Report

Impedance-based techniques 3-4-2014 ac source Impedance overview Potentiometer to null dc - Perturb cell w/ small magnitude alternating signal & observe how system handles @ steady state voltage - - cell R C V dc null detector RA Ru R RB I1 + I 2 – + dc null detector I1 I1RA = I2RB I1Ru = I2R Ru = R(RA/RB) I2 - Advantages: - High-precision (indef steady long term avg) - Theoretical treament - Measurement over wide time (104 s to ms) or freq range (10-4 Hz to MHz) Prototypical exp: faradaic impedance ,cell contains solution w/ both forms of redox couple so that potential of WE is fixed Cell inserted as unknown into one arm of impedance bridge & R, C adjusted to balance Determine values of R & C at measurement frequency Impedance measured as Z(w) Lock-in amplifiers, frequency response analyzers Interpret R, C in terms of interfacial phenom Faradaic impedance (EIS) high precision, eval heterogen charge-transfer parameters & DL structure ac voltammetry E t - 3 electrode cell (DME ac polaragraphy) - dc mean value Edc scanned slowly w/ time plus sine component (~ 5 mV p-to-p) Eac - Measure magnitude of ac component of current and phase angle w.r.t. Eac - dc potential sets surf conc. of O and R: CO(0,t) & CR(0,t) differ from CO* and CR* diffusion layer - Steady Edc thick diffusion layer, dimensions exceed zone affected by Eac CO(0,t) & CR(0,t) look like bulk to ac signal (DPP relies on same effect) - Start w/ solution containing only one Redox form & obtain contin plots of iac amp & phase angle vs. Edc represent Faradaic impedance at continuous ratios of CO(0,t) & CR(0,t) - EIS and ac voltammetry involve v. low amp excitation sig & depend on current-overpotential relation virtually linear @ low overpotential ac circuits e or i 2(p+f)/w 2p/w t e = E sin wt i = I sin (wt + f) w p/2 İ -p/2 Resistor 0 Capacitor q = Ce i = E/XC sin (wt + p/2) Ė=İR i = C(de/dt) XC = 1/wC i leads e p/w e or i f Ė e or i p p/w • Rotating vector (phasor) • Consider relationship between i, e rotating at w (2pf), separated by phase angle f. 2p/w t p/w 2p/w t İ Ė = –jXCİ = −1 ac circuits: RC Resistor ĖR = İ R Capacitor ĖC = –jXCİ Ė = ĖR + ĖC i = I sin (wt + f) Series Ė = İ (R – jXC) Ė=İZ XC = 1/wC Polar Form Z = Zejf Z(w) = ZRe – jZIm |Z|2 = R2 + XC2 = (ZRe)2 + (ZIm)2 tan f = ZIm/ZRe= XC/R = 1/wRC f = 0 R only f = p/2 C only Y = Ze –jf Y f admittance R –jXC = −1 f Z Bode plots RC parallel Ė = İ [RXC2/(R2 + XC2) – jR2XC/(R2 + XC2)] RC series -3 -2 -1 R = 100 W C = 1 mF Ė = İ (R – jXC) 2 1 log|Z| log|Z| 9 8 7 6 5 4 3 2 1 0 3 0 -3 0 1 2 3 4 5 6 0 1 -2 7 100 90 80 70 60 50 40 30 20 10 0 -1 0 f f -2 -1 2 3 4 5 6 7 -1 log f -3 -2 1 2 log f 3 4 5 6 7 -3 -2 100 90 80 70 60 50 40 30 20 10 0 -1 0 log f 1 2 log f 3 4 5 6 7 RC series Ė = İ (R – jXC) 18 16 14 12 10 8 6 4 2 0 RC parallel R = 100 W C = 1 mF Ė = İ [RXC2/(R2 + XC2) – jR2XC/(R2 + XC2)] 60 w 50 w 40 ZIm ZIm x 107 Nyquist plots 30 20 10 0 0 100 50 ZRe 150 0 103 104 105 20 40 60 ZRe 80 100 Equivalent circuit of cell i c Cd RW Zf ic + if if Zf = = Randles Equivalent Circuit - Frequently used - Parallel elements because i is the sum of ic, if - Cd is nearly pure C (charge stored electrostatically) - Faradaic processes cannot be rep by simple R, C which are independent of f (instead consider as general impedance Zf) Zw Rep charge transfer between Rct electrode-electrolyte Rs Cs Faradaic Impedance - Simplest rep as series resistance Rs, psuedocapacitance Cs - Alternative, pure resistance Rct and Warburg Impedance (kind of resistance to mass transfer) - Components of Zf not ideal (change with f) Equivalent Circuits - Rep cell performance at given f, not all f - Chief objective of faradaic impedance: discover f dependence of Rs, Cs apply theory to transform to chem info - Not unique Characteristics of equiv circuit i c Cd RW Zf ic + if if Zf = Rs Cs • Measurement of total impedance includes RW and Cd • Separate Zf from RW, Cd by considering f dependence or by eval RW and Cd in separate experiment w/o redox couple • Assume Zf can be expressed as Rs, Cs in series = + = + = sin = cos + sin = + Description of chemical system O + ne ⇄ R (O, R soluble) E = E[i, CO(0,t), CR(0,t)] = cos + sin = sin Because E is a function of 3 variables that depend on t, total differential is a combination of partial differentials (0, ) (0, ) = + + (0, ) (0, ) (0, ) (0, ) = + + (0, ) (0, ) by mass transfer considerations Find , = cos Initial conditions: CO(x,0) = CO*, CR(x,0) = CR* Recall from section 8.2.1: 0, = 1 2 ∗ 1+ 2 Notice the sign convention is opposite of usual 0, = − 1 2 ∗ 1+ 2 Determination of CO(0,t), CR(0,t) O + ne ⇄ R (O, R soluble) = cos + sin = sin E = E[i, CO(0,t), CR(0,t)] (0, ) (0, ) = cos + + Find (0, ) (0, ) by mass transfer considerations , Initial conditions: CO(x,0) = CO*, CR(x,0) = CR* Recall from section 8.2.1: 0, = 1 2 ∗ 1+ 2 0, = − 1 2 Recall Laplace Transform: () = ∞ − 0 = (s) Convolution integral: () = F t ∗ G(t) = − −1/2 ∞ = ∗ 1+ 2 −1/2 − 0 −1 () = f t ∗ g(t) = 0, = + 1 1 2 1 2 0 − 1/2 − 0 0 ∗ = −1/2 0, = ∗ − 1 1 1 2 0 2 − 1/2 Evaluation of O + ne ⇄ R − 0 1/2 (O, R soluble) = cos + sin = sin E = E[i, CO(0,t), CR(0,t)] (0, ) (0, ) = cos + + 0, = 0 ∗ + − = 1/2 1 1 2 0 1 2 0 − 1/2 0, = ∗ − 1 1 1 2 0 2 sin − 1/2 Recall trig identity sin w(t – u) = sin wt cos wu – cos wt sin wu Can be derived from Euler identity ejx = cos x – j sin x Also recall: sin x = (ejx – e–jx)/2j, cos x = (ejx + e–jx)/2 0 sin − = sin 1/2 0 cos − cos 1/2 0 sin 1/2 − 1/2 Evaluation of O + ne ⇄ R (O, R soluble) = sin E = E[i, CO(0,t), CR(0,t)] − 0 1/2 = cos + sin (0, ) (0, ) = cos + + 0, = 0 ∗ + 1 1 2 1 2 0 − 1/2 sin − = sin 1/2 0 0, = cos − cos 1/2 ∗ 0 − 1 1 1 2 0 2 − 1/2 sin 1/2 Now consider time range of interest. At t=0, CO(0, t) = CO* & CR(0, t) = CR* After few cycles: steady state is reached (no net electrolysis during any full cycle) Interest is in steady state Integrals rep transition from initial cond to steady state Because u–½ appears, integrands only significant at short times Obtain steady state by letting int limits go to − 0 1/2 Evaluation of O + ne ⇄ R (O, R soluble) = cos + sin = sin E = E[i, CO(0,t), CR(0,t)] (0, ) (0, ) = cos + + sin − = sin 1/2 ∞ 0 ∞ cos − cos 1/2 0 sin 1/2 Recall: sin x = (ejx – e–jx)/2j, cos x = (ejx + e–jx)/2 sin − = sin 1/2 −1/2 ∞ 0 ∞ = + − − cos 2 1/2 −1/2 − ∞ 0 + − 2 1/2 = −1/2 0 ∞ 0 + 2 1/2 ∞ 0 − = 1/2 2 − 1/2 + 1/2 1/2 2 1/2 1/2 = 2 1/2 − − 1/2 1/2 = − = 2 2 1/2 2 − 1/2 1/2 2 1/2 1/2 1/2 Can be derived from Euler identity ejx = cos x – j sin x Surface concentration expressions O + ne ⇄ R (O, R soluble) = cos + sin = sin E = E[i, CO(0,t), CR(0,t)] (0, ) (0, ) = cos + + 0, = ∗ 1 + 1 2 1 2 0 − 1/2 sin − = 2 1/2 1/2 = sin − 2 1/2 + 2 0, = 2 1 2 1 2 sin − cos sin + cos − 1 1 1 2 0 2 − 1/2 cos 0, 0, ∗ 0, = ∗ = ∗ − 2 0, =− 2 sin − cos 1 2 1 2 sin + cos Evaluation of Rs, Cs O + ne ⇄ R (O, R soluble) = cos + sin = sin E = E[i, CO(0,t), CR(0,t)] (0, ) (0, ) = cos + + 0, = 2 1 2 0, =− 2 sin + cos = + 1/2 cos + 1/2 sin Finding Rs, Cs depends on finding Rct, bO, bR Rct from heterogeneous charge-transfer kinetics s/w1/2 and 1/sw1/2 from mass-transfer effects = f dependent R = + − = + −1/2 − −1/2 Pseudo C ZW 1 2 1 2 − 1/2 sin + cos 1/2 Called pseudo C because energy is stored electrochemically (in rev faradaic redox reaction) rather than electrostatically (as in Cd) Kinetic parameters from impedance kf O+e⇄R kb (0, ) (0, ) = cos + + (O, R soluble) sine component is small, electrode’s mean potential at equilibrium use linearized i-h characteristic (see 3.4.30) to describe system (electronic current convention) (0, ) (0, ) = − + ∗ ∗ 0 = = cos + sin = 1 = = 0 1 2 2 1/2 + ∗ = ∗ = ∗ = + 1/2 cos + 1/2 sin k0 can be evaluated through i0 when Rs, Cs are known 1 1/2 ∗ R or XC − 0 Rct Rs 1/wCs Slope = s w–1/2 Limiting case: reversible system, fast charge transfer i0 , Rct 0, Rs s/w1/2 − = + = + − 2 = = 1 = = 0 1/2 = + −1/2 − −1/2 1/2 1 2 2 1/2 ∗ 1 = 1/2 ZW alone. Mass-transfer impedance (applies to any electrode reaction) minimum impedance. If kinetics are observable, Rct contributes and Zf is greater. + 1 1/2 ∗ Large concentrations reduce mass-transfer impedance Concentration ratio significantly different than one make s and Zf large Large transfer rates only achieved when concentrations are comparable (Zf minimal near E0’) Impedance measurements easiest near E0’ Limiting case: reversible system, fast charge transfer i0 , Rct 0, Rs s/w1/2 = = + −1/2 − −1/2 1 2 2 1/2 ∗ + 1 1/2 0 ≤ f ≤ 45o, always a component of iac inphase (0o) with Eac and can be measured with phase sensitive detector (lock-in amplifier) basis for discriminating against charging current in ac voltammetry 1 2 2 1/2 2 ∗ + 1 1/2 ∗ 2 = RW = s/w1/2 ∗ tan f = 1/wRsCs = (s/w1/2) / (Rct + s/w1/2) = 1 = = 0 f < 45o 1/wCs = s/w1/2 = + − − 1/2 Rct Electrochemical impedance spectroscopy ic Randles Equivalent Circuit - Frequently used - Parallel elements because i is the sum of ic, if - Cd is nearly pure C - Faradaic processes cannot be rep by simple R, C which are independent of f (instead consider as general impedance Zf) Cd RW Zf ic + if if Zf = = Rs Cs Rct Zw • Measurement of cell characteristics includes RW and Cd • Separate Zf from RW, Cd by considering f dependence (EIS) or by eval RW and Cd in separate experiment w/o redox couple (Impedance bridge) EIS: study the way Z = RB – j/wCB = ZRe – jZIm varies with f Extract RW, Cd, Rs, and Cs Eliminates need for separate measurements w/o redox species Eliminates need to assume redox species has no effect on nonfaradaic impedance Electrochemical impedance spectroscopy ic - Based on similar methods used to analyze circuits in EE practice - Developed by Sluyters and coworkers - Variation of total impedance in complex plane (Nyquist plots) 60 Cd RW Zf 50 ic + if if ZIm Zf = Rs Cs = = Ω + 2 + 2 1 = = 30 20 10 Measured Z is expressed as series RB + CB ZRe = RB , ZIm = 1/wCB w 40 2 + 2 + 2 0 103 104 105 0 20 40 60 80 100 ZRe See Section 10.1.2 Can be shown by E = ERW + ECd(ERs + ECs)/(ECd + ERs +ECs) ER = IR, EC = –j/wC A = Cd/Cs , B = wRsCd Variation of total impedance ic Cd RW Rs = Ω + = = + if = 1 = = 2 + 2 + 2 A = Cd/Cs , B = wRsCd Zf ic + if Zf = = Ω + 2 + 2 1 = 1/2 1/2 Cs + −1/2 1/2 + 1 + −1/2 1/2 + 1 2 2 2 + 2 2 + −1/2 + −1/2 1/2 + 1 + 2 2 + −1/2 2 2 Obtain chem info by plotting Zim vs. ZRe Impedance: low-frequency limit = Ω + + −1/2 1/2 + 1 2 + 2 2 + −1/2 2 = + −1/2 1/2 + 1 2 2 + −1/2 1/2 + 1 + 2 2 + −1/2 As w 0 = Ω + + −1/2 = − Ω − + 2 2 ZIm Slope = 1 ZRe = −1/2 + 2 2 - Linear w/ unit slope and extrapolated line intersects ZRe axis at Ω + − 2 2 - Indicative of diffusion-controlled electrode process (under mass transfer control) - As f increases, Rct and Cd become more important leading to departure from ideal behavior 2 Impedance: high-frequency limit Cd Cd As w RW Zf = Ω − − − Ω − 2 2 + = Ω + 2 = 2 2 w = 1/RctCd ZIm RW ZRe RW + Rct RW 1 + 2 2 2 Rct = 2 1 + 2 2 2 - Circular plot center at (RW + Rct/2, 0), r = Rct/2 - At high f, all i is ic and only impedance comes from RW - As f decreases, Cd significant ZIm - At v. low f, Cd high Z, i mostly through Rct and RW - Expect departure in low f because ZW is important there Impedance: applications to real systems w = 1/RctCd ZIm ZIm RW ZRe Kinetic control w = 1/RctCd ZIm RW ZRe Mass-transfer control RW + Rct ZRe RW + Rct In real systems, both regions may not be well defined depending on Rct and its relation to ZW (s). If system is kinetically slow, large Rct and only limited f region where mass transfer significant. If Rct v. small in comparison to RW and ZW over nearly all s, system is so kinetically facile that mass transfer always plays a role. Limits to measurable k0 by faradaic impedance ZIm Upper limit Mass-transfer Kinetic control - Rct must make sig contribution to Rs control (Rct ≥ s/w1/2) w = 1/RctCd - k0 ≤ (Dw/2)1/2 (assume DO=DR, CO* = CR*) - Highest practical w is determined by RuCd ≤ cycle period of ac stimulus - For UME, useful measurements at w ≤ 107 s-1, with D ~ 10-5 cm2/s, k0 ≤ 7 cm/s RW - Think aromatic species to cation/anion RW + Rct ZRe radicals in aprotic solvents (k0 > 1 cm/s) - Cs ≥ Cd and Rs ≥ RW 1 1 = = − 1/2 ∗ i0 = FAk0C (Eqn 3.4.7) ∗ 1/2 2 0 2 Lower limit - Large Rct, ZW negligible - Rct cannot be so large that all i through Cd (Rct ≤ 1/wCd) k0 ≥ RTCdw/F2C*A - For C* = 10-2 M and w = 2p x 1 Hz, T=298, Cd/A = 20 mF/cm2 k0 ≥ 3 x 10-6 cm/s EIS and other applications • More complicated systems (couple homogeneous reactions, adsorbed intermediates) can also be explored with EIS • General strategy: obtain Nyquist plots and compare to theoretical models based on appropriate eqns rep rates of various processes and contributions to i(t) • May be useful to represent system by equivalent circuit (R, C, L), but not unique and cannot be easily predicted from reaction scheme • Electrode surf roughness and heterogeneity can also affect ac response (smooth, homogeneous Hg electrodes generally better than solid electrodes) • Application to variety of systems: corrosion, polymer film, semiconductor electrodes Instrumentation • Impedance measurements made in either f domain with frequency response analyzer (FRA) or t domain using FT with a spectrum analyzer • FRA generates e(t) = D sin(wt) and adds to Edc – take care to avoid f and amplitude errors that can be introduced by the potentiostat, particularly at high f – V∝ i(t) to analyzer, mixed with input signal and integrated over several periods to give ZIm, ZRe – Frequency range of 10 mHz to 20 MHz • Spectrum analyzer: Echem system subjected to potential variation resultant of many frequency (pulse, white noise signal), and i(t) is recorded – Stimulus and response converted via FT to spectral rep of amp and f vs. f – Allow interpretation of experiments in which several different excitation signals applied to chem system at same time (multiplex advantage) – Responses are superimposed but FT resolves them Additional references/further reading • Sluyters-Rehbach, Pure & Appl. Chem. 1994, 66:1831-1891. • Orazem & Tribollet, Electrochemical Impedance Spectroscopy, 2008, John Wiley & Sons: Hoboken, NJ.