Electrochemical Impedance Spectroscopy University of Twente, Dept. of Science & Technology, Enschede, The Netherlands

Bernard A. Boukamp Nano-Electrocatalysis, U. Leiden, 24-28 Nov. 2008.

Research Institute for Nanotechnology

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My ‘where abouts’

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E-mail: [email protected] Address: University of Twente Dept. of Science and Technology P.O.Box 217 7500 AE Enschede The Netherlands www.ims.tnw.utwente.nl

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Electrochemical techniques

Time domain (incomplete!): • Polarisation,

(V – I )

• Potential Step, (ΔV – I (t) ) • Cyclic Voltammetry, (V f(t)- I(V ) ) • Coulometric Titration, (ΔV - ∫I dt )

steady state Next slide

relaxation dynamic relaxation

• Galvanostatic Intermittent Titration (ΔQ – V (t) ) transient Frequency domain: • Electrochemical Impedance Spectroscopy perturbation of equilibrium state (EIS)

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Time or frequency domain?

1.E-04

C u rre n t, [A ]

1.E-05

1.E-06

1.E-07

0

1000

2000

Time, [sec]

3000

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Advantages of EIS:

System in thermodynamic equilibrium Measurement is small perturbation (approximately linear) Different processes have different time constants Large frequency range, μHz to GHz (and up) • Generally analytical models available • Evaluation of model with ‘Complex Nonlinear Least Squares’ (CNLS) analysis procedures (later). • Pre-analysis (subtraction procedure) leads to plausible model and starting values (also later) Disadvantage:

rather expensive equipment, low frequencies difficult to measure

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Black box approach

Assume a black box with two terminals (electric connections). One applies a voltage and measures the current response (or visa versa). Signal can be dc or periodic with frequency f, or angular frequency ω=2πf , with: 0≤ ω< ∞

Phase shift and amplitude changes with ω!

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So, what is EIS?

Probing an electrochemical system with a small ac-perturbation, V0⋅ejωt, over a range of frequencies. The impedance (resistance) is given by:

V0 V (ω) V0 e jωt = = [ cos ϕ − j sin ϕ] Z (ω) = j ( ωt +ϕ) I (ω) I0 e I0

ω= 2πf j = √-1

The magnitude and phase shift depend on frequency. Also: admittance (conductance), inverse of impedance: I0 e j (ωt +ϕ) I0 1 = = [ cos ϕ + j sin ϕ] Y (ω) = jωt Z (ω) V0 e V0 “real +j. imaginary”

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Complex plane Impedance ≡ ‘resistance’ Admittance ≡ ‘conductance’:

Zre − jZim 1 Y (ω) = = 2 Z (ω) Zre + Zim2 hence:

Yre − jYim 1 Z (ω) = = 2 Y (ω) Yre + Yim2 Representation of impedance value, Z = a +jb, in the complex plane

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Adding impedances and admittances

A linear arrangement of impedances can be added in the impedance representation:

Ztotal = Z1 + Z2 + Z3 + ... = ∑ Zn n

A ‘ladder’ arrangement of admittances (inverse impedances) can be added in the admittance representation :

Ytotal = Y1 + Y2 + Y3 + ... = ∑Yn n

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Simple elements

The most simple element is the resistance:

1 ZR = R ; YR = R

(e.g.: electronic- /ionic conductivity, charge transfer resistance) Other simple elements: • Capacitance: dielectric capacitance, double layer C, adsorption C, ‘chemical C’ (redox) See next page • Inductance: instrument problems, leads, ‘negative differential capacitance’ !

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Capacitance?

Take a look at the properties of a capacitor: C = Charge stored (Coulombs): Change of voltage results in current, I:

Q = C ⋅V

Aε0ε d

dQ dV I= =C dt dt

dV0 ⋅ e jωt Alternating voltage (ac): I (ωt ) = C = jωC ⋅V0 ⋅ e jωt dt Impedance:

V (ω) 1 = ZC ( ω) = I (ω) jωC

Admittance:

YC ( ω) = Z (ω)−1 = jωC

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Combination of elements

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What is the impedance of an -R-Ccircuit?

1 Z (ω) = R + = R − j / ωC jωC

Admittance?

1 = Y (ω) = R − j / ωC ω2C 2 R ωC +j 2 2 2 1+ ω C R 1 + ω2C 2 R2 Semicircle

‘time constant’: constant’: ‘time RC ττ == RC

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A parallel R-C combination

The parallel combination of a resistance and a capacitance, start in the admittance representation:

R

1 Y (ω) = + jωC R

C

Transform to impedance representation:

1 1 1/ R − jωC Z (ω) = = ⋅ = Y (ω) 1/ R + jωC 1/ R − jωC R − jωR2C 1 − jωτ =R 2 2 2 1+ ω R C 1 + ω2 τ2 A semicircle in the impedance plane!

Plot next slide

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Impedance plot (RC)

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fmax = 1/(6.3x3⋅10-9x105)=530 Hz

-Zimag, [ohm]

6.0E+04

518 Hz

R = 100 kΩ C = 3 nF

4.0E+04

2.0E+04

1 MHz

0.0E+00 0.0E+00

2.0E+04

1 Hz 4.0E+04

6.0E+04

Zreal, [ohm]

8.0E+04

1.0E+05

1.2E+05

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Limiting cases

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What happens for ω > τ ? ω > τ : Z (ω) = R

1 − jωτ R R 1 1 ≈ 2 2−j ≈ 2 2−j 2 2 1+ ω τ ω τ ωτ ω RC ωC

This is best observed in a so-called Bode plot log(Zre), log(Zim) vs. log(f ), or log|Z| and phase vs. log(f )

Next slides

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Bode plot (Zre, Zim)

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Zreal Zimag

Z re a l, -Z im a g , [o h m ]

1.E+04

1.E+03

1.E+02

ω-1

1.E+01

ω-2

1.E+00

1.E-01

1.E-02 1.E+00

1.E+01

1.E+02

1.E+03

frequency, [Hz]

1.E+04

1.E+05

1.E+06

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Bode, abs(Z), phase

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90

1.E+05

abs(Z) Phase (°)

75

a b s (Z ), [o h m ]

45

1.E+03

30

15

1.E+02 1.E+00

1.E+01

1.E+02

1.E+03

Frequency, [Hz]

1.E+04

1.E+05

0 1.E+06

P h a s e (d e g r)

60

1.E+04

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Other representations

Capacitance: C(ω) = Y(ω) /jω

for an (RC) circuit:

1 ⎡1 ⎤ C (ω) = Y (ω) / jω = ⎢ + jωC ⎥ / jω = C − j ωR ⎣R ⎦ Dielectric:

ε(ω) = Y(ω) /jωC0

C0 = Aε0/d

σion d ε(ω) = Y (ω) ⋅ = ε′ − j Aε0 ωε0 Modulus:

M(ω) = Z(ω) ⋅jω

ω2CR 2 + jωR M (ω) = Z (ω) ⋅ jω = 1 + ω2C 2 R 2

d: thickness thickness d: A: surf. surf. area area A:

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Simple model

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Example: an ionically conducting solid, e.g. yttrium stabilized zirconia,

Zr1-xYxO2-½x . Apply two ionically blocking electrodes, in this case thick gold.

Schematic arrangement of sample and electrodes.

Measure the ‘resistance’ (impedance) as function of frequency:

1

Z (ω) = jωCg +

1 1 Rion + 1 2 jωCint

Equivalent circuit: (C[RC])

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Low & high f - response

Low frequency regime, series combination Rion-Cint: Z (ω) = Rion − j / 12 ωCint Straight vertical line in impedance plane. High frequency regime, parallel combination of Rion//Cgeom:

2 ωRion Cgeom

Rion Z (ω) = −j 2 2 2 2 2 1 + ω RionCgeom 1 + ω2 Rion Cgeom Semicircle through the origin.

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‘Debije’ model:

An ionic conductor between two blocking electrodes:

1 Y (ω) = Z (ω)

Impedance representation

Admittance representation

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Other representations

Zimag Zreal

‘Bode’ representation

Different Bode representation

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Diffusion, Warburg element

Semi-infinite diffusion, Flux (current) : J = −D ∂C (Fick-1) ∂x

x =0

RT Potential : E=E + ln C nF ac-perturbation: C(t ) = Co + c(t ) o

Fick-2 Boundary condition

2 ∂ C ∂ C : =D 2 ∂t ∂x

: C( x, t ) x→∞ = C

o

Redox Li-battery on inert cathode electrode.

Solution through Laplace transform: next page

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Warburg element, cont.

Laplace transform: c( x, t ) ⇒ C( x, p)

∂2C( x, p) Transform of Fick-2: p ⋅ C( x, p) = D ∂x2 General solution:

C( x, p) = A ⋅ cosh x p / D + B ⋅ sinh x p / D

RT Transform of V (t): E( p) = C( x, p) o nFC ∂C( x, p) Transform of I (t): I ( p) = −nFD ∂x x=0 (Fick-1)

Boundary Boundary condition: condition:

C( x, p ) x→∞ = 0

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Warburg impedance

Define impedance in Laplace space!

E( p) RT Z ( p) = = I ( p) (nF )2 Co D ⋅ p Take the Laplace variable, p, complex: p = s + j ω. Steady state: s ⇒ 0, which yields the impedance:

RT −1/ 2 −1/ 2 Z (ω) = = Z0 (ω − jω ) 2 o (nF ) C jωD with:

RT Z0 = (nF )2 Co 2D

In solution:

⎛ RT 1 1 Z 0 = (σ = ) 2 2 + * ⎜ * n F A 2 ⎜⎝ CO DO CR DR

⎞ ⎟ ⎟ ⎠

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Transmission line

Real life Warburg, semi-infinite coax cable with r Ω/m and c F/m:

ZW (ω) =

r jωc

Combination: • Electrolyte resistance, Re’lyte Equivalent • Double layer capacitance, Cdl circuit • Charge transfer resistance, Rct • Warburg (diffusion) impedance, Wdiff

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Equivalent Circuit Concept

se m

ic ir

cl e

ω

45°

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Instruments

Measurement methods Bulk, conductivity: • two electrodes • pseudo-four electrodes • true four electrodes Electrode properties: • three electrodes

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Frequency Response Analyser Multiplier: Vx(ωt)×sin(ωt) & Vx(ωt)×cos(ωt) Integrator: integrates multiplied signals Display result: a + jb = Vsign/Vref

But be aware of the input impedance of the FRA!

Impedance: Zsample = Rm (a + jb)

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Potentiostat, electrodes Vpwr.amp = A Σk Vk A= amplification Vwork – Vref = Vpol. + V3 + V4 Current-voltage converter provides virtual ground for Work-electrode.

General schematic

Source of inductive effects

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Data validation

Kramers-Kronig relations (old!) Real and imaginary parts are linked through the K-K transforms: Kramers-Kronig conditions: • causality • linearity • stability • (finiteness)

Response only Response due to input State of scales linearly signalmay system with input notsignal change during measurement

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Relations,

Putting ‘K-K’ in practice ∞

2ω Z re ( x) − Z re (ω) Real → imaginary: Z im (ω) = dx 2 2 ∫ π 0 x −ω

not a singularity!

∞

2 xZ im ( x) − ωZ im (ω) Imaginary → real: Z re (ω) = R∞ + ∫ dx 2 2 π0 x −ω Problem: Finite frequency range: extrapolation of dispersion ) assumption of a model.

[1] M. Urquidi-Macdonald, S.Real & D.D. Macdonald, Electrochim.Acta, 35 (1990) 1559. [2] B.A. Boukamp, Solid State Ionics, 62 (1993) 131.

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Linear KK transform

Linear set of parallel RC circuits:

τk = Rk⋅Ck

Create a set of τ values: τ1 = ωmax-1 ; τM = ωmin-1 with ~7 τ-values per decade (logarithmically spaced).

If this circuit fits the data, the data must be K-K transformable! [3] B.A.Boukamp, J.Electrochem.Soc, 142 (1995) 1885

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Actual test

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Fit function simultaneously to real and imaginary part:

1 − jωi ⋅ τ k Z KK (ωi ) = R∞ + ∑ Rk 2 2 1 + ω ⋅ τ k =1 i k M

Set of linear equations in Rk, only one matrix inversion! Display relative residuals:

Δreal =

Zre,i − Z KK, re (ωi ) Zi

, Δimag =

Shortcut to KKtest.lnk

It works like a ‘K-K compliant’ flexible curve

Zim,i − Z KK,im (ωi ) Zi

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Example ‘K-K check’

Impedance of a sample, not in equilibrium with the ambient.

χ2KK = 0.9·10-4 χ2CNLS = 1.4 ·10-4

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Finite length diffusion

Particle flux at x=0: ~ dC( x, t ) J (t ) = −D dt x=0

Fick’s 2nd law: dC( x, t ) ~ d2C( x, t ) =D dx2 dt

But now a boundary condition at x = L. Activity of A is measured at the interface at x=0. with respect to a reference, e.g. Amet

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Finite length diffusion Replace concentration by its perturbation:

c( x, t ) = C( x, t ) − C 0

Impermeable dC( x, t ) boundary at x =L:

dx

= 0 FSW x=l

Ideal source/sink with C = CL (=C0): C( x, t ) = Cl = C 0 x=l

(

General expression for permeable dC( x, t ) boundary:

dx

[

x =l

)

FLW

= −k C( x, t ) x=l − Cl

]

General!

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FLD, continued

Voltage with respect to reference C0 (a0):

RT ax=0 RT ⎡ d ln a ⎤ E(t ) = ln 0 = c( x, t ) x=0 0 ⎢ ⎥ nF a nFC ⎣ d ln C ⎦ Current through interface at x = 0:

~ dc( x, t ) I (t ) = nF ⋅ S ⋅ J (t ) = −nF ⋅ S ⋅ D dx x=0 Assumption: Δa 1, gradient search

Successfuliteration: iteration: Successful Sold new 3000 s (~ 0.3 mHz) fmin ~ 10 mHz

MEASURE RESPONSE RESPONSE IN IN MEASURE THE TIME TIME DOMAIN! DOMAIN! THE

10 10 99 88 -2 Current Current[μA.cm [μA.cm-2] ]

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77

4.05V 4.05V 4.10V 4.10V

66 4.00V 4.15V 4.00V 5 4.15V 5 44

4.20V 4.20V

3.95V 33 3.95V 22

3.90V 3.90V 3.85V 11 3.85V 3.80V 3.80V 0 0 1000 2000 00 1000 2000 Time[s] [s] Time

3000 3000

Current response of a 0.75μm RFfilm to sequential 50mV potential steps from 3.80V to 4.20V.

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Fourier transform

Fourier transform of a temporal function X (t): ∞

X (ω) = ∫ X (t ) ⋅ e− jωt dt 0

Impedance:

V (ω) Z (ω) = I (ω)

E.g. with a voltage step, V0:

V0 V (ω) = jω

Model function: Laplace transform of transport equations and boundary conditions, with p = s +j ω. Set s = 0: ⇒ impedance

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Fourier Transform

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Two problems with F-T:

X(t )=at + b

• Data is discrete: approximate by summation (X =at + b)

Xi -1

• Data set is finite (next slide)

Xi

Very Simple Summation Solution (VS3):

L M N LX cosωt − X − j∑ M N N

X (ω) = ∑ Xi sin ωti − Xi −1 sin ωti −1 + i =1

N

i =1

i

i

ti -1 ti

O b gP Q a O − b sin ωt − sin ωt g ω P ω Q

a cosωti − cosωti −1 ω −1 + ω

i −1 cosωti −1

i

i −1

−1

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Simple exponential extension

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Assume finite value, Q0, for t ⇒ ∞, this value can be subtracted before total FT. Fit exponential function to selected data set in end range: Q(t ) = Q0 + Q1 e−t /τ Full Fourier Transform:

z

z

tN

∞

Q0 − jωt X (ω) = [ X (t ) − Q0 ] e dt − j + Q1 e −t / τ e − jωt dt ω 0 t N

Analytical transform of exponential extension:

z ∞

Q1 e tN

−t / τ

e

− jωt

dt = Q1 ⋅ e

−t N / τ

R τ cos ωt − ω sin ωt ⋅S T ω +τ −1

N 2

−2

N

ω cos ωt N + τ −1 sin ωt N +j ω 2 + τ −2

U V W

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Fourier transformed data

Simple discrete Fourier transform: tN

X (ω) = ∫ X (t ) e− jωt dt ≈ 0

X (tk ) − X (tk −1 ) (cosωt − j sin ωt ) ∑ tk − tk −1 k =1 N

Correction / simulation for t→∞:

X (t ) = X 0 + X 1e −t / τ X 0 = leakage current.

V (t ) Impedance: Z (ω) = I (t )

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V-step experiment

Sequence of 10 mV step Fourier transformed impedance spectra, from 3.65 V to 4.20 V at 50 mV intervals. Fmin = 0.1 mHz

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CNLS-fit of FT-data Circuit Description R1 R2 Code:

R(RQ)OT

*)

Fit result: χ2CNLS = 3.7⋅10-5 *) O

= ‘FLW’

T = ‘FSW’

Q3, Y0 ,, n O4, Y0 ,, B T5, Y0 ,, B

: : : : : : : :

550 49 6.8⋅10-3 0.96 0.047 30 0.028 5.9

0.5 % 10 % 12 % 8 % 1.5 % 2.4 % 2.9 % 2.9 %

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Bode Graph Double logarithmic display almost always gives excellent result !

‘Bode plot’, Zreal and Zimag versus frequency in double log plot

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Conclusions

Electrochemical Impedance Spectroscopy: • Powerful analysis tool • Subtraction procedure reveals small contributions • Presents more ‘visual’ information than time domain • Almost always analytical expressions available • Equivalent Circuit approach often useful • Data validation instrument available (KK transform) • Also applicable to time domain data (FT: ultra low frequencies possible) • Able to analyse complex systems

Unfortunately, analysis requires experience!

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Not just electrochemistry!

Data analysis strategy is applicable to any system where: • a driving force • a flux can be defined/measured. Examples: • mechanical properties, e.g. polymers: G (ω) or J (ω) & γ • catalysis, pressure & flux, e.g. adsorption • rheology • heat transfer, etc.

No need to measure in the frequency domain!

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Last slide

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Effect of truncation

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Delta-Real Delta-Imag

8

tmax tmax = 100 = 100 s, s, τ =τ 20 = 30 sec, sec, Y Y(t(max tmax ) =) 0.67% = 3.6%

6

Delta-Real Delta-Imag

Δre, Δim , [%]

4 2

tmax = 100 s, τ = 40 sec, Y (tmax) = 8.2%

0 -2 -4 -6 -8 -10 0.01

Δim =

0.1

Zimag (ω) − Zim,tr (ω) Z (ω)

1

Frequency, [Hz]

10

Δre =

100

Zreal (ω) − Zre,tr (ω) Z (ω)

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More Fourier transform

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Method Martijn Lankhorst: ¾ fit polynomials to small sets m t of data points (sections): Pm (t ) t = ∑ Ak t k

piece wise wise piece integration integration

r

q

k =0

¾ analytical transformation to frequency domain: m i +1

P (ω) t = ∑ ∑ Ai tr

q

i = 0 k =1

(i − 1)! (i − 1 − k )!

⋅

t

i −1− k q

⋅e

− jωt q

−t

i −1− k r

⋅e

− jωt r

( jω) k +1

More general extrapolation function (stretched exponential):

Q(t ) = Q0 + Q1 ⋅ e

−(t / τ )α

, 0 ≤ α ≤1

(Fourier transform complicated, can be done numerically)

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Non linear effects

Electrode response based on Butler-Vollmer: ⎡ αRTa F η − (1−αRTa ) F η ⎤ −e I = I0 ⎢e ⎥ ⎣ ⎦ When the voltage amplitude is too large, the current response will contain higher harmonics (i.e. is not linear with V). Substituting a = αaF/RT, b = (1-αc)F/RT and a serial expression for exp(), we obtain:

0.05 0.05

Current, [A] Current, [A]

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0

0

I0 = 1 mA αa = 0.4 T = 23°C

-0.05 -0.05

-0.1 -0.1 -0.2 -0.2

-0.1 -0.1

0 0 Polarisation, [V] Polarisation, [V]

0.1 0.1

0.2 0.2

⎡ ⎤ a2η2 a3η3 b2η2 b3η3 + + ... −1+ bη− + + ...⎥ I = I0 ⎢1 + aη+ 2! 3! 2! 3! ⎣ ⎦ ⎡ ⎤ (a2 − b2 )η2 (a3 + b3 )η3 = I0 ⎢(a + b)η+ + + ...⎥ 2! 3! ⎣ ⎦

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Higher-order terms

At zero bias, with the perturbation voltage, ∆·ejωt, this equation yields:

I (t )

2 2 3 3 ⎡ ⎤ − + ( a b ) ( a b ) j 3ωt jωt j 2ωt = I0 ⎢(a + b)Δe + Δe + Δe + ...⎥ 2! 3! ⎣ ⎦

This clearly shows the occurrence of higher-order terms. When the polarization current is ‘symmetric’ the even terms will drop out as a = b. At a dc-polarization the response is more complex: 3 3 2 ⎧⎪⎡ ⎤ jωt ( a b ) + η 2 2 I (t ) = I0 ⎨⎢(a + b) + (a − b )η+ + ...⎥ Δe + 2! ⎪⎩⎣ ⎦

⎡ a2 − b2 (a3 + b3 )η ⎤ j 2ωt ⎡ a3 + b3 ⎤ j 3ωt +⎢ + + ...⎥ Δe + ⎢ + ...⎥ Δe + ... 2! ⎣ 2! ⎦ ⎣ 3! ⎦

}

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The derivatives!

Having the derivatives is essential! •

best method, calculate the derivatives on basis of the function: accuracy and speed.

•

Second best: numerical evaluation* (for proper derivatives we have to calculate F(xi,a1..M) 2M +1 times!!

F ( xi , a1.., a j + Δa j ,..aM ) − F ( xi , a1.., a j − Δa j ,..aM ) ∂ F ( xi , a1..M ) = ∂a j 2Δa j * This is actually an approximation