Handbook of Electrochemical Impedance Spectroscopy

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "Handbook of Electrochemical Impedance Spectroscopy"

Αναστασούλα Ελευθεριάδης
6 χρόνια πριν
Προβολές:

1 Handbook of Electrochemical Impedance Spectroscopy A B D Im Y * Im Y * Im Y * Im Y * e Y * e Y * e Y * e Y * IUITS made of ESISTOS, INDUTOS and APAITOS E@SE/EPMI J.-P. Diard, B. e Gorrec,. Montella Hosted by August 3, 2

2 2

3 ontents ircuits made of, and 5. +/ circuit ircuit Impedance educed impedance / circuit Impedance educed impedance / circuit ircuit Impedance educed impedance /+/ circuit Impedance educed impedance Nyquist impedance diagrams Inductive and capacitive Nyquist diagrams ρ = / 2 = Array of Nyquist impedance diagrams parallel circuit ircuit Admittance educed admittance Impedance educed impedance serie circuit ircuit Impedance educed impedance Admittance educed admittance parallel circuit ircuit Impedance Transformation formulae / + / 2 r + parallel 7 3

4 4 ONTENTS 2 Quartz resonator 9 2. BVD equivalent circuit Admittance educed admittance haracteristic frequencies

5 hapter ircuits made of, and. +/ circuit.. ircuit Fig... Figure.: ircuit +/...2 Impedance e Zω = Zω = i ω + + i ω 2 2 ω 2, Im Zω = ω +..3 educed impedance ω 2 + Z u = Zu = i T u + + i u, u = ω, T = e Z u = u 2 +, Im Z u = u T u 2 + educed characteristic angular frequency u c = with: T < : e Zu c = /2, Im Zu c = T /2 5 2.

6 6 HAPTE. IUITS MADE OF, AND u Im Z= = T T, e Zu Im Z= = T. reduced angular frequency at the apex : u a = 2T + 8T + 2 T with: e Zu a = 4 8T + + Im Zu a = T 8T + 3 2T + 8T + 2 8T + u c u a u Im Z.5.5 Figure.2: Nyquist diagram of the reduced impedance for the +/ circuit Fig.., Eq..2 plotted for T =.2 left and T =.,.2,.5,,.5 right. The line thickness increases with increasing T. Dots: reduced characteristic angular frequency u c = right..2 ++/ circuit Fig..3. Figure.3: ircuit ++/.

7 .3. +/ IUIT 7.2. Impedance e Zω = educed impedance Zω = + i ω + + i ω 2 2 ω 2, Im Zω = ω ω 2 + Z u = Zu = ρ + i T u + + i u, u = ω, ρ =, T = e Z u = ρ + u 2 +, Im Z u = u educed characteristic angular frequency u c = with: T < : u Im Z= = T u 2 + e Zu c = ρ + /2, Im Zu c = T /2 T T, e Zu Im Z= = ρ + T. reduced angular frequency at the apex : u a = 2T + 8T + 2 T 2.2 with: e Zu a = ρ + 4 8T + + Im Zu a = T 8T + 3 2T + 8T + 2 8T +.3 +/ circuit.3. ircuit Fig Impedance Zω = i ω + i ω + i ω e Zω = 2 ω 2 2 ω 2, Im Zω = 2 ω ω ω

8 8 HAPTE. IUITS MADE OF, AND u c u a u Im Z Ρ Ρ 2 Ρ Ρ Ρ 2 Ρ Figure.4: Nyquist diagram of the reduced impedance for the ++/ circuit Fig.., Eq..2 plotted ρ =.2 and T =.2 left and T =.,.2,.5,,.5 right. The line thickness increases with increasing T. Dots: reduced characteristic angular frequency u c = right. Figure.5: ircuit +/..3.3 educed impedance Z u = Zu = i T u + i u + i u, u = ω, T = 2 e Z u = u2 u 2 +, Im Z u = u u 2 + T u educed characteristic angular frequency u c = with:.3 T > : u Im Z= = e Zu c = /2, Im Zu c = /2 /T T, e Zu Im Z= = T. reduced angular frequency at the apex : u a = 2 T + T + 8 T + 2 T

9 .4. /+/ IUIT 9 u Im Z u c ua.5.5 Figure.6: Nyquist diagram of the reduced impedance for the +/ circuit Fig..5, Eq..3 plotted for T = 5 left and T =.66,, 2, 5, right. The line thickness increases with increasing T. Dots: reduced characteristic angular frequency u c = right. with: Im Zu a = T e Zu a = + 4 T T +8 T 2T + T T + T + T + 8 T +8 T +2 T.4 /+/ circuit Fig Figure.7: ircuit /+/..4. Impedance Zω = i ω 2 + i ω i ω + = τ i ω + τ i ω τ 2 i ω, τ =, τ 2 = 2 2.4

10 HAPTE. IUITS MADE OF, AND e Zω = τ 2 + ω2 + lim Zω = 2, ω.4.2 educed impedance Z u = Zω 2 e Z u = = ρ 2 τ2 2, Im Zω = ω2 + lim ω Zω = τ ω τ 2 ω2 + i u + i u + + T i u, u = ω τ, ρ =, T = τ 2 2 τ u2 ρ u T 2 u 2 +, Im Z u =.4.3 Nyquist impedance diagrams T >, Fig..8. uρ u 2 + T u T 2 u 2 + 2τ 2 ω τ 2 2 ω2 + T Ρ T Ρ Ρ Ρ Figure.8: T >. Nyquist diagrams of the impedance for the /+/ circuit Fig..7, Eq..4 plotted for : top : ρ < ρ =.5, bottom : ρ > ρ =.5. T T = 2 left and increasing values of T right. The line thickness increases with increasing T. T <, Fig..9. T =, Fig... T = Z u = + ρ i u + i u

11 .4. /+/ IUIT T Ρ T Ρ Ρ Ρ Figure.9: T <. Nyquist diagrams of the impedance for the /+/ circuit Fig..7, Eq..4 plotted for : top : ρ < ρ =.5, bottom : ρ > ρ =.5. T T = 2 left and increasing values of T right. The line thickness increases with increasing T. Ρ Ρ Ρ Figure.: T =. Nyquist diagrams of the impedance for the /+/ circuit. eft: ρ <, middle: ρ =, right: ρ >..4.4 Inductive and capacitive Nyquist diagrams T > and T < ρ < T or T < and T < ρ < T T ρ u Im= =, e Im = = + ρ T T ρ + T Fig...

12 2 HAPTE. IUITS MADE OF, AND T T u Im u Im Ρ Ρ T Ρ Ρ T Figure.: Inductive and capacitive Nyquist diagrams. eft : T > and T < ρ < T, right : T < and T < ρ < T..4.5 ρ = / 2 = Fig..2. Z u = i u + i u + + T i u T T u u 2 T 2 T Figure.2: ρ = / 2 =. Nyquist diagram: full circle. eft: T >, right T <..4.6 Array of Nyquist impedance diagrams Fig parallel circuit.5. ircuit Fig..4.

13 .5. PAAE IUIT 3 logρ log T Figure.3: Array of Nyquist impedance diagrams for the /+/ circuit. T = ρ = Z u =, u. Figure.4: ircuit //..5.2 Admittance Y ω = i ω + i ω + i ω + + i ω2 = i ω e Y ω =, Im Y ω = ω + ω e Y ω is constant, lim ω Im Y ω =, lim ω Im Y ω = Nyquist diagram of Y ω is a vertical straight line.

14 4 HAPTE. IUITS MADE OF, AND.5.3 educed admittance i u + i u Y u = Y u = + Λ, u = ω, Λ = e Y u =, Im Y u = Λ u u u c3 Im Y u c2 u c e Y Figure.5: Nyquist diagram of the // circuit reduced admittance. u c = Λ 2 /2 Λ, u c2 =, u c3 = Λ 2 /2 Λ..5.4 Impedance Zω = Y ω = i ω + i ω + 2 ω 2 e Zω = 2 ω 2 + ω 2 = i ω i ω + + i ω 2 2 ω ω 2 2, Im Zω = 2 ω ω 2 2 The Nyquist diagram of Y ω is a vertical straight line the Nyquist diagram of Zω is a full circle..5.5 educed impedance Z u = Zu = i u Λ + i u + Λ i u 2, u = ω, Λ = e Z u = u 2 u 2 + Λ 2 u 2 2, Im Λ u u 2 Z u = u 2 + Λ 2 u serie circuit.6. ircuit Fig..7.

15 .6. SEIE IUIT 5.5 u c3 u u c2.5 u c Figure.6: Nyquist diagram of the // circuit reduced impedance. u c = Λ 2 /2 Λ, u c2 = u r =, u c3 = Λ 2 /2 Λ u c3 u c = Λ. Figure.7: ircuit Impedance Zω = + i ω + + i ω + i ω2 = i ω i ω e Zω =, Im Zω = ω + ω e Zω is constant, lim ω Im Zω =, lim ω Im Zω = Nyquist diagram of Zω is a vertical straight line..6.3 educed impedance Z u = Zu = + Λ i u + i u e Z u =, Im Z u = Λ, u = ω, Λ = u u.6.4 Admittance Y ω = Zω = i ω + i ω + i ω 2 2 ω 2 ω ω 2 e Y ω = 2 2 ω ω 2 2, Im Y ω = + ω ω 2 The Nyquist diagram of Zω is a vertical straight line the Nyquist diagram of Y ω is a full circle.

16 6 HAPTE. IUITS MADE OF, AND u c u c2 u c3 Figure.8: Nyquist diagram of the ++ circuit reduced impedance. u c = Λ Λ 2 /2, u c2 =, u c3 = Λ Λ 2 / educed admittance Y Λ i u u = Y u = + Λ i u + i u 2, u = ω, Λ = e Y u = u 2 Λ 2 + u 4 + u Λ 2, Im Y u Λ u 2 u = + u 4 + u Λ 2.5 u c Im Y u u c2.5 u c3 e Y Figure.9: Nyquist diagram of the ++ circuit reduced admittance. u c = Λ Λ 2 /2, u c2 = u r =, u c3 = Λ Λ 2 /2, u c3 u c = Λ..7 + parallel circuit.7. ircuit Fig Impedance Zω = + i ω i ω + + i ω 2.5

17 .8. TANSFOMATION FOMUAE / + / 2 + PAAE 7 Figure.2: + parallel circuit. Z u = Zu = ρ + i u Λ + i u + Λ i u 2, ρ =, u = ω, Λ =.5 u c3 u u c2.5 u c Ρ Ρ Figure.2: Nyquist reduced impedance diagram of the + parallel circuit. u c = Λ 2 /2 Λ, u c2 = u r =, u c3 = Λ 2 /2 Λ u c3 u c = Λ..8 Transformation formulae / + / 2 r + parallel r + r 2 /l 2 /c 2 parallel circuit is not-distinguishable from / + / 2 circuit for 2 2 / 2 > Fig p 2 + 2p + Zp =.6 2 p 2 + p r c 2 l 2 p 2 + l2pr+r2 r r 2 + zp = c 2 l 2 p 2 + l2p.7 r c 2 = 2 2, l 2 = 2, 2 2 r 2 =

18 8 HAPTE. IUITS MADE OF, AND c 2 2 r l 2 r 2 Figure.22: / + / 2 / + 2 / 2 circuit with = 2 and r + r 2 /l 2 /c 2 parallel circuit.

19 hapter 2 Quartz resonator 2. BVD equivalent circuit Fig. 2., [3, 5, 6, ]. Figure 2.: BVD Butterworth-van Dyke-equivalent circuit of a quartz resonator. 2.2 Admittance Y ω = + i ω + + i ω = i ω i ω 2 ω 2 e Y ω = 2 2 ω ω 2 2, Im Y ω = ω + i ω i ω + + ω 2 + ω ω educed admittance Y Λ i u u = Y u = + Λ i u + i u 2 + γ i u, u = ω, Λ =, γ = e Y u 2 Λ 2 u = + u 4 + u Λ 2, Im Y u Λ u 2 u = u γ + + u 4 + u Λ 2 9

20 2 HAPTE 2. QUATZ ESONATO B u Im Y * u m u p u s u r u 2 e Y * Figure 2.2: Definitions for u, u m, u r, u s, u 2 and u p. A B Im Y * Im Y * e Y * e Y * D Im Y * Im Y * e Y * e Y * Figure 2.3: hange of admittance diagram with γ. Λ =, γ = 2 A, 2 B, /3, /2 D haracteristic frequencies Maximum of the real part of Y for: u r = e Y u r =, Im Y u r = γ

21 2.3. EDUED ADMITTANE 2 Zero-phase reduced angular frequencies: u s and u p defined for γ < /2 + Λ :. γ < 2 + Λ Λ γ 2 + Λ u s = 2 Λ 2 γ Λ + γ Λ 2, 2 γ e Y u s = + γλ + γλ 2 γλ γ + Λ γ Λ u p = 2 + Λ 2 γ Λ + γ Λ 2, 2 γ e Y u p = + γλ γλ 2 γλ γ = 2 + Λ u s = u p = γλ2 + 2γ + Λ e Y u s = e Y u p = + γλ Fig γ > 2 + Λ no zero-phase reduced angular frequency Fig. 2.3D. eal quartz : 2 F, 4 F, Ω, 2 H Λ 4 and γ 2 [4, 2]. 2γ

22 22 HAPTE 2. QUATZ ESONATO

23 Bibliography [] Arnau, A., Sogorb, T., and Jiménez, Y. A continuous motional series resonant frequency monitoring circuit and a new method of determining Butterworth-Van Dyke parameters of a quartz crystal microbalance in fluid media. ev. Sci. Instrum. 7 2, [2] Bizet, K., Gabrielli,., Perrot, H., and Terrasse, J. Validation of antibody-based recognition by piezoelectric transducers through electroacoustic admittance analysis. Biosensors & Bioelectronics 3 998, [3] Butterworth, S. Proc. Phys. Soc. ondon 27 95, 4. [4] Buttry, D. A., and Ward, M. D. Measurement of interfacial processes at electrode surfaces with the electrochemical quartz crystal microbalance. hem. ev , [5] VanDyke, K. S. Phys. ev , 895. [6] VanDyke, K. S. In Proceeding of the 928 IEEE International Frequency ontrol Symposium New York, 928, vol. 6, IEEE, p

Σχετικά έγγραφα

Handbook of Electrochemical Impedance Spectroscopy

Handbook of Electrochemical Impedance Spectroscopy Im Z u c T u c T Re Z CIRCUITS made of RESISTORS and INDUCTORS ER@SE/LEPMI J.-P. Diard, B. Le Gorrec, C. Montella Hosted by Bio-Logic @ www.bio-logic.info