Handbook of Electrochemical Impedance Spectroscopy

Transcription

1 Handbook of Electrochemical Impedance pectroscopy DITRIBUTED and MIXED IMPEDANCE LEPMI J.-P. Diard, C. Montella Hosted by eptember 7, 205

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3 Contents Introduction 5. Lumped vs. distributed systems Lumped systems Distributed systems Mixed lumped-distributed systems Examples in electrochemistry Lumped systems Distributed systems Mixed lumped-distributed systems Impedance containing th th Electrochemical reaction Electrochemical impedance Reduced impedance Graphs of the reduced impedance Pole-zero map th + α th Electrochemical reaction Reduced Faradaic impedance ( Nyquist diagrams α th ) th + β th Electrochemical reaction Reduced concentration impedance Nyquist diagrams Mixed impedance 5 th 3. + α Pole-zero map Nyquist diagrams

4 4 CONTENT α th + β Electrochemical reaction: Volmer-Heyrovský (V-H) Reduced concentration impedance of adsorbed species Nyquist diagrams α + β th Electrochemical reaction: catalytic copper deposition Reduced concentration impedance of adsorbed species Nyquist diagrams th + α + β th Electrochemical reaction: catalytic copper deposition Reduced concentration impedance of soluble species Cl Nyquist diagrams α th + β + γ th Electrochemical reaction: E-EAR reaction Reduced concentration impedance of adsorbed species Nyquist diagrams ( + α ) th + β + γ th Electrochemical reaction: E-EAR reaction Reduced concentration impedance of soluble species R Nyquist diagrams A ome rational fractions in 33 A. Introduction A A.3 + ( ) ( ) 2 A.4 ( + α( )2 ) A.5 ( ) 2 ( α( )2 ) B Reactions involving adsorbed and soluble species 39 Bibliography 42

5 Chapter Introduction. Lumped vs. distributed systems.. Lumped systems The transfer functions of systems modeled by ordinary differential equations, often called lumped-parameter systems, are rational functions (i.e. a ratio of two polynomials in s, the Laplace variable) [, 2]...2 Distributed systems The transfer functions of distributed parameter systems are irrational functions. The analysis of rational and irrational transfer functions differ in a number of important aspects. The most obvious differences between rational and irrational transfer functions are the poles and zeros. Irrational transfer functions often have infinitely many poles and zeros [2]...3 Mixed lumped-distributed systems The transfer functions of mixed lumped-distributed systems contain rational and irrational functions in s..2 Examples in electrochemistry.2. Lumped systems Faradaic impedance Z f and impedance Z of electrochemical adsorption reaction (EAR) are lumped systems [3, 4]. Eq. (.) is a rational fraction in s ( ). Z(s) = + R ct C ads s s ((C dl + C ads ) + sr ct C dl C ads ) (.) Z(s) = K + α s s ( + β s), K =, α = R ct C ads, β = R ctc dl C ads (.2) C dl + C ads C dl + C ads Replacing a capacitor, for example C dl, by a CPE [5] transforms a lumped impedance in a distributed impedance. This case is not subsequently envisaged. 5

6 6 CHAPTER. INTRODUCTION C dl C ads R ct Figure.: Equivalent circuit for electrochemical adsorption reaction (EAR)..2.2 Distributed systems The Faradaic impedance of a corroding electrode with mass transfer limitation (Fig.2)) is a rational fraction in s, i.e. an irrational fraction in s. Z f (s) = Z f (s) = R σ σ + R s K + α s, K = R, α = R σ (.3) (.4) R ct Figure.2: Equivalent circuit for a corroding electrode with mass transfer limitation. W: Warburg element for semi-innite linear diffusion [6]..2.3 Mixed lumped-distributed systems The impedance of the Randles equivalent circuit [4, 6, 7] (Fig..3) is a mixed lumped and distributed system: Z(s) = R ct + R d th τ d s τd s + R ct C dl s + C dl s R d th τ d s τd s (.5) Z(s) = K + α th τ d s τd s + β s + γ s th τ d s τd s, K = R ct, α = R d R ct, β = R ct C dl γ = C dl R d (.6)

7 .2. EXAMPLE IN ELECTROCHEMITRY 7 C dl R ct Figure.3: Randles equivalent circuit for a redox reactions studied on a rotating disk electrode. W δ : bounded diffusion impedance [6].

8 8 CHAPTER. INTRODUCTION

9 Chapter 2 Impedance containing th 2. th 2.. Electrochemical reaction Redox reaction [4, 8, 9]: O + e R studied on a rotating disk electrode with mass transfer limitation Electrochemical impedance 2..3 Reduced impedance Z Wδ (s) = R d th τ s τ s Z Wδ (s) = R d th τ s τ s ( ) (2.) Z() = Z W δ (s) R d 2..4 Graphs of the reduced impedance 2..5 Pole-zero map Infinite product expansion [2 6]: th = Thanks to Eq. (2.3) + 4 π 2 k= = th, = τ s = Σ + i u (2.2) + (k π) 2 = P 4 (2.3) + ((2 k + ) π) 2 This expression could be replaced by a more accurate one [0, ]. This case is not subsequently envisaged. 9

10 0 CHAPTER 2. IMPEDANCE CONTAINING TH infinity of interlaced real poles and zeros (Fig. 2.). s p = 4 ((2k + ) π)2, k = (2.4) s Z = (k π) 2, k = (2.5) Figure 2.: Pole-zero map of th s. Figure 2.2: 3D plot of the modulus of th.

11 TH + α TH th + α th 2.2. Electrochemical reaction Corroding electrode with mass transfert limitation: M M n+ + n e O e + 4 H + 2 H 2 O Reduced Faradaic impedance Z() = th Nyquist diagrams + α th, u c = 2.54, u c2 = α 2 (2.6) Figure 2.3: Nyquist diagrams calculated from Eq. (2.6). Red dots : u c = 2.54, black dots : u c2 = α 2. u c u c2, 2.54 α 2 u c u c2, 2.54 α 2 quarter of a lemniscate, quarter of a circle (see Annex A.2).

12 2 CHAPTER 2. IMPEDANCE CONTAINING TH 2.3 ( + α th ) + β th th 2.3. Electrochemical reaction EE reaction [7]: R X + e X O + e studied on a rotating disk electrode with D R = D X = D O Reduced concentration impedance Concentration impedances of soluble species: ( + α th ) th Z Xi () = Nyquist diagrams Figs. 2.4, 2.5 and β th, α, β 0 (2.7) Figure 2.4: ome amazing Nyquist diagrams calculated from Eq. (2.7), α =, β = 0 (left), β = 0 4 (right). Red dots : u c = 2.54, black dots : u c2 = β 2. a: α =, β = 0, b: α =, β = 0 4, c: α = 2.5, β = 0 5, d: α = 2.5, β = 0 (Hokusai s great wave).

13 2.3. ( + α TH ) TH + β TH 3 Figure 2.5: Array of impedance diagrams calculated from Eq. (2.7). Red dots : u c = 2.54, black dots : u c2 = β 2.

14 4 CHAPTER 2. IMPEDANCE CONTAINING TH Figure 2.6: Array of impedance diagrams calculated from Eq. (2.7). Red dots : u c = 2.54.

15 Chapter 3 Impedance containing lumped and distributed elements 3. th + α Characteristic frequencies: u c = 2.54 u c2 = /α 3.. Pole-zero map ame zeros as th Z() = th + α (3.) ame poles as th plus one real pole ( ) (Figs ). α 3..2 Nyquist diagrams Figs u c u c2 Z() th u c u c2 Z() + α 5

16 6 CHAPTER 3. MIXED IMPEDANCE 4Π 2 3Π 2 2Π 2 Π 2 Im s 0 7Π 2 4 5Π 2 4 Re s 3Π 2 4 Π2 4 Im H Re H Figure 3.: Pole-zero map and Nyquist diagram of Red dot: u c = 2.54, black dot: u c2 = /α. + α th. α = Π 2 3Π 2 2Π 2 Π 2 Im s 0 7Π 2 4 5Π 2 4 Re s 3Π 2 4 Π2 4 Im H Re H Figure 3.2: Pole-zero map and Nyquist diagram of Red dot: u c = 2.54, black dot: u c2 = /α. + α th. α = 0 2.

17 3.. TH + α 7 4Π 2 3Π 2 2Π 2 Π 2 Im s 0 7Π 2 4 5Π 2 4 Re s 3Π 2 4 Π2 4 Im H Re H Figure 3.3: Pole-zero map and Nyquist diagram of ( α > π2 4 ). Red dot: u c = 2.54, black dot: u c2 = /α. + α th. α = 4Π 2 3Π 2 2Π 2 Π 2 Im s 0 7Π 2 4 5Π 2 4 Re s 3Π 2 4 Π2 4 Im H Re H Figure 3.4: Pole-zero map and Nyquist diagram of ( α > π2 4 ). Red dot: u c = 2.54, black dot: u c2 = /α. th. α = α s

18 8 CHAPTER 3. MIXED IMPEDANCE th Figure 3.5: Change of Nyquist diagram of with increasing value + α of α from 0 3 to Decimal logarithm of α reported on the Nyquist diagrams. Red dots: u c = 2.54, black dots: u c2 = /α.

19 α TH + β + α th + β 3.2. Electrochemical reaction: Volmer-Heyrovský (V-H) Electrochemical reaction: Volmer-Heyrovský (V-H) [4, 8] A + + s + e A,s A + + A,s + e A 2 + s Reduced concentration impedance of adsorbed species Z θ () = + α th + β = + β + α th + β 9 (3.2) α Z θ () α th u c = 2.54 (3.3) α 0 Z θ () + β i u u c2 = β (3.4) Nyquist diagrams Figs. 3.6 and 3.7. Figure 3.6: Impedance diagrams calculated from Eq. (3.2)). a : α =, β = 0 5, b : α = 0 3, β = 0 3, c : α = 0 3, β = 0 3, d : α = 0 3, β =. Characteristic dimensionless frequencies: red dots : u c = 2.54, black dots : u c2 = /β.

20 20 CHAPTER 3. MIXED IMPEDANCE Figure 3.7: Array of impedance diagrams calculated from Eq. (3.2). Impedance diagrams are made of one or two arcs. Characteristic dimensionless frequencies: red dots : u c = 2.54, black dots : u c2 = /β.

21 α + β TH 2 + α + β th 3.3. Electrochemical reaction: catalytic copper deposition Cu 2+ +Cl + s + e CuCl,s CuCl,s + e Cu + Cl + s Hypotheses: no mass transfer limitation for Cu 2+, (Cu 2+ (0, t) Cu 2+ ), kinetic irreversibility of the two steps [8, 9] Reduced concentration impedance of adsorbed species Z() = Poles and zeros of Z () are real and interlaced. + α + β th (3.5) Zeros of Z() are the poles of th : 4 ((2k + ) π)2, k =. Characteristic dimensionless frequencies: u c = 2.54, u c2 = /α, u c3 = /β Nyquist diagrams Figs. 3.8 and 3.9.

22 22 CHAPTER 3. MIXED IMPEDANCE 7Π 2 4 5Π 2 4 3Π 2 4 Π2 4 Im s 0 Re s 0.5 Z Θ Im Z Re Z Figure 3.8: Pole-zero map and impedance diagrams calculated from Eq. (3.5). α = 0 2, β =, red dot : u c = 2.54, black dot: u c2 = /α, blue dot: u c3 = /β.

23 α + β TH 23 Figure 3.9: Graphics array representation of the impedance diagram, calculated form Eq. (3.5) and plotted using the Nyquist representation (orthonormal scales) for catalytic copper deposition. Characteristic dimensionless frequencies: red dots : u c = 2.54, black dots: u c2 = /α, blue dots: u c3 = /β.

24 24 CHAPTER 3. MIXED IMPEDANCE 3.4 th + α + β th 3.4. Electrochemical reaction: catalytic copper deposition Cu 2+ +Cl + s + e CuCl,s CuCl,s + e Cu + Cl + s Hypotheses: no mass transfer limitation for Cu 2+, (Cu 2+ (0, t) Cu 2+ ), kinetic irreversibility of the two steps [9] Reduced concentration impedance of soluble species Cl Z() = Nyquist diagrams Poles and zeros of Z() are real. th + α + β th (3.6) Zeros of Z() are the zeros of th ( Z = (k π) 2, k = ) plus one zero at the origine of the complex plane (derivator). Characteristic dimensionless frequencies: u c = 2.54, u c2 = /α, u c3 = /β, u c4 = (β/α) 2. Fig. 3.0.

25 3.4. TH + α + β TH 25 Figure 3.0: Graphics array representation of the reduced impedance diagram calculated from Eq. (3.6) and plotted using the Nyquist complex plane representation (orthonormal scales) for catalytic copper deposition. Characteristic dimensionless frequencies: red dots : u c = 2.54, black dots: u c2 = /α, blue dots: u c3 = /β, green dots: u c4 = (β/α) 2.

26 26 CHAPTER 3. MIXED IMPEDANCE α th + β + γ th 3.5. Electrochemical reaction: E-EAR reaction [20] R + s O + s + n e A + s A,s + n 2 e Reduced concentration impedance of adsorbed species Z() = Nyquist diagrams + α th + β + γ th, α, β, γ 0 (3.7) Figure 3.: Graphics array representation of the Nyquist diagram for the impedance calculated from Eq. (3.7) and plotted using the Nyquist representation (orthonormal scales). Characteristic dimensionless angular frequencies: red dots: u c = 2.54, black dots: u c2 = /β. γ = 0 2.

27 α TH + β + γ TH 27 Figure 3.2: Graphics array representation of the Nyquist diagram for the impedance calculated from Eq. (3.7) and plotted using the Nyquist representation (orthonormal scales). Characteristic dimensionless angular frequencies: red dots: u c = 2.54, black dots: u c2 = /β. γ = 0.

28 28 CHAPTER 3. MIXED IMPEDANCE Figure 3.3: Graphics array representation of the Nyquist diagram for the impedance calculated from Eq. 3.7 and plotted using the Nyquist representation (orthonormal scales). α, β, γ < 0. Characteristic dimensionless angular frequencies: red dots: u c = 2.54, black dots: u c2 = / β. α, β, γ < 0, γ = 0.

29 ( + α ) TH + β + γ TH 29 ( + α ) th + β + γ th 3.6. Electrochemical reaction: E-EAR reaction [20] R + s O + s + n e A + s A,s + n 2 e Reduced concentration impedance of soluble species R Z() = Nyquist diagrams Figs. 3.4 and 3.5. ( + α ) th + β + γ th, α, β, γ 0 (3.8)

30 30 CHAPTER 3. MIXED IMPEDANCE Figure 3.4: Graphics array representation of the Nyquist diagram for the impedance calculated from Eq. (3.8) and plotted using the Nyquist representation (orthonormal scales). Characteristic dimensionless angular frequencies: red dots: u c = 2.54, black dots: u c2 = /β. γ = 0 3.

31 3.6. ( + α ) TH + β + γ TH 3 Figure 3.5: Graphics array representation of the Nyquist diagram for the impedance calculated from Eq. (3.8) and plotted using the Nyquist representation (orthonormal scales). α, β, γ < 0. Characteristic dimensionless angular frequencies: red dots: u c = 2.54, black dots: u c2 = / β. γ = 0 3.

32 32 CHAPTER 3. MIXED IMPEDANCE

33 Appendix A ome rational fractions in A. Introduction The use of a rational fraction in Z( N m=0 ) = b m( ) m P p=0 a p( ) p (A.) has been proposed by Pintelon et al. [2, 22]. ome rational fraction in are studied below. A.2 + H(u) = + i u 2 u + 2 u Re H(u) = 2 ( u + 2 ), Im H(u) = ( ) u + 2 u + 2 u + (A.2) (A.3) H(u) (/2 + i/2) = 2 (Re H(u) /2) 2 + (Im H(u) /2) 2 = 2 2 circle, radius = 2 Nyquist diagram: Fig. A.. (A.4) A.3 + ( ) 3 H(u) = + (i u) (3/2) (A.5) 33

34 34 APPENDIX A. OME RATIONAL FRACTION IN Figure A.: Nyquist diagram of Im H(u c ) = ( ) i u. One quarter circle. u c =, Re H(u) = 2u 3/2 2 2 ( 2u 3/2 u 3 ), Im H(u) = u 3/2 2u 3/2 + 2u (A.6) H(u) (/2 i/2) = 2 (Re H(u) /2) 2 + (Im H(u) /2) 2 = 2 2 circle, radius = 2 (A.7) Remarkable frequencies u = , u 2 =, u 3 = 3 2, u 4 = (A.8) Nyquist diagram: Fig. A.2. Figure A.2: Nyquist diagram of. Three quarter circle. + (i u) 3/2

35 A.4. A.4 + ( ) 2 ( + α( )2 ) + ( ) 2 ( + α( )2 ) 35 H(u) = + ( iu) 2 iu ( + α ( iu)2 ) (A.9) Re H(u) = u(α(u ) + ) + u(αu + α ), Im H(u) = 2 u (α2 u 2 + ) 2 u (α2 u 2 + ) (A.0) Three different limiting cases α, Nyquist and Bode diagrams: Fig. A.3 α =, H(u) = iu u = u Im H=0 = α 2 6α + α + 2α u 2 = u Im H=0 = + α 2 6α + α + 2α α (A.) (A.2) α, Nyquist diagram: Fig. A.4 u = u ReH=0 = α 2 6α + + α 2α α α2 6α + + α u 2 = u ReH=0 = 2α (A.3) (A.4) A.5 ( ) 2 ( α( )2 ) Re H(u) = H(u) = ( iu) 2 iu ( α ( iu)2 ) u(α + αu ) + u(α + α( u) ), Im H(u) = 2 u (α2 u 2 + ) 2 u (α2 u 2 + ) (A.5) (A.6) Three different limiting cases α, Nyquist diagram: Fig. A.5 α =, H(u) = iu u = u Re H=0 = α 2 6α + α + 2α α2 6α + α + u 2 = u Re H=0 = 2α α (A.7) (A.8) α.

36 36 APPENDIX A. OME RATIONAL FRACTION IN Figure A.3: Nyquist and Bode (modulus) diagram of a. Red dot: u 2 = /α, black dot: u = 0. + ( iu) 2 iu ( + α ( iu)2 ). Figure A.4: Nyquist and Bode (modulus) diagram of a. Red dot: u Im H=0 /α, black dot: u Im H=0. + ( iu) 2 iu ( + α ( iu)2 ).

37 A.5. ( ) 2 ( α( )2 ) 37 Figure A.5: Nyquist and Bode (modulus) diagram of a. Red dot: u 2 = /α, black dot: u = 0. ( iu) 2 iu ( α ( iu)2 ).

38 38 APPENDIX A. OME RATIONAL FRACTION IN

39 Appendix B Impedance structure of reactions involving both adsorbed and soluble species Table B.: Impedance structure of reactions involving both adsorbed and soluble species. First order denominator. Expression Reaction Impedance + α th + β (V-H) A + + s + e A,s A + + A,s + e A 2 + s Z θ Table B.2: Impedance structure of reactions involving both adsorbed and soluble species. econd order denominator. Expression Reaction Impédance + α s + β th + δ + ϵ 2 + γ s th (V-H) with desorption A + + s + e A,s A + + A,s + e A 2,s A 2,s A 2 + s Z θ 39

40 40APPENDIX B. REACTION INVOLVING ADORBED AND OLUBLE PECIE Table B.3: Impedance structure of reactions involving both adsorbed and soluble species. Dénominateur : + α + β th Expressions Reactions Impédances + α + β th (catalytic) A 2+ + B + s + e AB,s AB,s + e A + B + s Hyp. A 2+ (0, t) = cte Z θ th + β s + γ th (catalytic) A 2+ + B + s + e AB,s AB,s + e A + B + s Hyp. A 2+ (0, t) = cte Z B Table B.4: Impedance structure of reactions involving both adsorbed and soluble species. Dénominateur : + β + γ th. Expressions Reactions Impedances + α + β + γ th (V-H)#2 A + + s + e A,s A,s + e A + s Z θ + α th + β + γ th (E-EAR) R + s O + s + n e A + s A,s + n 2 e Z θ ( + α ) th + β + γ th (E-EAR) R + s O + s + n e A + s A,s + n 2 e (V-H)#2 A + + s + e A,s A,s + e A + s Z R Z A + Table B.5: Impedance structure of reactions involving both adsorbed and soluble species. Dénominateur : + β s + γ th + δ th. Expression Reaction Impedance + α th s + β + γ th + δ s th s (DP3) M,s M 2+ + s + 2 e M,s + A 2 MA,s + 2 e MA,s + B MAB + s Hyp. A 2 (0, t) = A 2 Z s

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