Supplementary material
Detection of nonlinearities in electrochemical
impedance spectra by Kramers Kronig Transforms
Authors: Fathima Fasmina, Ramanathan Srinivasana*
Address:
a
Department of Chemical Engineering, Indian Institute of Technology-Madras, Chennai
600036, India.
Phone : +91 44 2257 4171
Fax: +91 44 2257 0509
Email: srinivar@iitm.ac.in
* corresponding author
Index:
1. Fig. S1. (a) Sum of squares vs. number of Voigt elements for the data in Fig. 7. Normalized
residual error resulting from a measurement model fit with 95% confidence intervals (b) Vac0 =
300 mV, real and (c) Vac0 = 300 mV, imaginary parts.
2. Fig. S2. (a) Log |iF| as a function of dc potential for the data in Fig. 8 (b) sum of squares vs.
number of Voigt elements. Normalized residual error resulting from a measurement model fit
with 95% confidence intervals (c) Vac0 = 300 mV, real and (d) Vac0 = 300 mV, imaginary parts. (e)
Residual error resulting from linear KKT fit at Vac0 = 300 mV.
3. Fig. S3. Impedance spectrum for three step reaction with k10 = 10-10 mol cm-2 s-1, b1 =15 V-1,
k20 = 10-12 mol cm-2 s-1, b2 = 30 V-1, k30 = 2 ×10-8 mol cm-2 s-1, b3 = 0 V-1, Γ = 10-8 mol cm-2 and
Vdc = 0.25 V. (a) Complex plane plots of impedance spectra Vac0 = 1 mV and 150 mV (b) log|i F|
as a function of dc potential. Bode plots of (c) |Z| and (d) at Vac0 = 150 mV with results of direct
integration of KKT. (e) sum of squares vs. number of Voigt elements. Normalized residual error
resulting from a measurement model fit with 95% confidence intervals (f) Vac0 = 150 mV, real
and (g) Vac0 = 150 mV, imaginary part. (h) Residual error resulting from linear KKT fit at Vac0 =
300 mV.
4. Fig. S4. Example illustrating that when log|iF| vs. dc potential is linear, direct integration KKT
and measurement model analysis do not flag the nonlinearity, but linear KKT successfully flags
the nonlinearity. Impedance spectrum for two step reaction with k10 = 10-12 mol cm-2s-1, b1 =10 V, k20 = 10-9mol cm-2s-1, b2 = 9 V-1, and Γ = 10-8mol cm-2. (a) Complex plane plots of impedance
1
spectra Vac0 = 1 mV and 250 mV with Vdc = 0.7 V (b) log |iF| as a function of dc potential. Bode
plots of (c) |Z| and (d)  at Vac0 = 250 mV with results of direct integration of KKT (e) Residual
error resulting from measurement model fit at Vac0 = 250 mV. The residuals were normalized with
the magnitude of impedance (f) Residual error resulting from linear KKT fit at Vac0 = 250 mV.
5. Fig. S5 Example illustrating that when log|iF| vs. dc potential is nonlinear, all three data
validation methods successfully flag the nonlinearity. Impedance spectrum for two step reaction
with k10 = 10-9 mol cm-2s-1, b1 =3 V-1, k20 = 10-11 mol cm-2s-1, b2 = 15 V-1, and Γ = 10-8 mol cm-2. (a)
Complex plane plots of impedance spectra Vac0 = 1 mV and 300 mV with Vdc = 0.5 V (b) log |iF|
as a function of Vdc. Bode plots of (c) |Z| and (d)  at Vac0 = 300 mV with results of direct
integration of KKT (e) Sum of squares vs. number of Voigt elements. Normalized residual error
resulting from a measurement model fit with 95% confidence intervals (f) Vac0 = 300 mV, real
and (g) Vac0 = 300 mV, imaginary parts. (h) Residual error resulting from linear KKT fit at Vac0 =
300 mV.
6. Fig. S6. Data validation for the spectra given in Fig 9b after adding an equivalent resistance of
100 -cm2 in series. Bode plots of (a) the magnitude of admittance |Y| and (b) at Vac0 = 1 mV,
with results of direct integration of KKT using data in the admittance form.
7. Fig. S7. Data validation for the spectra given in Fig. 9b. (a) Residual errors in measurement
model fit at Vac0 = 300 mV. The residuals were normalized with the magnitude of impedance (b)
Residual errors resulting from linear KKT fit performed in admittance mode, at Vac0 = 300 mV
8. Table S1. Summary of the results
Fig. S1.
6
10
Sum of squares
Vac0 = 1 mV
Vac0 = 300 mV
4
10
2
10
0
10
-2
10
1
2
3
4
Number of Voigt elements
Fig.S1 (a)
5
Vac0 = 300 mV
(ZIm - ZMM
) / ZIm
Im
(ZRe - ZMM
) / ZRe
Re
0.2
2 Voigt elements
0
-0.2 -3
10
10
0
frequency/ Hz
Fig.S1 (b)
10
3
Vac0 = 300 mV
2 Voigt elements
0
-5 -3
10
10
0
10
3
frequency/ Hz
Fig.S1 (c)
Fig. S2.
10
10
-2
log ( iF / A cm )
Sum of squares
-6
5
10
-9
Vac0 = 1 mV
Vac0 = 300 mV
0
10
-12
-5
10
0.1
0.5
V /V
0.9
1
2
3
4
5
Number of Voigt elements
dc
Fig.S2 (a)
Fig.S2 (b)
1.5
/ ZIm
Vac0 = 300 mV
3 Voigt elements
ZMM
)
Im
0
0
(ZIm -
(ZRe - ZMM
) / ZRe
Re
0.5
Vac0 = 300 mV
3 Voigt elements
-0.5 -3
10
10
0
10
3
frequency/ Hz
real, imag/ %
Vac0 = 300 mV
0
real part
imaginary part
0
Fig.S2 (e)
0
10
Fig.S2 (d)
20
10
frequency/ Hz
10
frequency/ Hz
Fig.S2 (c)
-75 -3
10
-1.5 -3
10
3
10
3
Fig. S3.
230
-6
log ( iF / A cm )
Vac0 = 1mV
-ZIm / cm
2
-2
Vac0 = 150mV
-9
-12
0
-20
0
Z
Re
/ cm
0.1
230
2
0.5
V /V
0.9
dc
Fig.S3 (a)
Fig.S3 (b)
3
10
130
simulated data
KKT by integration
|Z| / cm
2
simulated data
KKT by integration

Vac0 = 150 mV
0
10 -3
10
0
0
-3
10
3
10
10
frequency / Hz
6
Sum of squares
Vac0 = 1 mV
Vac0 = 150 mV
2
10
0
10
-2
10
1
2
3
10
Fig.S3 (d)
10
4
0
10
frequency / Hz
Fig.S3 (c)
10
Vac0 = 150 mV
3
4
Number of Voigt elements
Fig.S3 (e)
5
5
Vac0 = 150 mV
(ZIm - ZMM
) / ZIm
Im
(ZRe - ZMM
) / ZRe
Re
0.05
3 Voigt elements
0
-0.05 -3
10
10
0
10
3
frequency/ Hz
real, imag/ %
real part
imaginary part
0
Vac0 = 150 mV
0
Fig.S3 (h)
0
-5 -3
10
10
0
10
Fig.S3 (g)
10
10
frequency/ Hz
3 Voigt elements
frequency/ Hz
Fig.S3 (f)
-5 -3
10
Vac0 = 150 mV
3
10
3
Fig. S4.
600
-6
-2
log ( iF / A cm )
Vac0 = 1 mV
-ZIm / cm
2
Vac0 = 250 mV
-10
-14
0
0
Z
Re
/ cm
0.1
600
2
0.5
V /V
0.9
dc
Fig.S4 (a)
Fig.S4 (b)
130
3
simulated data
KKT by integration
simulated data
KKT by integration
|Z| / cm
2
10

Vac0 = 250 mV
0
10 -3
10
0
0
-3
10
3
10
Vac0 = 250 mV
10
frequency / Hz
0
3
10
10
frequency / Hz
Fig.S4 (c)
Fig.S4 (d)
1
real part
imaginary part
0
Vac0 = 250 mV
real, imag/ %
real, imag/ %
1
real part
imaginary part
0
Vac0 = 250 mV
2 Voigt elements
-1 -3
10
0
10
frequency/ Hz
Fig.S4 (e)
3
10
-1 -3
10
0
10
frequency/ Hz
Fig.S4 (f)
3
10
Fig S5.
-5
300
-2
log ( iF / A cm )
Vac0 = 1mV
-ZIm / cm
2
Vac0 = 300mV
-8
-11
0
0
Z
Re
/ cm
2
0.1
300
0.5
V /V
0.9
dc
Fig.S5 (a)
Fig.S5 (b)
4
10
130
simulated data
KKT by integration
|Z| / cm
2
simulated data
KKT by integration

Vac0 = 300 mV
0
10 -3
10
0
0
-3
10
3
10
10
frequency / Hz
Fig.S5 (d)
6
Sum of squares
10
4
10
2
Vac0 = 1 mV
Vac0 = 300 mV
0
10
-2
10
1
2
0
10
frequency / Hz
Fig.S5 (c)
10
Vac0 = 300 mV
3
Number of Voigt elements
Fig.S5 (e)
4
3
10
10
Vac0 = 300 mV
(ZIm - ZMM
) / ZIm
Im
(ZRe - ZMM
) / ZRe
Re
0.08
2 Voigt elements
0
-0.08 -3
10
10
0
10
3
frequency/ Hz
real, imag/ %
Vac0 = 300 mV
real part
imaginary part
0
10
frequency/ Hz
Fig.S5 (h)
0
-10 -3
10
10
0
10
Fig.S5 (g)
0
-20 -3
10
2 Voigt elements
frequency/ Hz
Fig.S5 (f)
5
Vac0 = 300 mV
3
10
3
Fig. S6.
V
ac0
-1
V
= 1 mV
ac0
200
100
-2
10

|Y| / -1 cm-2
10
= 1 mV
Simulated data +100 
KKT by Integration
-3
10
-3
10
0
10
Frequency / Hz
Fig.S6 (a)
3
10
0
-100
Simulated data +100 
KKT by Integration
-3
10
0
10
Frequency / Hz
Fig.S6 (b)
3
10
Fig. S7.
real, imag/ %
35
real part
imaginary part
Vac0 = 300 mV
0
3 Voigt elements
-35 -3
10
0
10
frequency/ Hz
Fig.S7 (a)
3
10
Fig.S7 (b)
Table S1.
Parameter
sets
Is log(iF)
vs. Vdc
nonlinear?
Does KKT
detect
nonlinearity?
Does linear
KKT detect
nonlinearity?
True
Does
measurement
model detect
nonlinearity?
True
Fig. S3
True
Fig. S4
Fig. S5
Estimated
True
80.8
False
False
True
True
True
True
True
True
240.0
225.5
Rct,NL (cm)