Basics of Impedance Spectroscopy

Transcription

1 Basics of Impedance Spectroscopy (<1% of the entire topic!) B. Markovsky, Summer Courses, Bar-Ilan University, September 2014

2 The main goal of this presentation is a brief Introduction to EIS

3 Electrochemical Techniques

4 CVs of LiMO 2 electrode. Dynamic technique. t 50 V/s V 10 V/s 3.69 V 10 T=30 0 C

5 Time domain. Current = f(time) Relaxation: Current decays with time

6 An Electrochemical Impedance Spectrum 1 µhz Frequency domain from high to low frequencies 100 khz

7 The number of papers on EIS has doubled every 4 5 years!

8 AC vs. DC methods Potential Current time time Once we apply DC methods, the cell is totally changed. Surface and volume changes Phase transitions Electrolyte oxidation/reduction Potential time Current time For AC methods, very small perturbation is applied. Nearly non-destructive! Cell is unchanged!

9 Alternating current (ac) methods: the merits To grasp the entire features of the system: Sometimes we need X-ray to see the inside of our body. Sometimes we need ac methods to see the inside of an electrochemical cell.

10 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY Electrochemical Impedance Spectroscopy (EIS) is actually a special case among electrochemical techniques. It is based on the perturbation of an equilibrium state, while the standard techniques are dynamic (e.g. CV) or are based on the change from an initial equilibrium state to a different, final state (e.g. potential step, chronocoulometry). Hence, EIS is a small-signal technique where, in the analysis of the impedance spectra, a linear current-voltage relation is assumed.

11 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY Impedance spectroscopy is a non-destructive technique and so can provide time dependent information about the properties of a system but also about ongoing processes such as: - corrosion of metals, - discharge and charge of batteries, - electrochemical reactions in fuel cells, capacitors or any other electrochemical process.

12 Resistance. Ideal Resistor Everyone knows about the concept of electrical resistance. What is resistance? It is the ability of a circuit element to resist the flow of electrical current. Ohm's law defines resistance R in terms of the ratio between voltage, E, and current, I. R Et () It ()

13 Ideal Resistor R This relationship is limited to only one circuit element ---- the ideal resistor! An ideal resistor has several simplifying properties: It follows Ohm's law at all current and voltage levels. It's resistance value is independent on frequency. AC current and voltage signals through a resistor are in phase with each other. Et () It ()

14 In a Real World: Circuit elements exhibit much more complex behavior. In place of resistance, we use impedance, which is a more general circuit parameter. Like resistance, impedance is a measure of the ability of a circuit to resist the flow of electrical current. Electrochemical Impedance is normally measured using a small excitation signal (3 10 mv). This is done so that the cell's response is pseudo-linear. In a pseudo-linear system, the current response to a sinusoidal potential will be a sinusoid at the same frequency but shifted in phase.

15 Sinusoidal Current Response in a Linear System Phasor (Vector) diagram for an ac-voltage E time I time - phase angle Phaseshift Phase-shift A purely sinusoidal voltage: E t =E 0 sin t E is the amplitude of the signal, and is the radial (angular) frequency. (in radians/second) and frequency f (in Hertz (1/sec) are related as: =2 f

16 Phasor (Rotating Vector) diagram

17 Response di of de from the Current / Potential relation: We can disturb an electrical element at a certain potential E with a small perturbation de and we will get at the current I a small response perturbation di. In the first approximation, as the perturbation de is small, the response di will be a linear response as well. An oval is plotted. This oval is known as a "Lissajous figure".

18 Complex Numbers In Electrical Engineering to add together resistances, currents or DC voltages real numbers are used. But real numbers are not the only kind of numbers we need to use especially when dealing with frequency dependent sinusoidal sources and vectors. Complex Numbers were introduced to allow complex equations to be solved with numbers that are the square roots of negative numbers, -1. i= -1

19 Complex Number = Real number + Imaginary number In electrical engineering, -1 is called an imaginary number and to distinguish an imaginary number from a real number the letter j known commonly in electrical engineering as the j-operator, is used. The letter j is placed in front of a real number to signify its imaginary number operation. Examples of imaginary numbers are: j3, j12, j100 etc. A complex number consists of two distinct but very much related parts, a Real Number plus an Imaginary Number.

20 Complex Numbers. Complex Plane Complex Numbers represent points in a two-dimensional complex plane that are referenced to two distinct axes. The horizontal axis is called the Real Axis while the vertical axis is called the Imaginary Axis. The real and imaginary parts of a complex number, Z are abbreviated as Re(z) and Im(z).

21 Two Dimensional Complex Plane (Four Quadrant Argand Diagram) Z= -8 j5 Z = 5 + j0 Z = 0 + j4 Negative Imaginary Axis

22 Complex Numbers. Complex Plane Imaginary axis i= -1 Real axis The plane of complex numbers spanned by the vectors 1 and i, where i is the imaginary number. Every complex number corresponds to a unique point in the complex plane (Argand or Gauss plane).

23 Complex writting Using Euler s relationship exp( i ) cos isin it is possible to express the impedance as a complex function. The potential is described as, Z() t E( t) E0 exp( j t) and the current response as, 0 I( t) I exp( i t i ) The impedance is then represented as a complex number: E Z Z0exp( i ) Z0(cos isin ) I 0 E( t) E cos( t) cos( t) Z0 I( t) I cos( t ) cos( t ) 0

24 Imaginary part Data Presentation: Nyquist Plot with Impedance Vector The expression for Z( ) is composed of a real and an imaginary part. If the real part is plotted on the X axis and the imaginary part on the Y axis of a chart, we get a "Nyquist plot. (Harry Nyquist, ). Real part E Z Z0exp( i ) Z0(cos isin ) I C R Z R i C The Nyquist plot results from the RC circuit. The semicircle is characteristic of a single "time constant".

25 Semicircle in Nyquist plot E t =E 0 sin t (1) I t =I 0 sin ( t + ) (2) The impedance of an ohmic resistance R and a capacitance C in parallel can be written as follows: R C General formulae of a circle: X 2 + Y 2 =r 2 (r is the radius)

26 The Bode Plot Another popular presentation method is the "Bode plot". The impedance is plotted with log frequency (log ) on the X-axis and both the absolute value of the impedance ( Z =Z 0 ) and phase-shift on the Y-axis. Unlike the Nyquist plot, the Bode plot explicitly shows frequency information.

27 A parallel R-C combination The parallel combination of a resistance and a capacitance, start in the admittance representation: Y ( ) 1 R j C R Transform to impedance representation: C 1 1 1/ R j C Z( ) Y ( ) 1/ R j C 1/ R j C 2 R j R C 1 j R RC 1 A semicircle in the impedance plane! General formulae of a circle: X 2 + Y 2 =r 2 (r is the radius)

28 Semicircles in Nyquist plots The semicircle is characteristic of a single RC-constant. Electrochemical impedance plots often contain several semicircles. Often only a portion of a semicircle is seen.

29 Zreal, -Zimag, [ohm] Bode plot (Z re, Z im ) 1.E+05 1.E+04 Zreal Zimag 1.E+03 1.E+02 1.E+01 1.E+00 1.E-01 1.E-02 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 frequency, [Hz]

30 abs(z), [ohm] Phase (degr) Bode plot: absolute (Z), phase vs. frequency 1.E abs(z) Phase ( ) 75 1.E E E E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 Frequency, [Hz]

31 Different Bode representations Z imag Z real

32 Other representations

33 Bode graph. Double log plot

34 Z R ; Y R R 1 R

35 1 Z( ) R R j / C j C 1 Y ( ) R j / C 2 2 C R C j 1 C R 1 C R Semicircle time constant : = RC

36 Warburg Impedance The rate of an electrochemical reaction: charge-transfer, diffusion Whenever diffusion effects completely dominate the electrochemical reaction mechanism, the impedance is called the Warburg Impedance. For diffusion-controlled electrochemical reaction, the current is 45 degrees out of phase with the imposed potential. In this case, (45 0 ), the real and imaginary components of the impedance vector are equal at all frequencies. In terms of simple equivalent circuits, the behavior of Warburg impedance (a 45 0 phase shift) is midway between that of a resistor (a 0 0 phase shift) and a capacitor (90 0 phase shift).

37 Warburg impedance

38 Diffusion: Warburg element Semi-infinite diffusion, Flux (current) : First Fick s Law Potential : ac-perturbation: Second Fick s Law : Boundary condition : C J D x x 0 RT E E ln C nf C( t) C c( t) C t C( x, t ) x 2 C D 2 x C

39 PITT and Impedance spectroscopy PITT Small potential steps E from E eq, I vs. t is measured. Kinetic limitations other than diffusion are ignored. = l 2 /D It 1/2 is the time invariant at t<< (short-time domain) D(E)=[ 1/2 l It 1/2 /Q m X(E)] 2 = [( 1/2 l (It 1/2 / E)/ C int (E)] 2 EIS the semi-infinite (Warburg) domain for finite-space diffusion response: Z vs. Z at the low frequency is analyzed. D = 0.5 l 2 [C int A w ] -2 PITT and EIS for the same electrode potential, should provide the constant: A w [(It 1/2 )/ E]= (2 ) -1/2 This is a proof that the measurements are correct. -1/2

40 Real thin (1500 ) cathode Li x V 2 O , Warburg element M. Levi, Z. Lu, D. Aurbach, JPS, 2001, 97, 482

41 Diffusion time. Li + diffusion coefficient in LiV 2 O 5

42 Li + diffusion coefficient in thin film graphite electrodes M. Levi, D. Aurbach, J. Phys. Chem., 1997, 101, 4641

43 Li+ intercalation cathode LixCoO2

44 Li + diffusion in Li x CoO 2 at low frequencies

45 Thin Na-V 2 O 5 electrodes, 3, 1 2 mg/cm , Warburg element

46 Thin Na-V 2 O 5 electrodes. Li + diffusion coefficient

47 Equivalent Circuit Concept R sol 45 R sol

48 Analysis and Modeling. Data Validation Before starting the analysis and modeling of the experimental results one should be certain that the impedances are valid. There is a general mathematical procedure (Kramers-Kronig), which allows for the verification of the impedance data. The impedance measured is valid provided that the following 4 criteria are met: linearity, causality, stability, finiteness.

49 1. Linearity: A system is linear when its response to a sum of individual input signals is equal to the sum of the individual responses. Electrochemical systems are usually highly non-linear and the impedance is obtained by linearization of equations for small amplitudes. For the linear systems the response is independent of the amplitude. It is easy to verify the linearity of the system: if the obtained impedance is the same when the amplitude of the applied ac-signal is halved then the system is linear.

50 Electrochemical systems are, in general, not linear. A very small portion of the I vs. V curve appears to be linear (pseudo-linear) In normal EIS practice, a small (1 to 10 mv) AC signal is applied to the cell. With such a small potential signal, the system is pseudo-linear.

51 AC Current / A AC Current / A Lissajous plot AC Potential / V A typical Lissajous plot for a linear system A Lissajous plot showing a non-linear response

52 2. Causality: The response of the system must be entirely determined by the applied perturbation. The impedance measurements must also be stationary. The measured impedance must not be time dependent!

53 3. Stability: The stability of a system is determined by its response to inputs. The system is stable if its response to the impulse excitation approaches zero at long times. The measured impedance must not be time dependent. This condition may be easily checked by repetitive recording of the impedance spectra; then the obtained Bode plots should be identical.

54 Steady-State Systems Measuring an impedance spectrum takes time (minutes - hours). The system being measured must be at a steady-state throughout the time required to measure the spectrum. A common cause of problems in EIS measurements and analysis is drift in the system being measured. Standard EIS analysis tools may give wildly inaccurate results on a system that is not at steady-state.

55 Possible tests for the validity of EIS data Kronig-Kramers (KK) test The Kronig-Kramers (KK) relations are mathematical properties which connect the real and imaginary parts of any complex function. During the KK test, the experimental data points are fitted using a special model circuit which always satisfies the KK relations.

56 Is the impedance data stable? Application of K-K test for system stability

57 4. Finiteness: The real Z real and imaginary Z im components of the impedance must be finite-valued over the entire frequency range 0 < ω <. In particular, the impedance must tend to a constant real value for ω 0 and ω.

58 Instrumentation: 3-electrode cell in a thermostat Potentiostat Frequency Response Analyzer

59 40 years ago.. From Prof. B. Boukamp s lecture, Intern. Symp. on EIS, 2008

60 Measuring impedance by means of oscilloscopes, Sept. 1960

61 Impedance analysis in the old days I

62 PC FRA BTU

63 RE CE WE Pouch-cell

64 Frequency response analyzer (FRA) osc. t cos( t) sin( t) R(t) cos( t) R(t) sin( t) Z" Z' V o sin( t) Cell R( t) I0 sin( t ) A sin( k t ) noise( t) This is necessary. Harmonic components k k k 1 T int 1 T int T 0 T 0 int int R( t)cos t dt R( t)sin t dt 2 Io 2V o 2 Io 2V o Z Z o o Io sin ( ) 2V cos ( ) Vanishes by orthogonality 2 Io 2V 64 o 2 o Z Z re im ( ) ( ). S/N increase by repeated measurements

65 More applications for Li-Batteries

66 -Z / Ohm Impedance spectra of Lithium electrodes LiAsF 6 1 M LiAsF M LiAsF 6 1 M +200 ppm H 2 O Z / Ohm Z / Ohm 3 hours aging 6 days aging Z / Ohm D. Aurbach, E. Zinigrad, A. Zaban, J. Phys. Chem., 100, 1996, 3091

67 Li-Intercalation Electrodes Li[Mn-Ni-Co]O 2 Z" / Ohm 50 khz Z" / Ohm Z" / Ohm D Graph 5 2D Graph khz khz 0 Initial state, 30 0 C 12.6 Hz 20 Hz 2D Graph Z' / Ohm D Graph st Semi circle 32 mhz 20 Hz 2-nd Semi circle 158 mhz R1C1 R2C2 5 mhz 5 mhz W V V 4.4 V After aging 4 weeks at 60 0 C 20 Hz 2.5 Hz 32 mhz 5 mhz 5 mhz 5 mhz Z' / Ohm

68 Uncoated material Li[Ni-Mn-Co]O 2 AlF 3 -coated material Z'' / Ohm mhz 5 mhz 5 mhz 5 mhz mhz 5 mhz 5 mhz 5 mhz 5 mhz Hz Hz Cycle number Hz Z' / Ohm Z" / Ohm 2D Graph Hz Hz 10 Cycle number Hz Z' / Ohm 293-A15-2imp,Z' vs Col 17 vs 293-A15-2imp,Z'' 293-A15-3imp,Z' vs Col 18 vs 293-A15-3imp, Z'' 293-A15-4imp,Z' vs Col 19 vs 293-A15-4imp, Z'' 293-A15-5imp, Z' vs Col 20 vs 293-A15-5imp,Z'' A15-6imp, Z' vs Col 21 vs 293-A15-6imp,Z'' R sf / Ohm.cm Uncoated AlF3-coated 2imp,Z' vs Col 17 vs 2imp,Z'' 3imp,Z' vs Col 18 vs 3imp, Z'' 4imp,Z' vs Col 19 vs 4imp, Z" 5imp,Z' vs Col 20 vs 5imp,Z'' 6imp, Z' vs Col 21 vs 6imp, Z" Cycles F. Amalraj, B. Markovsky et al., JES, 160, A2220, 2013

69 Thin-film Li[Ni-Mn-Co]O 2 electrode (no PVdF, CB), cycled, E=4.3 V. Impedance data fitting ZView 1-st 2-nd Warburg element, Li + Solid-state diffusi Slope=-1.076

70 Equivalent circuit Why we use it? Intuitive Practical Relatively easy to model Disadvantages Ambiguities Does not tell the mechanism of the reaction Explanation is difficult!

71 A Typical Equivalent Circuit R el CPE sf CPE ct Z w R sf R ct

72 Intercalation electrodes can be described by the following equivalent analog Solution resistance Surface films Interfacial charge transfer Solid state diffusion W Intrcalation capacitance

73 Building blocks of equivalent circuits

74 Fuel cell: Equivalent circuit analog

75 Conclusions-1 Impedance is not a physical reality It is an alternating current technique. It is in the frequency domain. It is rigorously generalized concept that contains whole features of an electrochemical system. 75

76 Conclusions-2 In many cases data fitting is not needed. Exact plotting is essential. Graphical analysis is very useful. Experience is needed for data fitting. Anyway, enjoy EIS measurements! It is non-destructive technique. Good to understand your system systematically.

77 The main question: So, What is Electrochemical Impedance Spectroscopy? Probing an electrochemical system with small ac-perturbation over a range of frequencies.

78 Literature 1. Barsukov, E. and Macdonald, J. R Impedance Spectroscopy, 2nd ed. Wiley-Interscience, New York. 2. Conway, B. E Electrochemical Supercapacitors, Kluwer Academic/Plenum, New York. 3. A. Bard and L. Faulkner, Electrochemical Methods. 4. Orazem, M. and Tribollet, B Electrochemical Impedance Spectroscopy (The ECS Series of Texts and Monographs) Wiley-Interscience, New York. 5. Solartron Analytical Frequency Response Analyzer (FRA). Available at AFRAPage.htm. 6. Lectures by Prof. Bernard Boukamp, University of Twente, Dept. of Science &Technology, Enschede, The Netherlands.

79 ..Sunrise in Ein-Gedi, Dead Sea