer resistance

increases with an increase in Vac0 whereas it was decreasing with Vac0 for the parameters

employed for fig. 7.

3.2.3. Kramers Kronig Transform

The data in fig. 7 is subjected to KKT and the results, along with the original data are

presented in fig.9. Since the impedance data for Vac0 = 1mV and 10 mV are practically

identical, the data for Vac0 = 1mV is not presented here. Although nonlinear effects are

clearly visible, there are no significant differences seen between the original data and the

transformed results. Previous reports also indicate that KKT does not always identify

violations of linearity [22, 25]. For the case of direct electron transfer, the conditions

under which nonlinearities cause deviations between the original data and the KK

transformed data have been identified [19,20]. Specifically, if the solution resistance is

significant and at least a part of the spectra is acquired in the frequencies above a

transition frequency, KKT can identify the deviation from linearity. Here, “the transition

frequency marks the transition from low frequency nonlinear behavior to high frequency

linear behavior” [19]. However, if the solution resistance is negligible, then for this

system the violations of non linearity are not identified by KKT [25]. The data in fig. 8 is

subjected to KKT and the results are presented in fig. 10. Clearly, the transformed data

for Vac0 = 100 mV or 200 mV show significant differences between the original data and

the transformed results, in both the real and imaginary component of the impedance. The

data for Vac0 = 10 mV does not show any significant deviation, as expected. For the case

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of Vac0 = 100 mV, the deviations begin at around 10 Hz, while for Vac0 = 200 mV, the

deviations begin at around 100 Hz.

The reason why KKT is able to diagnose violation of linearity for this set of parameters,

while not being able to identify the same for the set of parameters employed in fig. 5, is

not clear. In EIS, the nonlinearity with respect to the ac potential perturbation arises

since the rate constants depend exponentially on the applied potential. In case of a

reaction with adsorbed intermediates, the surface coverage values of the intermediates are

limited between 0 and 1. When an ac potential perturbation is applied, the θ will also

vary with time, perhaps with some phase shift with respect to the applied potential. In the

linear regime, after stabilization of θav , the variation of θ for a species may be written as

θ = θav +θac0 sin(ωt+φ), where θac0 is the amplitude of oscillation in θ and φ is the phase

shift. A larger Vac0 would change the value θav as seen in the results here. A larger Vac0

would also normally lead to a larger θac0, even if the relationship between θac0 and Vac0 is

not linear. However, when the θav of one or more species is close to the extreme values of

0 or 1, applying a large Vac0 will not lead to correspondingly large θac0, since θ can not go

beyond the physical limit of 0 or 1. This also would result in a nonlinear response for θ

and hence in the current. In fact, the increase in the charge transfer resistance at higher

Vac0 seen in fig. 8 may be a result of this trimming of the surface coverage oscillations.

Based on the values of θav observed for various species in these two systems, the

following hypothesis is proposed. It is possible that the exponential dependence of the

rate constant on the ac perturbation potential may not lead to deviations in KK

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transformed results, even though the relationship is non linear. On the other hand, if

nonlinearity arises due to trimming of the surface coverage oscillations, then the

transformed data may not match with the original data. However, this is based on a

comparison of a limited set of numerical simulation results and there is no analytical

proof for this hypothesis yet.

4. Conclusions

In summary, during the measurement of impedance, the average surface coverage (θav )

of an adsorbed intermediate species in an electrochemical reaction drifts during the initial

stages of measurement. At high frequencies and at long times, the θav values calculated

by numerical simulations and by analytical method match well. The Faradaic impedances

measured are affected by this drift at the high and mid-frequency regimes, particularly for

the large amplitude perturbation. Since the double layer capacitance dominates at the

high frequency regime, the total impedance is affected by this drift mainly at the mid-

frequency regime. The data acquired during the drift do not represent the system correctly

and the EIS data would not be valid. A comparison of repeat runs will not identify this

drift in θav, but KKT is effective in detecting the deviations due to this drift. The drift

effect can be minimized by applying a small amplitude perturbation. It can also be

overcome by applying the perturbation (small or large amplitude) for a sufficiently long

time and acquiring the data afterwards for each frequency.

Since the kinetic parameters depend exponentially on the potential, large amplitude

perturbations will cause the current profile to deviate from a sinusoidal profile, but KKT

21

does not always identify the violations of linearity conditions. To study the ability of

KKT to detect violations of linearity criteria, the response of a metal dissolving via two

adsorbed intermediate species to ac potential perturbation was numerically simulated and

the impedance was calculated, after allowing the θav values to stabilize. KKT is able to

detect the nonlinear effects for the case where the surface coverage values of one or more

species is near the extremes of 0 or 1. It is likely that KKT identifies the violations caused

by the trimming of the surface coverage values. The exponential dependence of the rate

constants on the perturbation potential also violates the linearity criterion, but that alone

does not appear to cause the data to be KKT noncompliant. Considering that KKT is

successful in identifying the spectra acquired in unsteady state and that all the data

analyses in literature assume that the systems under investigations are under steady state

conditions, it is recommended that any EIS data must be tested for KKT compliance

before further analysis.

Acknowledgements: We would like to thank Prof. D.D. Macdonald

(macdonald@matse.psu.edu ) for the Kramers-Kronig Transform software.

References:

1. J.-P. Diard, B. Le Gorrec, C. Montella, Electrochim. Acta 42 (1997) 1053.

2. K. Darowicki, Electrochim. Acta 40 (1995) 439.

3. E. Barsoukov, J.R. Macdonald, Impedance Spectroscopy (second ed.), John Wiley

and Sons, NJ, 2005.

22

4. M. Orazem, B. Tribollet, Electrochemical Impedance Spectroscopy, John Wiley

and Sons, NJ, 2008.

5. DD Macdonald, Electrochim. Acta 51 (2006) 1376.

6. J.R. Macdonald, Electrochim. Acta 35 (1990) 1483.

7. C. Gabrielli, M. Keddam, J.F. Lizee, J. Electroanal. Chem. 163 (1984) 419

8. M. Keddam, O.R. Mattos, H. Takenouti, Electrochim. Acta 31 (1986), 1147.

9. M. Bojinov, Electrochim. Acta 42 (1997) 3489.

10. D.D. Macdonald, S. Real, S.I. Smedley, M. Urquidi-Macdonald, J. Electrochem.

Soc. 135 (1988) 2410.

11. J. Gregori J.J. Garcia-Jareno, D. Gimenez-Romero, F. Vicente, J. Solid-State

Electrochem. 9 (2005) 83.

12. A.J. Bard, L.R. Faulkner, Electrochemical methods, Fundamentals and

applications (second ed.), John Wiley and Sons, NJ, 2001.

13. J. Maddala, S. Krishnaraj, V. Vinod Kumar, S. Ramanathan, J. Electroanal.Chem.

638 (2010) 183.

14. D. D. Macdonald, M. Urquidi-Macdonald, J. Electrochem. Soc. 132 (1985) 2316.

15. K. Darowicki, Corr. Sci. 37 (1995) 913.

16. K. Darowicki, Electrochim. Acta 42 (1997) 1073.

17. K. Darowicki, Electrochim. Acta 42 (1997) 1781.

18. D.A. Harrington, Can. J. Chem. 75 (1997) 1508.

19. B. Hirschorn, B. Tribollet, M.E. Orazem, Isr. J. Chem. 48 (2008) 133.

20. B. Hirschorn, M.E. Orazem, J. Electrochem. Soc. 156 (2009) C345.

23

21. M. Urquidi-Macdonald, S. Real, D.D. Macdonald, J. Electrochem. Soc. 133

(1986) 2018.

22. M. Urquidi-Macdonald, S. Real, D.D. Macdonald, Electrochim. Acta 35 (1990)

1559.

23. A. Sadkowski, M. Dolata, J.-P. Diard, J. Electrochem. Soc. 151 (2004) E20.

24. A. Sadkowski, J.-P. Diard, C. Montella, J. Electrochem. Soc. 156 (2009) F7.

25. S. Noyel Victoria, S. Ramanathan, Electrochim. Acta 56 (2011) 2606.

26. M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with

formulas, graphs and mathematical tables, Dover Publications, NY, 1972.

27. J.P. Diard, B. Le Gorrec, C. Montella, J. Electroanal. Chem. 326 (1992) 13.

28. JP Diard, B Le Gorrec C Montella C. Montero-Ocampo, J. Electroanal. Chem.

352 (1993) 1.

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30. D. D. Macdonald, M. Urquidi-Macdonald, J. Electrochem. Soc. 137 (1990) 515.

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518.

24

Appendix. A

Consider the mass balance eqn. [6]

( )1 21M

M M

dk k

dt

θτ θ θ

+

+ += − − [6]

with ( ) ( )1 0 sin sin

ac ib V t t

i iDC iDCk k e k e

ω α ω= =

It can be rewritten as M

M

dP Q

dt

θθ

+

++ = [A1]

where

( ) ( )1 2sin sin

1 2 t t

DC DCk e k e

P

α ω α ω

τ

+= and

( )1 sin

1 t

DCk e

Q

α ω

τ= [A2]

The solution can be written as

Pdt Pdt Pdt

e Qe dt Ceθ− −∫ ∫ ∫= +∫ where the constant C is found based on the initial

conditions.

( )( ) ( )

( )( ){ }

( ) ( )( ){ }

( )( ) ( )

( )( ){ }

( ) ( )( ){ }

1

2 1 1

0 1

01

2 1

1

1

2 1 2

0 2

02

2 2

1

112 cos 2 1

2 1

112 sin 2

2

112 cos 2 1

2 1

112 sin 2

2

m

m

mDC

m

m

m

m

m

mDC

m

m

m

II t m t

mkPdt

Im t

m

II t m t

mk

Im t

m

αα ω

ω

τ αω

ω

αα ω

ω

τ αω

ω

+∞

+

=

∞

=

+∞

+

=

∞

=

−+ +

+ =

− +

−+ +

+ +

− +

∑∫

∑

∑

∑

[A3]

At long times (t >> 0) and high frequencies (ω >> 1)

25

( )

( ) ( )( )( ){ }

( ) ( )( ){ }

1

2 1 1

0

0 1

2 1

1

112 cos 2 1

2 1 >>

112 sin 2

2

m

m

m

m

m

m

Im t

mI t

Im t

m

αω

ωα

αω

ω

+∞

+

=

∞

=

−+

+

− +

∑

∑

[A4]

( )

( ) ( )( )( ){ }

( ) ( )( ){ }

1

2 1 2

0

0 2

2 2

1

112 cos 2 1

2 1 >>

112 sin 2

2

m

m

m

m

m

m

Im t

mI t

Im t

m

αω

ωα

αω

ω

+∞

+

=

∞

=

−+

+

− +

∑

∑

[A5]

In that case, eqn. [A3] can be simplified as follows:

( ) ( )1 0 1 2 0 2DC DC

k I t k I tPdt

α α

τ

+∫ ≃

( ) ( ) ( ) ( ) ( )1 0 1 2 0 2 1 0 1 2 0 21 sin

1

DC DC DC DCk I k I k I k Itt tPdt Pdt

DCk e

e Qe dt e e dt

α α α αα ωτ τ τ τ

τ

− + + − ∫ ∫∫ ∫≃ [A6]

This will oscillate about

( ) ( )( )

( ) ( )1 0 1 2 0 2 1 0 1 2 0 2

1 0 1I

DC DC DC DCk I k I k I k It t

DCk

e e dt

α α α α

τ τ τ τα

τ

− + + ∫≃ [A7]

( )

( ) ( )1 0 1

1 0 1 2 0 2

I

I I

DC

DC DC

k

k k

α

α α+≃ [A8]

At sufficiently long time, 0Pdt

e−∫ → . Hence, the solution at high frequencies and at long

time will oscillate about

( )

( ) ( )

( )( )

( )( )

2 0 2 01 0 1 0

2

1 0 1 0 2 0 2 0

0 01

1

1

DC acDC acf

DC ac DC ac

iDC i aci

k I b Vk I bV

k I bV k I b V

k I bV

θ

=

= =+

∑ [A9].

26

It may be noted that the restriction of Langmuir isotherm model allows for the calculation

of fθ at the limiting case of long time and high frequencies.

27

Figure Captions. (Color online for all figures)

Fig. 1. The transient and average surface coverage of the adsorbed intermediate as a

function of time for various frequencies. The parameter values are: k10 = 10−11

mol cm−2

s−1

, b1 = 15 V-1

, k20 = 10−9

mol cm−2

s−1

, b2 = 5 V−1

, τ = 10−7

mol cm−2

, Vdc = 0.5 V and

Vac0 = 10 mV. For 10 Hz and 100 Hz frequencies, the inset shows the transient surface

coverage, at the initial stage. Frequency is (a) 100 Hz (b) 10 Hz (c) 1 Hz (d) 100 mHz .

Average surface coverage

Transient surface coverage

Fig.2. The average surface coverage av

Mθ + after 25 s, as a function of frequency for two

different perturbation amplitudes. Parameters employed are the same as those for fig. 1.

Continuous lines are the analytical results and points are the numerical results.

Fig. 3. Complex plane plots of impedance spectra for two different conditions. Cdl = 10−6

F cm−2

. All other parameters employed are the same as those in fig. 1. Vac0 is (a) 10 mV

(b) 100 mV.

Stable conditions

Drifting conditions

Fig. 4. Bode plots of Kramers-Kronig Transforms of the impedance data for the systems

shown in fig. 3. Continuous lines are the original data and the points are the transformed

results. (a) Vac0 = 10 mV, data acquired under drifting conditions (b) Vac0 = 10 mV, data

28

acquired under stable conditions (c) Vac0 = 100 mV, data acquired under drifting

conditions, (d) Vac0 = 100 mV, data acquired under stable conditions

Fig. 5. Average surface coverage of adsorbed intermediates and vacant metal sites as a

function of time for mechanism in eqn. [6], for ac perturbation at 100 Hz frequency. The

parameter values are : k10 = 4 x 10−12

mol cm−2

s−1

, b1 = 17 V-1

, k20 = 10−11

mol cm−2

s−1

,

b2 = 15 V−1

, k30 = 3 x 10−11

mol cm−2

s−1

, b3 = 13 V−1

, τ = 10−7

mol cm−2

, Vdc = 0.5 V.

Vac0 = 100 mV. Continuous lines are the numerical results and points are the heuristic

results.

Fig. 6. Average surface coverage of adsorbed intermediates and vacant metal sites as a

function of time for mechanism in eqn. [6], for ac perturbation at 100 Hz frequency. The

parameter values are : k10 = 3 x 10−14

mol cm−2

s−1

, b1 = 30 V-1

, k20 = 10−9

mol cm−2

s−1

,

b2 = 15 V−1

, k30 = 10−9

mol cm−2

s−1

, b3 = 6 V−1

, τ = 10−7

mol cm−2

, Vdc = 0.5 V. Vac0 =

100 mV. Continuous lines are the numerical results and points are the heuristic results.

Fig.7. Complex plane plots of impedance spectra for the system corresponding to fig. 5.

Cdl = 10−6

F cm−2

. Vac0 is varied and all other parameters are the same as those in Fig. 5.

The inset shows the low frequency impedance for Vac0 of 200 mV, expanded for clarity

X Vac0 = 1 mV

Vac0 = 10 mV

Vac0 = 100 mV

Vac0 = 200 mV

29

Fig. 8. Complex plane plots of impedance spectra for the system corresponding to fig. 6.

Cdl = 10−6

F cm−2

. Vac0 is varied and all other parameters are the same as in fig. 6.

X Vac0 = 1 mV

Vac0 = 10 mV

Vac0 = 100 mV

Vac0 = 200 mV

Fig 9. Bode plots of Kramers Kronig Transforms of the impedance data shown in fig. 7.

Continuous lines are the original data and points are the transformation results

Fig. 10. Bode plots of Kramers Kronig Transforms of the impedance data shown in fig. 8.

Continuous lines are the original data and points are the transformation results

30

Fig. 1 (a)

0.5972

0.5988

0 10 20 30t / s

θθ θθ

V ac0 = 10 mV

f = 100 Hz

av

Mθ + M

θ +

Fig. 1 (b)

0.5972

0.5988

0 10 20 30

t / s

θθ θθ

V ac0 = 10 mV

f = 10 Hz

av

Mθ +

Mθ +

31

Fig. 1 (c)

0.597

0.6

0 1 2 3t / s

θθ θθ

Starting PointV ac0 = 10 mV

f = 1 Hz

av

Mθ +

Mθ +

Fig. 1 (d)

0.585

0.615

0 10 20 30t / s

θθ θθ

V ac0 = 10 mV

f = 100 m Hz

av

Mθ +

Mθ +

32

Fig. 2

0.596

0.599

0.001 10 100000f / Hz

θθ θθav-f

inal

0.55

0.75

10 mV

100 mV

θa

v-f

inal

V ac0 = 10 mV

V ac0 = 100 mV

33

Fig. 3 (a)

0

50

100

0 50 100 150 200

Z re /ohm

-Zim

/ o

hm

V ac0 = 10 mV

Fig. 3 (b)

0

50

100

0 50 100 150 200

Z re / ohm

-Zim

/o

hm

V ac0 = 100 mV

34

Fig. 4 (a)

Drifting

0

200

0.001 10 100000f /Hz

Imp

ed

an

ce /

oh

m V ac0 = 10 mVZre

-Zim

Fig. 4 (b)

Stable

0

200

0.001 10 100000

f /Hz

Imp

ed

an

ce

/o

hm V ac0 = 10 mVZre

-Zim

35

Fig. 4 (c)

Drifting

0

200

0.001 10 100000f / Hz

Imp

ed

an

ce

/ o

hm

V ac0 = 100 mV

Zre

-Zim

Fig. 4 (d)

.

Stable

0

200

0.001 10 100000f / Hz

Imp

ed

an

ce /

oh

m

V ac0 = 100 mVZre

-Zim

36

Fig. 5

0.27

0.37

0 10 20 30t / s

θθ θθ

V ac0 = 100 mV

f = 100 Hz Heuristic Results

av

Mθ +

2

av

Mθ +

av

Mθ

37

Fig. 6

av

Mθ +

0

0.025

0 1 2t / s

θθ θθ

V ac0 = 100 mV

f = 100 Hz

Heuristic Result

0.8

0.96

0 1 2t / s

θθ θθ

0

0.18

V ac0 = 100 mV

f = 100 Hzθθ θθ

Heuristic Results

2

av

Mθ +

av

Mθ

38

Fig. 7

0

60

0 60Z re / ohm

-Zim

/ o

hm

100 mV

200 mV

10 mV

1 mV

200 mV

39

Fig. 8

-5

30

0 35Z re / ohm

-Zim

/ o

hm

100 mV

200 mV

10 mV

1 mV

40

Fig. 9

0

60

0.001 10 100000f / Hz

Zre

/ o

hm

Z(Re)KKT10mv

Z(Re)KKT100mv

Z(Re)KKT200mv

V ac0 = 10 mV

V ac0 = 100 mV

V ac0 = 200 mV

0

25

0.001 10 100000

f / Hz

- Zim

/ o

hm

Z(Im)KKT10mv

Z(Im)KKT100mv

Z(Im)KKT200mv

V ac0 = 10 mV

V ac0 = 100 mV

V ac0 = 200 mV

41

Fig. 10

0

60

0.001 10 100000

f / Hz

Zre

/ o

hm

10mvKKT

100mvKKT

200mvKKT

V ac0 = 10 mV

V ac0 = 100 mV

V ac0 = 200 mV

-5

0

5

10

15

0.001 10 100000

f / Hz

- Zim

/ o

hm

10mvKKT

100mvKKT

200mvKKT

V ac0 = 10 mV

V ac0 = 100 mV

V ac0 = 200 mV