Handbook of Electrochemical Impedance Spectroscopy

Handbook of Electrochemical Impedance Spectroscopy Im Z Re Z ELECTRICAL CIRCUITS CONTAINING CPEs ER@SE/LEPMI J.-P. Diard, B. Le Gorrec, C. Montella Hosted by Bio-Logic @ www.bio-logic.info March 9, 3

Contents Circuits containing one CPE 5. Constant Phase Element CPE, symbol Q............ 5. Circuit R+Q............................ 5.. Impedance.......................... 5.. Reduced impedance..................... 6.3 Circuit R/Q............................. 7.3. Impedance.......................... 7.3. Reduced impedance..................... 7.3.3 Pseudocapacitance #.................... 7.3.4 Pseudocapacitance #.................... 8.4 Circuit R/Q+R/Q+.. Voigt................. 9.5 Circuit R +R /Q........................ 9.5. Impedance...........................5. Reduced impedance......................6 Circuit R /R +Q.........................6. Impedance...........................6. Reduced impedance......................7 Transformation formulae betweenr+r/q and R/R+Q.............................7. α = α............................7. α α........................... Circuits made of two CPEs 3. Circuit Q +Q........................... 3.. α = α = α......................... 3.. α α............................ 3..3 Reduced impedance..................... 4. Circuit Q /Q........................... 6.. α = α = α......................... 6.. α α............................ 6..3 Reduced impedance..................... 6 3 Circuits made of one R and two CPEs 9 3. Circuit R /Q + Q....................... 9 3.. α = α = α......................... 9 3.. α α............................ 3. Circuit R + Q /Q....................... 3.. α = α = α......................... 3

4 CONTENTS 3.. α α............................ 4 Circuits made of two Rs and two CPEs 3 4. Circuit R /Q +R /Q..................... 3 4. Circuit R +R /Q /Q..................... 4 4.3 Circuit Q +R /Q /R..................... 5 4.4 Circuit Q +R /R /Q..................... 6 A Symbols for CPE 9

Chapter Circuits containing one CPE. Constant Phase Element CPE, symbol Q Figure.: Most often used symbol for CPE see also the Appendix A. Z = Q i ω α, Re Z = c α Q ω α, Im Z = s α Q ω α c α = cos π α, s α = sin π α Z = Q ω α, φ Z = π α The Q unit Fcm s α depends on α.. Circuit R+Q.. Impedance R + Different equations for CPE: Z = Q iω α, Re Z = R + c α Q ω α, Im Z = s α Q ω α Q i ω α [5], Z = Q i ω α [6]. 5

6 CHAPTER. CIRCUITS CONTAINING ONE CPE Im Z Α Π Re Z Im Y Α Π Re Y Figure.: Nyquist diagram of the impedance and admittance for the CPE element, plotted for α =.8. The arrows always indicate the increasing frequency direction. R Q Figure.3: Circuit R+Q... Reduced impedance The τ unit depends on α: u τ = s α. Z ω = Zω R = + τ i ω α, τ = R Q Z u = + i u α, u = ω τ/α - Im Z * u c = Im Y *.5 u c = Re Z * Re Y * Figure.4: Nyquist diagram of the reduced impedance and admittance Y = R Y for the R+Q circuit, plotted for α =.8.

.3. CIRCUIT R/Q 7.3 Circuit R/Q Q R Figure.5: Circuit R/Q..3. Impedance R + τ i ω α ; τ = R Q R + τ ω α c α R τ ω α s α Re + τ ω α + τ ω α ; Im c α + τ ω α + τ ω α c α.3. Reduced impedance Z ω = Zω R = + τ i ω α ; τ = R Q Re Z + τ ω α c α ω = + τ ω α + τ ω α ; Im Z τ ω α s α ω = c α + τ ω α + τ ω α c α dim Z ω dω = α τ ω +α + τ ω α s α + τ ω α + τ ω α c α = ω α c = /τ [6] Fig..6 s α Re Z ω c = /, Im Z ω c = + c α α = π arccos + + 4 Im Z ω c Z u = + i u α, u = ω τ /α.3.3 Pseudocapacitance # The value of the pseudocapacitance C C/F cm for the R/C circuit giving the same characteristic frequency than that of the R/Q circuit Fig..7 is obtained from: ω c = R Q = /α R C C = Q α R α

8 CHAPTER. CIRCUITS CONTAINING ONE CPE - Im Z *.5 u c = Im Y * u c = Re Z * Re Y * Figure.6: Nyquist diagram of the reduced impedance depressed semi-circle [] and admittance Y = R Y for the R/Q circuit, plotted for α =.8. Q C R R Ω c RQ Α Ω c RC R R Figure.7: R/Q and R/C circuits with the same characteristic frequency at the apex or summit of impedance arc..3.4 Pseudocapacitance # The value of the pseudocapacitance C C/F cm for the R C /C circuit giving the same impedance for the characteristic frequency of the R Q /Q circuit Fig..7 is obtained from [, 9]: with: C = Q /α R /α Q sinα π/, R C = τ RC/C = R Q Q /α tanα π/4 R Q cosα π/4

.4. CIRCUIT R/Q+R/Q+.. VOIGT 9 Q C R Q R C Im Z Ω c R Q Q Α R C R Q Re Z Figure.8: R Q/Q and R C/C circuits with the same impedance for the characteristic frequency of the R Q/Q circuit..4 Circuit R/Q+R/Q+.. Voigt Re n RQ i= n RQ i= n RQ Im R i + τ i i ω αi ; τ i = R i Q i i=.5 Circuit R +R /Q R i + τ i ω αi c αi + τ i ω αi + τ i ω αi c αi R i τ i ω αi s αi + τ i ω αi + τ i ω αi c αi Q R R Figure.9: Circuit R +R /Q.

CHAPTER. CIRCUITS CONTAINING ONE CPE.5. Impedance R + i ω α Q + R R + R + i ω α τ + i ω α τ, τ = R Q, τ = R R Q R + R.5. Reduced impedance Z u = Zu + T i uα =. R + R + i u α u = τ /α ω, T = τ /τ = R /R + R < Re Z u = T c α u α + c α u α + Tu α + c α u α + u α + Im Z u = s Tuα α c α u α + u α + Im Z Τ Α Ω c Τ Α Im Z T Α u c R R R Re Z T Re Z Figure.: Nyquist diagrams of the impedance and reduced impedance for the R +R /Q circuit..6 Circuit R /R +Q R Q R Figure.: Circuit R /R +Q.

.7. TRANSFORMATION FORMULAE BETWEENR+R/Q AND R/R+Q.6. Impedance R + τ i ω α, τ = R + R Q, τ = R Q + τ i ω α Re R cos πα τ + τ ω α + τ τ ω α + τ τ ω α + cos πα ω α + ω α sin πα R τ τ Im τ τ ω α + cos πα ω α +.6. Reduced impedance Z u = Zu R = + T i uα + i u α u = τ /α ω, T = τ /τ = R /R + R < cf. Eq.. and Fig....7 Transformation formulae betweenr+r/q and R/R+Q.7. α = α Q R R Q R R Figure.: The R+R/Q and R/R+Q circuits are non-distinguishable for α = α []. Transformations formulae R+R/Q R/R+Q R = R + R, R = R Q R + R, Q = R R + R Transformations formulae R/R+Q R+R/Q Q = Q R + R R, R = R R, R = R + R R R + R.7. α α The R+R/Q and R/R+Q circuits Fig.. are distinguishable for α α

CHAPTER. CIRCUITS CONTAINING ONE CPE

Chapter Circuits made of two CPEs. Circuit Q +Q Q Q Figure.: Circuit Q +Q... α = α = α + Q Q i ω α = Q i ω α, Q = Q Q Q + Q cf..... α α Impedance Q i ω + α Q i ω = Q i ω α + Q i ω α α Q Q i ω α+α ω α ω α Re cos πα Q + cos Im sin πα ω α sin Q Z Q = Z Q ω = ω c = Q Q πα Q πα Q ω α α α 3

4 CHAPTER. CIRCUITS MADE OF TWO CPES α < α Figs.. and.3 α > α ω Zω ω Zω Q i ω α, ω Zω Q i ω α Q i ω α, ω Zω Q i ω α 4 Im Z log Im Z Re Z 4 log Re Z Figure.: Nyquist and log Nyquist [8] diagrams of the impedance for the Q +Q circuit, plotted for Q = F cm s α, Q = F cm s α, α =.6, α =.9 α < α. Dots: ω c = Q /Q /α α. 5 Α 45 Α Π log Z Α Φ Z 5 5 Q Q Α Α 9 Α Π 5 5 Q Q Α Α logω rd s logω rd s Figure.3: Bode diagrams of the impedance for the Q +Q circuit. Same values of parameters as in Fig... α < α...3 Reduced impedance Z u = Q ω α c i u α + i u α, u = ω ω c

.. CIRCUIT Q +Q 5 3 Im Z log Im Z 3 Re Z log Re Z Figure.4: Nyquist and log Nyquist [8] diagrams of the reduced impedance for the Q +Q circuit, plotted for α =.6, α =.9 α < α. Dots: u c =. 3 Α 45 Α Π log Z Α Φ Z 3 3 3 9 Α Π 3 3 log u log u Figure.5: Bode diagrams of the impedance for the Q +Q circuit. Same values of parameters as in Fig..4. α < α.

6 CHAPTER. CIRCUITS MADE OF TWO CPES. Circuit Q /Q Q Q Figure.6: Circuit Q /Q... α = α = α cf... Q + Q i ω α = Q i ω α, Q = Q + Q.. α α Impedance Q i ω α + Q i ω α cos πα Q ω α + cos πα Q ω α Re Q + Q ωα + cos ωα π α α Q Q ω α+α sin πα Q ω α + sin πα Q ω α Im Q + Q ωα + cos ωα π α α Q Q ω α+α α < α Figs..7 and.8 α > α ω Zω ω Zω Q i ω α, ω Zω Q i ω α Q i ω α, ω Zω Q i ω α..3 Reduced impedance Z u = Q ωc α i u α + i u, u = ω α ω c

.. CIRCUIT Q /Q 7 3 Im Z 5 log Im Z 5 Re Z 3 log Re Z Figure.7: Nyquist and log Nyquist [8] diagrams of the impedance for the Q /Q circuit plotted for Q = F cm s α, Q = F cm s α, α =.6, α =.9 α < α. Dots: ω c = Q /Q /α α. 5 45 Α Π log Z Α Φ Z Α 5 5 Q Q Α Α 9 Α Π 5 5 Q Q Α Α logω rd s logω rd s Figure.8: Bode diagrams of the impedance for the Q /Q circuit. Same values of parameters as in Fig..7. α < α. Im Z log Im Z Re Z log Re Z Figure.9: Nyquist and log Nyquist [8] diagrams of the reduced impedance for the Q /Q circuit, plotted for α =.6, α =.9 α < α. Dots: u c =.

8 CHAPTER. CIRCUITS MADE OF TWO CPES 3 45 Α Π log Z Α Φ Z 3 Α 3 3 9 Α Π 3 3 log u log u Figure.: Bode diagrams of the impedance for the Q /Q circuit. Same values of parameters as in Fig..9. α < α.

Chapter 3 Circuits made of one R and two CPEs 3. Circuit R /Q + Q Q Q R Figure 3.: Circuit R /Q +Q. 3.. α = α = α Impedance + + Q i ω R α Q i ω α + i ω α τ i ω α Q + i ω α τ, τ = R Q, τ = Q + Q R, τ < τ Re cos πα τ τ ω α + ω α + cosπατ + τ Q τ τ ω α + cos πα ωα + Im sin πα τ τ ω α + ω α + sinπατ Q τ τ ω α + cos πα ωα + 9

CHAPTER 3. CIRCUITS MADE OF ONE R AND TWO CPES Reduced impedance Z u = Zu R = T + T i u α i u α + i u α 3. u = ω τ /α, T = τ /τ = + Q /Q > Re Z u = u α T + cosαπu α + Tu α + cos απ T cos απ uα + u α + Im Z u = u α T u α απ cos απ sin uα + u α + Im Z Re Z Figure 3.: Nyquist diagram of the reduced impedance for the R /Q +Q circuit Fig. 3., Eq. 3., plotted for T = 4, 9,9 and α =.85. The line thickness increases with increasing T. Dots: reduced characteristic angular frequency u c = ; circles: reduced characteristic angular frequency u c = /T /α φ uc = φ uc. 3.. α α Impedance Re cos πα ω α + Q Im + Q + Q i ω R α i ω α R cos πα Q R ω α + Q R Q R ω α + cos πα sin πα Q R ωα Q R Q R ω α + cos πα sin πα ω α + ω α + Q ω α

3.. CIRCUIT R + Q /Q Q Q R Figure 3.3: Circuit R +Q /Q. 3. Circuit R + Q /Q 3.. α = α = α Impedance i ω α Q + R + i ω α Q + Q R i ω α = i ω α Q + Q + i ωα Q Q R Q + Q + τ i ω α i ω α Q + Q + i ω α τ, τ = Q Q R, τ = Q R Q + Q Re ω α cosπαω α + τ ω α + cos πα τ ω α + cos πα ωα + ω α + Q + Q τ Im ω α sin πα cos πα ω α + τ ω α + cos πα ωα + ω α + Q + Q τ Reduced impedance Z u = Zu = T R T + T i u α i u α + i u α 3. u = ω τ /α, T = τ /τ = + Q /Q > Re Z u = T u α T + cosαπu α + Tu α + cos απ T cos απ uα + u α + Im Z u = T u α cos απ u α + Tu α + sin απ T cos απ uα + u α + 3.. α α + R i ω α Q i ω α Q + + R i ω α Q i ω α Q

CHAPTER 3. CIRCUITS MADE OF ONE R AND TWO CPES Im Z Re Z Figure 3.4: Nyquist diagram of the reduced impedance for the R +Q /Q circuit Fig. 3.3, Eq. 3., plotted for T = 4, 9,9 and α =.85. The line thickness increases with increasing T. Dots: reduced characteristic angular frequency u c = ; circles: reduced characteristic angular frequency u c = /T /α φ uc = φ uc. + τ i ω α i ω α Q + i ω α Q + τ i ω α+α Q, τ = R Q Re ω α c α + τ ω α + τ ω α c α Q + ω α τ ω α + c α Q / ω α + τ ω α + τ ω α c α Q + ω α+α τ ω α c α + c αmα Q Q + ω α Q π α α c αmα = cos Im ω α + τ ω α + τ ω α c α Q s α ω α Q s α / ω α + τ ω α + τ ω α α Q + ω α+α τ ω α c α + c αmα Q Q + ω α Q

Chapter 4 Circuits made of two Rs and two CPEs 4. Circuit R /Q +R /Q Q Q R R Figure 4.: Circuit R /Q +R /Q. i ω α Q + R + i ω α Q + R R R +, τ + i ω α τ + iω α = R Q, τ = R Q τ Re R + R + i ω α R τ + i ω α R τ + i ω α τ + i ω α τ R + ω α c α τ + ω α τ c α + ω α τ + R + ω α c α τ + ω α τ c α + ω α τ ω α R s α τ Im + ω α τ c α + ω α τ ω α R s α τ + ω α τ c α + ω α τ 3

4 CHAPTER 4. CIRCUITS MADE OF TWO RS AND TWO CPES Im Z.3 u c u c.5 Re Z Figure 4.: Nyquist diagrams of the reduced impedance for the R /Q +R /Q circuit Fig. 4.. R = R, α = α, Q Q..5 u c u c.5 u c u c Im Z Im Z.5 Re Z Π 4 Π 4.5 Re Z Figure 4.3: Unusual Nyquist diagrams of the reduced impedance for the R /Q +R /Q circuit Fig. 4.. R = R, Q = Q, α =. Left: α =.3, right: α =.5. 4. Circuit R +R /Q /Q i ω α Q + R + i ω α Q + R R + R + i ω α Q R R + i ω α Q R + R + i ω α Q R + i ω α+α Q Q R R Q Q R R Figure 4.4: Circuit R +R /Q /Q.

4.3. CIRCUIT Q +R /Q /R 5 Re R + R + ω α Q R + ω α C α Q R R + ω α C α Q R + R + ω α C α Q R R + R + ω α C α Q R + R / + ω α Q + ω α Q R C α + ω α Q R R + ω α Q R + R C α + ω α Q R + R + ω α Q R C α + ω α Q C αmα R + C α R C α + ω α Q R + R π α α c αmα = cos Im ω α Q ω α Q R R ω α C α Q R R R + R R + R S α ω α Q R S α / + ω α Q + ω α Q R C α + ω α Q R R + ω α Q R + R C α + ω α Q R + R + ω α Q R C α + ω α Q C αmα R + C α R C α + ω α Q R + R 4.3 Circuit Q +R /Q /R R Q Q R Figure 4.5: Circuit Q +R /Q /R. R + i ω α Q + i ω α Q + R R + i ω α Q R + i ω α Q R + i ω α Q R + R + i ω α Q R + i ω α+α Q Q R R

6 CHAPTER 4. CIRCUITS MADE OF TWO RS AND TWO CPES Re R + ω α Q R C α + ω α Q R + ω α Q R R + R + ω α C α Q R + ω α Q R C α + ω α C αmα Q R + C α R + ω α Q R C α + ω α Q R / + ω α Q + ω α Q R C α + ω α Q R R + ω α Q R + R C α + ω α Q R + R + ω α Q R C α + ω α Q C αmα R + C α R C α + ω α Q R + R Im ω α Q R S α + ω α Q R C α + ω α Q R S α + ω α Q R S α / + ω α Q + ω α Q R C α + ω α Q R R + ω α Q R + R C α + ω α Q R + R + ω α Q R C α + ω α Q C αmα R + C α R C α + ω α Q R + R R + τ i ω α + τ i ω α + + R /R τ iω α + τ i ω α + τ τ R /R i ω α+α τ = Q R, τ = Q R 4.4 Circuit Q +R /R /Q Q R Q R Figure 4.6: Circuit Q +R /R /Q. i ω α Q + + R + R i ω α Q R + iω α Q R + i ω α Q R + i ω α Q R + i ω α Q R + i ω α+α Q Q R R

4.4. CIRCUIT Q +R /R /Q 7 Re R + ω α Q ω α Q R R + R + C α R + R + ω α C α Q R + ω α Q R C α + ω α Q R / + ω α Q R + R C α + ω α Q R + R + ω α Q R + ω α Q R C α + ω α Q R + ω α Q R C α + ω α Q C αmα R + C α C α R + ω α C α Q R R + R Im R ω α Q + ω α Q R C α + ω α Q R S α ω α Q S α / + ω α Q R + R C α + ω α Q R + R + ω α Q R + ω α Q R C α + ω α Q R + ω α Q R C α + ω α Q C αmα R + C α C α R + ω α C α Q R R + R R + i ω α τ + i ω α τ + + R /R i ω α τ + i ω α+α τ τ τ = Q R, τ = Q R

8 CHAPTER 4. CIRCUITS MADE OF TWO RS AND TWO CPES

Appendix A Symbols for CPE 9

3 APPENDIX A. SYMBOLS FOR CPE A B C CPE D E F G H Q I J K L M N O P Q R S T Figure A.: Some CPE symbols, taken from A: [3], B: [9], C: [3], D: [5], E: [], F: [7], G: [8], H: [], I: [], J: [5], K: [], L: [, 9], M: [], N: [3, 4], O: [4], P: [4], Q, R [5], S [7], T [6].

Bibliography [] Berthier, F., Diard, J.-P., and Montella, C. Distinguishability of equivalent circuits containing CPEs. I. Theoretical part. J. Electroanal. Chem. 5,. [] Boillot, M. Validation expérimentale d outil de modélisation d une pile à combustible de type PEM. PhD thesis, INPL, Nancy, Oct. 5. [3] Bommersbach, P., Alemany-Dumont, C., Millet, J., and Normand, B. Formation and behaviour study of an environment-friendly corrosion inhibitor by electrochemical methods. Electrochim. Acta 5, 6 5, 76 84. [4] Bommersbach, P., Alemany-Dumont, C., Millet, J.-P., and Normand, B. Hydrodynamic effect on the behaviour of a corrosion inhibitor film: Characterization by electrochemical impedance spectroscopy. Electrochimica Acta 5, 9 6, 4 48. [5] Brug, G. J., van den Eeden, A. L. G., Sluyters-Rehbach, M., and Sluyters, J. H. The analysis of electrode impedance complicated by the presence of a constant phase element. J. Electroanal. Chem. 76 984, 75 95. [6] Chabli, A., Diaco, T., and Diard, J.-P. Détermination de la séquence de pondération d une pile leclanché à l aide d une méthode d intercorrélation. J. Applied Electrochem. 98, 66. [7] Ding, S.-J., Chang, B.-W., Wu, C.-C., Lai, M.-F., and Chang, H.- C. Impedance spectral studies of self-assembly of alkanethiols with different chain lengths using different immobilization strategies on au electrodes. Analytica Chimica Acta 554 5, 43 5. [8] Fournier, J., Wrona, P. K., Lasia, A., Lacasse, R., Lalancette, J.-M., Menard, H., and Brossard, L. J. Electrochem. Soc. 39 99, 37. [9] Franck-Lacaze, L., Bonnet, C., Besse, S., and Lapicque, F. Effect of ozone on the performance of a polymer electrolyte membrane fuel cell. Fuel Cells 9 9, 56 569. [] Han, D. G., and Choi, G. M. Computer simulation of the electrical conductivity of composites: the effect of geometrical arrangement. Solid State Ionics 6 998, 7 87. 3

3 BIBLIOGRAPHY [] Hernando, J., Lud, S. Q., Bruno, P., Gruen, D. M., Stutzmann, M., and Garrido, J. A. Electrochemical impedance spectroscopy of oxidized and hydrogen-terminated nitrogen-induced conductive ultrananocristalline diamond. Electrochim. Acta 54 9, 99 95. [] Holzapfel, M., Martinent, A., Alloin, F., B. Le Gorrec, Yazami, R., and Montella, C. First lithiation and charge/discharge cycles of graphite materials, investigated by electrochemical impedance spectroscopy. J. Electroanal. Chem. 546 3, 4 5. [3] Jansen, R., and Beck, F. Electrochim. Acta 39 994, 9. [4] Jonscher, A. K., and Bari, M. A. Admittance spectroscopy of sealed primary batteries. J. Electrochem. Soc. 35 988, 68 65. [5] Kulova, T. L., Skundin, A. M., Pleskov, Y. V., Terukov, E. I., and Kon kov, O. I. Lithium insertion into amorphous silicon thin-film electrode. J. Electroanal. Chem. 6 7, 7 5. [6] Laes, K., Bereznev, S., Land, R., Tverjanovich, A., Volobujeva, O., Traksmaa, R., Raadik, T., and Öpik, A. The impedance spectroscopy of photoabsorber films prepared by high vacuum evaporation technique. Energy Procedia,, 9 3. [7] Matthew-Esteban, J., and Orazem, M. E. On the application of the Kramers-Kronig relations to evaluate the consistency of electrochemical impedance data. J. Electrochem. Soc. 38 99, 67 76. [8] Messaoudi, B., Joiret, S., Keddam, M., and Takenouti, H. Anodic behaviour of manganese in alkaline medium. Electrochim. Acta 46, 487 498. [9] Orazem, M. E., Shukla, P. S., and Membrino, M. A. Extension of the measurement model approach for deconvolution of underlying distributions for impedance measurements. Electrochim. Acta 47, 7 34. [] Quintin, M., Devos, O., Delville, M. H., and Campet, G. Study of the lithium insertion-deinsertion mechanism in nanocrystalline γ-fe O 3 electrodes by means of electrochemical impedance spectroscopy. Electrochim. Acta 5 6, 646 6434. [] Seki, S., Kobayashi, Y., Miyashiro, H., Yamanaka, A., Mita, Y., and Iwahori, T. Degradation mechanism analysis of all solid-state lithium polymer rechargeable batteries. In IMBL Meeting 4, The Electrochemical Society. Abs. 386. [] Sluyters-Rehbach, M. Impedance of electrochemical systems: terminology, nomenclature and representation-part I: cells with metal electrodes and liquid solution IUPAC Recommendations 994. Pure & Appl. Chem. 66 994, 83 89. [3] Tomczyk, P., and Mosialek, M. Investigation of the oxygen electrode reaction in basic molten carbonates using electrochemical impedance spectroscopy. Electrochim. Acta 46, 33 33.

BIBLIOGRAPHY 33 [4] VanderNoot, T. J., and Abrahams, I. The use of genetic algorithms in the non-linear regression of immittance data. J. Electroanal. Chem. 448 998, 7 3. [5] Yang, B. Y., and Kim, K. Y. The oxidation behavior of Ni-5%Co alloy electrode in molten Li + K carbonate eutectic. Electrochim. Acta 44 999, 7 34. [6] Zoltowski, P. J. Electroanal. Chem. 443 998, 49.