Detection of Nonlinearities in Electrochemical Impedance Spectra by Kramers Kronig Transforms

Supplementary material

Detection of nonlinearities in electrochemical impedance spectra by Kramers Kronig Transforms

Authors:Fathima Fasmina, Ramanathan Srinivasana*

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a Department of Chemical Engineering, Indian Institute of Technology-Madras, Chennai 600036, India.

Phone: +91 44 2257 4171

Fax: +91 44 2257 0509

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* 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|iF| 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-1, k20 = 10-9mol cm-2s-1, b2 = 9 V-1, and Γ = 10-8mol cm-2. (a) Complex plane plots of impedance 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. S5Example 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 = 10-11mol cm-2s-1, b2 = 15 V-1, and Γ = 10-8mol 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.

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Table S1.