Electrochemical Impedance Spectroscopy. Part 1: Polarization Resistance: Familiar parameter measured in a new way June 6, 2008

Transcription

1 Electrochemical Impedance Spectroscopy Part 1: Polarization Resistance: Familiar parameter measured in a new way June 6, 2008

2 Objective The purpose of this lecture series is to generate a set of notes on Impedance Spectroscopy which are easy to use yet powerful enough that a researcher will be able to decide whether or not to pursue the technique more in depth. All figures are drawn by Matthew Taylor of the Georgia Institute of Technology, except where noted. This work may be used for any purpose, so long as reproduced figures are attributed to the author. 2

3 What is Electrochemical Impedance Spectroscopy? Electrochemical reactions are those that involve electron transfer. Corrosion Scientists are primarily interested in this type of phenomena. Impedance is a measure of resistance to response to an outside stimulus, such as a fluctuation in potential, and has both real and imaginary parts A Spectrum of data is collected by collecting measured impedances at individual frequencies and combined to form a spectrum of data. 3

4 Why would I want to use EIS? (I have DC) DC methods drive the system from equilibrium Reaction parameters are time-dependent and may be altered during DC. Observing the system tends to change it. There is more data avaliable (Rp as opposed to Rp+Rs can be measured) Excitation System Response DC Technique Input Output Potentiostatic / Pulsed Potential E I Cyclic Voltammetry E I E Potentiodynamic Scan Galvanostatic/ Pulsed Current E I E log(i) 4

5 Introduction of the Equivalent Circuit Describe a system in terms of impedance not a model, but an analog Possible to describe a variety of systems acoustics electrochemical charge transfer reactions fluids in a container mechanical materials systems Practically any system has an impedance associated with it F Analogous Systems Mechanical Electrical M R m F C R m M C m Mass Inductor Spring Capacitor Normal Force Electrical Potential Dashpot Resistor 5

6 Polarization Resistance Useful for predicting a corrosion rate under steady-state conditions Commonly determined from Taefel slopes on the potentiodynamic scan Our model circuit / system for consideration 6

7 DC determination of Polarization Resistance Measure R p (Polarization Resistance)by scanning potential away from Ecorr In DC, the circuit describing the corroding surface (for a simple case) is nothing but a resistor. The Impedance being measured is entirely real. From Ohm s law, we apply an overpotential and measure the current change, deriving Rp Any capacitance in the system may not be measured, as a capacitor acts as an open circuit under DC conditions! The solution Resistance is measured as a part of R p and must be accounted for. R p I net = η R p 7

8 AC Determination of Polarization Resistance Impedance has real and imaginary components, both of which depend on the AC frequency. R s R p Solution resistance is measured separately from polarization resistance C dl Capacitance of the ionic double layer may be measured Measured by applying a small amplitude AC signal over a range of appropriate frequencies Soluion Bulk Ionic Double Layer Metal (conductor) Figure: Randles Cell 8

9 Simple Passive Circuit Elements Element Impedance Symbol Resistor Z Ri (ω) =R i R i Capacitor Z Ci (ω) = j ωc C dl Note: j = 1 9

10 Rules for Circuit Analysis Impedances in series add directly Impedances in parallel add inversely Combination Circuit Equation Series Z 1 Z 2 Z series = Z 1 + Z 2 Parallel Z 1 Z 2 Z parallel = 1 1 Z Z 2 10

11 Impedance of the Randles Cell consider the capacitor and R p in parallel Z p = 1 1 R p + 1 j ωc dl R s R p C dl A little complicated, but we can multiply by the complex conjugate to of the bottom to seperate real and imaginary (multiplied by j) terms. Z p = R p 1+ω 2 CR 2 p j ωc dl R 2 p 1+ω 2 C 2 dl R2 p Adding R s and then Separating into Z (real) and Z (imaginary), we have: Z = R s + R p 1+ω 2 CR 2 p ωc dlr 2 p Z = 1+ω 2 Cdl 2 R2 p 11

12 Continued Z and Z are related to Z by the pythagorean theorem, and to ϕ by the tangent function Z = Z + jz Z = Z 2 + Z 2 φ = tan 1 ( Z Plotting Z versus frequency yields a plot, from which graphical extraction of analog parameters may be performed. Along with ϕ vs. f, this is known as a Bode Plot. Z ) Z,! R P R S log(f), hz f =2πω 12

13 Graphical Evaluation of the Randles Cell The high frequency intercept is equivalent to the solution resistance, Rs The low frequency intercept is equivalent to the polarization resistance plus the solution resistance Capacitance may be determined ω max = 1 R p C dl. -Z'', Imaginary Impedance,! High Frequency " max Low Frequency Z', Real Impedance,! ωmax is the frequency where -Z is at its maximum This geometrical - mathematical technique can be applied to any combination of RC circuits. 13

14 Electrochemical Impedance Spectroscopy Part 2: Advanced Circuit Elements June 20, 2008

15 Overview Part 1: Dive in with polarization resistance and simple circuit analysis DC vs. AC techniques Simple analog circuits Resistance Polarization by AC methods Part 2: The heavy basis for EIS: Linear Systems Theory and how we actually measure spectra Measurement of Impedance Advanced Circuit elements Geometrical Extrapolation* Part 3: History of EIS, experimental and analysis considerations and pitfalls, and data validation (KK transforms, stability diagrams) Part 4: Reaction Mechanism Determination 15

16 Refresher/Clean-Up Electrochemical systems can be described by analog circuits Impedance is a complex measure of resistance to change in the system, related by ohm s law Individual processes may be delineated from one another using EIS RP (Ω) Corrosion Rate (mm/y) , Example Mild Steel / Strong Acid Mild Steel / Natural water Mild Steel / Inhibited Water 1,000, Passive Metal 16

17 Why EIS? (II) The power of EIS arises from: (i) it is a linear technique and hence the results are readily interpreted in terms of Linear Systems Theory; (ii) if measured over an infinite frequency range, the impedance (or admittance) contains all of the information that can be gleaned from the system by linear electrical perturbation/response techniques; (iii) the experimental efficiency(amount of information transferred to the observer comparedto the amount produced by the experiment) is extraordinarily high; (iv) the validity of the data is readily determined using integral transform techniques (the Kramers Kronig transforms) that are independent of the physical processes involved. Macdonald. Reflections on the history of electrochemical impedance spectroscopy. Electrochimica Acta (2006) vol. 51 (8/9) pp

18 Advanced Passive Circuit Elements Element Impedance Symbol Pseudo- Inductor Z L = jωl L i Warburg Z W = σ (1 j) ω! i Constant Phase Element Z = 1 (jωa) 1 α (Q i,! i ) Note: j = 1 18

19 Pseudo Inductance No physical basis for inductance in an electrochemical system Yet it is observed! More on this later 19

20 Warburg Diffusion Impedance Element Discovered by Warburg Describes diffusion controlled electron transfer reactions σ can be used to calculate the diffusivity of the element in question, knowing more about the system. -z'' 45º Z W = σ (1 j) ω Z' Randles Cell with Warburg Impedance Element Included 20

21 Constant Phase Element A Generalized Impedance Element For values of α=0, the CPE behaves as a perfect Capacitor and A=C Z = 1 (jωa) 1 α (Q i,! i ) For α=2.0, the CPE behaves as a perfect inductor and A=L Sometimes, 1-α is written as α, so it is importatnt to pay attention, as it changes the actual capacitance measured. Explained by inhomogeneities in the surface C dl = A dl (ω max adjust for true capacitance ) α 1 Surface roughness uneven distribution of charge Observed more often than perfect capacitance, but easily misinterpereted effect of α 21

22 Linear Systems Theory Criteion for a linear system: The system is described by linear functions The system must be causal Response is generated only due to an imposed stimulus The system is reversible output (i) excitation (E) Removal of stimulus causes the system to relax to its previous state Reversing the singal to the starting point gives no hysteresis log(i) The system is Finite No Infinite Values of impedance are allowed e 0 E 22

23 OR Constraints of Linear Systems Theory (i) the response of the system must be described by linear (differential) equations and hence the superposition principle must hold (ii) the system must be stable, i.e., upon removal of the perturbation the system should relax to its initial state (iii) the system must be causal, that is, the system must not produce a response before t = 0 (the time at which the perturbation is applied) (iv) the impedance must be finite (physical systems cannot contain singularities in the evolution of their properties). Macdonald. Reflections on the history of electrochemical impedance spectroscopy. Electrochimica Acta (2006) vol. 51 (8/9) pp

24 X-Y Single Beam Oscilloscope Method Potentiostatic EIS Choose potential of interest Δe is chosen small for LST (around 10 mv) Plot V(t) vs. I(t) as a parametric function (as seen on the Gamry insturments in our lab) Geometric examination extracts parameters to calculate immittance (Y) Immittance is the inverse of impedance. Current Potential! Z = e i e 0 sin (φ) = i i = α β i e #e " Lissajous figure #i' #i Z = Z cos (φ) Z = Z sin (φ) 24

25 Dual Beam Oscilloscope Method Measure the potential drop across a resistor, R s Display input potential and output potential response on a dual-beam oscilloscope The phase angle can be directly observed by measuring peak to peak. b Z = R s e (jω) e R (jω) a Z = Z cos (φ) Z = Z sin (φ) e e R input (potential) response (potential drop) 25

26 Curve Fitting Complex Nonlinear Least Squares Know thy system Make your model physical Change a parameter until goodness of fit decreases. Search for the closest achievable fit Caution: Fits are not always what they seem! Excellent (within 1 order of magnitude) first guess Degenerate Circuits are possible Local solutions may be found, which are not the global solution! 26

27 Degenerate Circuits Analogs must be based on physical phenomena, otherwise, BAD assumptions can be made A circuit without physical basis is worthless from a prediction standpoint. Degenerate circuits exist for all systems, where data will fit equally well to any of the (incorrect) degenerate cases as it will to the correct one. Even worse than degenerate cases are circuits consisting of too many elements. Adding enough elements will fit practically ANY curve. 3 circuits with two time constants which can be equally well fit to an experimental spectrum Fletcher, S., Tables of Degenerate Electrical Networks for Use in the Equivalent- Circuit Analysis of Electrochemical Systems. Journal of The Electrochemical Society, (7): p

28 Advanced Extrapolation Techniques Passive materials with very high polarization resistances may stymie completion of the capacitive semicircle seen in the Randles circuit due to time requirements of measuring at low frequencies Curve Fits are questionable because an excellent first guess is required for a proper fit. -Z'' '#####$ (#####$ Z' Geometric method works well for one time constant. See attached excel sheet for automated extrapolation )#####$ #$!)#####$!(#####$!'#####$!&#####$!%#####$!"#####$ %######$ #$ %#####$ )######$ )%#####$ (######$ (%#####$ '######$ '%#####$ &######$ &######$ *+,-*.,$/,01,-$ 2*1*$ '######$ (######$ #$!'######$ #$ '######$ %######$ )######$ *######$!(######$!'######$!&######$!%######$!"######$ Extrapolated data for a real system +,-+.-,/01$ +02/03$ 4,/,$ 567$ 82/0390+/9$ 28

29 End of Part II 29

30 Reading Book Chapters Mansfeld, Florian Analytical Methods in Corrosion Science and Engineering (2005) Ch. 13 pp Books Cottis et al. Corrosion Testing Made Easy: Electrochemical Impedance and Noise. (1999) Barsoukov and J.R. MacDonald, Impedance Spectroscopy: Theory, Experiment, and Applications. (2005) pp. 608 Baboian. Electrochemical Techniques for Corrosion Engineering. (1986) Peer Reviewed Papers Macdonald, Digby D. Reflections on the history of electrochemical impedance spectroscopy. Electrochimica Acta (2006) vol. 51 (8/9) pp

31 Reading: Solartron Analytical TB/ANALYTICAL/001 High frequency, high current impedance spectroscopy: Experimental protocols enabling measurement up to 1MHz at high current densities By John Harper, Mike Rust, Brian Sayers and Andrew Savage TB/ANALYTICAL/2 Use of auxiliary channels for impedance analysis: Detecting failure mechanisms within a fuel cell / battery stack By J Harper and B Sayers TB/ANALYTICAL/3 Parallel Channel Combinations By J Harper and B Sayers TB/ANALYTICAL/004 Solartron CellTest System Impedance measurement techniques Brian Sayers Technote 04 Identification of Electrochemical Processes by Frequency Response Analysis By C Gabrielli Technote 06 An Introduction to Electrochemical Impedance Measurement By N D Cogger and N J Evans Technote 10 Frequency Response Analysis By N D Cogger and R V Webb Technote 17 Understanding Electrochemical Cells By A M Kaufmann Technote 24 Use and Applications of Electrochemical Impedance Techniques By C Gabrielli Technote 26 Analysis and Interpretation of EIS Data for Metals and Alloys By F Mansfeld Technote 29 The Application of Impedance Spectroscopy to Cementitious Systems By W. J. McCarter Technote 31 Electrochemical Impedance Spectroscopy (EIS) for Battery Research and Development By Hong Shih and Tai-Chin Lo Technote 33 The Potentiodynamic Polarization Scan By D G Evans and L L Scribner 1470 Tech Instrumentation for the Characterization of Energy Storage Devices and Multi-Cell Systems By B Sayers and A Hinton Coatings Tech Determination of Coating Adhesion Using Electrochemical Impedance Spectroscopy By A Hinton Advanced Instrumentation for Solid State Applications By A Hinton and B Sayers Beyond the Limits: 1296 Dielectric Interface By A Hinton and B Sayers Advanced Instrumentation for Civil Engineering Applications By A Hinton and B Sayers Advanced Instrumentation for Bioimpedance Measurements By A Hinton and B Sayers ImpedTech 1 Impedance Measurement Techniques: Sine Correlation By B Sayers and A Hinton 31

32 Reading: Gamry Instruments Basics of EIS Electrochemical Applications Equivalent Circuit Modeling in EIS EIS on Painted Samples Multiplexed EIS on Painted Samples DC Corrosion Techniques Determination of Double-Layer Capacitance from a CPE Rapid Electrochemical Assessment of Paints (REAP) Potentiostat Fundamentals Electrochemical Instrumentation Compliance Voltage - How Much is Enough? Reference Electrodes Care of Vycor Porous Glass Frits Trouble-Shooting Your Gamry Potentiostat with the Universal Dummy Cell 3 Measurement of Small Electrochemical Signals Accuracy Contour Plots EIS Measurement of a Very Low Impedance Lithium Ion Battery Verification of Low Impedance EIS Using a 1 mohm Resistor NEW! Open Source Scripting Understanding ir Compensation You can also find a list of Frequently Asked Questions here. 32