Atlas of Electrochemical Nyquist Diagrams. I. One semicircle and straight line A guide to select equivalent circuits

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1 Atlas of Electrochemical Nyquist Diagrams. I. One semicircle and straight line A guide to select equivalent circuits Π 4 Π 4 September 28, 211

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3 Contents 1 Nyquist diagrams made of one straight line Capacitive straight line element circuit Q1 element Q1 circuit Q1 element, α1 = 1/ W1 element Q1 circuit, α1 = 1/ W1 circuit Inductive straight line L1 element L1 circuit Q1 element, α1 < Q1 element, α1 < L1+ circuit L1+ circuit Nyquist diagrams made of one semicircle One capacitive semicircle / circuit /C2 circuit /(+C2) circuit One inductive semicircle /L1 circuit /L1 circuit, < /L2 circuit /L2 circuit, < Nyquist diagrams made of one semicircle and straight line Capacitive semicircle and straight line /(C2+) circuit /C2 circuit /(C2+) circuit /C2 circuit /(Q2+) circuit Q1+/C2 circuit /(Q2+) circuit

4 4 CONTENTS Q1+/C2 circuit /(+W2) circuit /(+W2) circuit W1+/C2 circuit W1+/C2 circuit Capacitive semicircle and inductive straight line L1+/C2 circuit L1+/C2 circuit Q1+/C2 circuit, α1 < Q1+/C2 circuit, α1 < Inductive semicircle and straight line /L1+L2 circuit L1/(+L2) circuit /L2+L3 circuit L2/(+L3) circuit /L1+Q1 circuit, α1 < /L2+Q1 circuit, α1 < Inductive semicircle and capacitive straight line /L1+ circuit /L2+ circuit /L1+Q1 circuit /L2+Q1 circuit /L1+W1 circuit /L2+W1 circuit

5 Chapter 1 Nyquist diagrams made of one straight line 1.1 Capacitive straight line Table 1.1: Six different Nyquist diagrams made of one capacitive straight line. cf , p. 5 cf , p. 6 cf , p. 6 cf , p. 6 cf , p. 6 cf , p. 6 cf , p. 6 cf , p. 7 Π 4 Π 4 ( 1 ) element Fig Figure 1.1: element. For an electrochemical system denotes the electrochemical double layer capacity C dl. 1 Nyquist diagrams are always plotted using convention of electrochemists: orthonormal parametric plot Im Z vs. Re Z. The arrows always indicate the increasing frequency direction. 5

6 6 CHAPTER 1. NYQUIST DIAGRAMS MADE OF ONE STRAIGHT LINE circuit Fig Figure 1.2: + circuit. For an ideally polarized electrode denotes the electrochemical double layer capacity C dl and the ohmic (R Ω ) or uncompensated resistance (R u ) [1] Q1 element Fig Q1 Figure 1.3: Q1 element (CPE element, cf. Handbook of Electrochemical Impedance Spectroscopy. Electrical circuits containing CPEs: Q1 circuit Fig. 1.4 [1, 2]. Q1 Figure 1.4: +Q1 circuit [1, 2] Q1 element, α1 = 1/2 Fig. 1.3 with α1 = 1/ W1 element Fig Q1 circuit, α1 = 1/2 Fig. 1.4 with α1 = 1/2.

7 1.2. INDUCTIVE STRAIGHT LINE 7 W1 Figure 1.5: W1 element (Warburg impedance, cf. Handbook of Electrochemical Impedance Spectroscopy. Diffusion impedances: W1 circuit Fig W1 Figure 1.6: +W1 circuit. 1.2 Inductive straight line Table 1.2: Four different Nyquist diagrams made of one inductive straight line and two diagrams made of one capacitive and one inductive straight line. cf , p. 7 cf , p. 8 cf , p. 8 cf , p. 8 cf , p. 8 cf , p L1 element Fig L1 Figure 1.7: L1 element.

8 8 CHAPTER 1. NYQUIST DIAGRAMS MADE OF ONE STRAIGHT LINE L1 circuit Fig L1 Figure 1.8: +L1 circuit Q1 element, α1 < Fig. 1.3 with α1 < Q1 element, α1 < Fig. 1.4 with α1 < L1+ circuit Fig L1 L1 Figure 1.9: L1+ circuit L1+ circuit Fig L1 L1 Figure 1.1: +L1+ circuit.

9 Chapter 2 Nyquist diagrams made of one semicircle 2.1 One capacitive semicircle Table 2.1: Four different Nyquist diagrams made of one capacitive semicircle. cf , p. 9 cf , p. 9 cf , p. 11 cf , p. 11 cf , p / circuit Figs. 2.1, 2.2 ( 1 )., > Fig <, > Fig Replacing C by Q (CPE element) in a parallel RC circuit changes a semicircle in a depressed semicircle arc, cf. Handbook of Electrochemical Impedance Spectroscopy. Electrical circuits containing CPEs: 9

10 1 CHAPTER 2. NYQUIST DIAGRAMS MADE OF ONE SEMICIRCLE Figure 2.1: / circuit. For an electrochemical system denotes the electrochemical double layer capacity and the charge transfer resistance R ct. Ω c 1 Ω c 1 Figure 2.2: Nyquist diagrams of the impedance for the / circuit. Left:, >, right <, >. Voigt circuit with one time constant Figs. 2.3, 2.4. Z(ω) = n i=1 n Ri 1 + i ω τ = i=1 Ri, τ = Ri Ci 1 + i ω τ C2 C3 Cn R3 Rn Figure 2.3: Voigt circuit /C2 circuit Figs. 2.5 ( 2 ). 2 The +/C2 and /(+C2) circuits are non-distinguishable, i.e. these circuits can be interchanged (cf. Handbook of Electrochemical Impedance Spectroscopy. Circuits made of resistors and capacitors:

11 2.1. ONE CAPACITIVE SEMICIRCLE 11 Ω c 1 RiCi Ri Figure 2.4: Nyquist diagram of the impedance for Voigt circuit with one time constant. C2 C2 Figure 2.5: +/C2 circuit. For an electrochemical system C2 denotes the electrochemical double layer capacity C dl, the ohmic (R Ω) or uncompensated resistance (R u) and the charge transfer resistance R ct.,, C2 > Fig Ω c 1 C2 Ω c 1 C2 Figure 2.6: Nyquist impedance diagrams for the +/C2 circuit. Left:,, >, right: <, >, >. >, <, C2 > Fig. 2.6.

12 12 CHAPTER 2. NYQUIST DIAGRAMS MADE OF ONE SEMICIRCLE /(+C2) circuit Figs Ω c 1 C2 C2 C2 Figure 2.7: /(+C2) circuit and Nyquist impedance diagram.,, >

13 2.2. ONE INDUCTIVE SEMICIRCLE One inductive semicircle Table 2.2: Four different Nyquist diagrams made of one inductive semicircle. cf , p. 13 cf , p. 13 cf , p. 14 cf , p /L1 circuit Figs. 2.8, 2.9. L1 Figure 2.8: /L1 circuit. Ω c L1 Ω c L1 Figure 2.9: Nyquist impedance diagram for the /L1 circuit. Left: >, right: < /L1 circuit, < Fig. 2.8, <.

14 14 CHAPTER 2. NYQUIST DIAGRAMS MADE OF ONE SEMICIRCLE /L2 circuit Figs. 2.1, L2 L2 Figure 2.1: + /L2 circuit. Ω c L2 Ω c L2 Figure 2.11: Nyquist impedance diagrams for the +/L2 circuit. Left: >, right: < /L2 circuit, < Figs. 2.1, 2.11.

15 Chapter 3 Nyquist diagrams made of one semicircle and one straight line 3.1 Capacitive semicircle and straight line Table 3.1: Six different Nyquist diagrams made of one capacitive semicircle and one straight line (limiting cases). cf , p. 15 cf , p. 16 cf , p. 17 cf , p. 17 cf , p. 18 cf , p. 19 cf , p. 15 cf , p. 17 cf , p. 17 cf , p. 17 cf , p. 18 cf , p /(C2+) circuit Fig. 3.1 ( 1 ) /C2 circuit Fig /(C2+) and +/C2 circuits are non-distinguishable, i.e. these circuits can be interchanged (cf. Handbook of Electrochemical Impedance Spectroscopy. Circuits made of resistors and capacitors: 15

16 16CHAPTER 3. NYQUIST DIAGRAMS MADE OF ONE SEMICIRCLE AND STRAIGHT LINE C2 C2 Figure 3.1: /(C2+) circuit and Nyquist impedance diagrams. C2 C2 Figure 3.2: +/C2 circuit /(C2+) circuit Fig C2 C2 Figure 3.3: +/(C2+) circuit and Nyquist impedance diagrams.

17 3.1. CAPACITIVE SEMICIRCLE AND STRAIGHT LINE /C2 circuit Fig C2 C2 Figure 3.4: ++/C2 circuit /(Q2+) circuit Fig Q2 Q2 Figure 3.5: /(Q2+) circuit Q1+/C2 circuit Fig /(Q2+) circuit Fig Q1+/C2 circuit Fig. 3.8.

18 18CHAPTER 3. NYQUIST DIAGRAMS MADE OF ONE SEMICIRCLE AND STRAIGHT LINE Q1 C2 Q1 C2 Figure 3.6: Q1+/C2 circuit and Nyquist impedance diagrams. Q2 Q2 Figure 3.7: +/(Q2+) circuit and Nyquist impedance diagrams. C2 Q1 Q1 C2 Figure 3.8: +Q1+/C2 circuit /(+W2) circuit Fig /(+W2) circuit Fig W1+/C2 circuit Fig

19 3.1. CAPACITIVE SEMICIRCLE AND STRAIGHT LINE 19 W2 W2 Figure 3.9: /(+W2) circuit (Randles circuit) and Nyquist impedance diagrams. (cf. Handbook of Electrochemical Impedance Spectroscopy. Diffusion impedances: W2 W2 Figure 3.1: +/(+W2) circuit and Nyquist impedance diagrams W1+/C2 circuit Fig

20 2CHAPTER 3. NYQUIST DIAGRAMS MADE OF ONE SEMICIRCLE AND STRAIGHT LINE W1 C2 W1 C2 Figure 3.11: W1+C2/ and Nyquist impedance diagrams. W1 C2 W1 C2 Figure 3.12: +W1+/C2 circuit and Nyquist impedance diagrams.

21 3.2. CAPACITIVE SEMICIRCLE AND INDUCTIVE STRAIGHT LINE Capacitive semicircle and inductive straight line Table 3.2: Four different Nyquist diagrams made of one capacitive semicircle and one inductive straight line (limiting cases). cf , p. 21 cf , p. 21 cf , p. 21 cf , p L1+/C2 circuit Fig L1 C2 L1 C2 Figure 3.13: L1+/C2 circuit and Nyquist impedance diagrams L1+/C2 circuit Fig Q1+/C2 circuit, α1 < Fig

22 22CHAPTER 3. NYQUIST DIAGRAMS MADE OF ONE SEMICIRCLE AND STRAIGHT LINE C2 L1 L1 C2 Figure 3.14: +L1+/C2 circuit and Nyquist impedance diagrams. Q1,Α1 C2 Q1 C2 Figure 3.15: Q1+/C2 circuit. α1 < Q1+/C2 circuit, α1 < Fig

23 3.2. CAPACITIVE SEMICIRCLE AND INDUCTIVE STRAIGHT LINE 23 Q1,Α1 Q1 C2 C2 Figure 3.16: +Q1+/C2 circuit and Nyquist impedance diagrams. α1 <.

24 24CHAPTER 3. NYQUIST DIAGRAMS MADE OF ONE SEMICIRCLE AND STRAIGHT LINE 3.3 Inductive semicircle and straight line Table 3.3: Four different Nyquist diagrams made of one inductive semicircle and one inductive straight line (limiting cases). cf , p. 24 cf , p. 24 cf , p. 25 cf , p. 25 cf , p. 24 cf , p /L1+L2 circuit Fig L1 L2 L1 L2 Figure 3.17: /L1+L2 circuit and Nyquist impedance diagrams L1/(+L2) circuit Fig ( 2 ) /L2+L3 circuit Fig The /L1+L2 and L1/(+L2) circuits are non-distinguishable, cf. Handbook of Electrochemical Impedance Spectroscopy. Circuits made of resistors and inductors:

25 3.3. INDUCTIVE SEMICIRCLE AND STRAIGHT LINE 25 L1 L2 L1 L2 Figure 3.18: L1/(+L2) circuit and Nyquist impedance diagrams. L2 L3 L2 L3 Figure 3.19: +/L2+L3 circuit and Nyquist impedance diagrams L2/(+L3) circuit Fig /L1+Q1 circuit, α1 < Fig /L2+Q1 circuit, α1 < Fig

26 26CHAPTER 3. NYQUIST DIAGRAMS MADE OF ONE SEMICIRCLE AND STRAIGHT LINE L2 L3 L2 L3 Figure 3.2: +L2/(+L3) circuit and Nyquist impedance diagrams. L1 Q1,Α1 L1 Q1 Figure 3.21: /L1+Q1 circuit (α1 < ) and Nyquist impedance diagrams. L2 Q1,Α1 L2 Q1 Figure 3.22: +/L2+Q1 circuit (α1 < ) and Nyquist impedance diagrams.

27 3.4. INDUCTIVE SEMICIRCLE AND CAPACITIVE STRAIGHT LINE Inductive semicircle and capacitive straight line Table 3.4: Six different Nyquist diagrams made of one inductive semicircle and one capacitive straight line. cf , p. 27 cf , p. 27 cf , p. 27 cf , p. 27 cf , p. 28 cf , p /L1+ circuit Fig L1 L1 Figure 3.23: /L1+ and Nyquist impedance diagrams /L2+ circuit Fig /L1+Q1 circuit Fig /L2+Q1 circuit Fig

28 28CHAPTER 3. NYQUIST DIAGRAMS MADE OF ONE SEMICIRCLE AND STRAIGHT LINE L2 L2 Figure 3.24: +/L2+ and Nyquist impedance diagrams. L1 Q1 L1 Q1 Figure 3.25: /L1+Q1 circuit and Nyquist impedance diagrams. L2 L2 Q1 Q1 Figure 3.26: +/L2+Q /L1+W1 circuit Fig /L2+W1 circuit Fig

29 3.4. INDUCTIVE SEMICIRCLE AND CAPACITIVE STRAIGHT LINE 29 L1 W1 L1 W1 Figure 3.27: /L1+W1 circuit and Nyquist impedance diagrams. L2 L2 W1 W1 Figure 3.28: +/L2+W1 and Nyquist impedance diagrams.

30 3CHAPTER 3. NYQUIST DIAGRAMS MADE OF ONE SEMICIRCLE AND STRAIGHT LINE

31 Bibliography [1] 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. 176 (1984), [2] Lonné, Q., Glandut, N., Labbe, J.-C., and Lefort, P. Fabrication and characterization of ZrB 2 -SiC ceramic electrodes coated with a proton conducting, SiO 2 -rich glass layer. Electrochimica Acta In Press, Corrected Proof (211),. 31

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