Impedance-based techniques

Impedance-based techniques

3-4-2014

Impedance overviewac source

cellR

C dc null detector

ac null detector

Potentiometer to null dc voltage

RB

dc null detector

RA

V

– +

Ru

RI1 + I2

I1

I2

I1RA = I2RB

I1Ru = I2R

Ru = R(RA/RB)

- Perturb cell w/ small magnitude alternating signal & observe how system handles @ steady state

- 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

t

E

- 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• Rotating vector (phasor)• Consider relationship between

i, e rotating at w (2pf), separated by phase angle f.

p/w 2p/w

2(p+f)/w

t

e or

i

e = E sin wti = I sin (wt + f)

fĖ

İ

w

/2p

p

- /2p

0 Ė = İ RResistor

p/w 2p/w

t

eor

i

Capacitor

p/w 2p/w

te

or i

q = Ce i = C(de/dt)i = E/XC sin (wt + p/2)

İ

Ė = –jXCİ

i leads e

𝑗=√−1

XC = 1/wC

ac circuits: RCĖR = İ RResistor

Capacitor

Ė = ĖR + ĖC

ĖC = –jXCİ

Ė = İ (R – jXC)

Ė = İ Z

Z(w) = ZRe – jZIm

|Z|2 = R2 + XC2 = (ZRe)2 + (ZIm)2

tan f = ZIm/ZRe= XC/R = 1/wRC

i = I sin (wt + f)

f = 0 R onlyf = p/2 C only

R

–jXC

𝑗=√−1Z

Y

f

f

Series

Z = Zejf Y = Ze –jf

admittance

Polar Form

XC = 1/wC

Bode plots

0123456789

-3 -2 -1 0 1 2 3 4 5 6 7

log|Z|

log f

0102030405060708090

100

-3 -2 -1 0 1 2 3 4 5 6 7

f

log f

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3 4 5 6 7

log|Z|

log f

0102030405060708090

100

-3 -2 -1 0 1 2 3 4 5 6 7

f

log f

RC series

Ė = İ (R – jXC)R = 100 W C = 1 mF

RC parallelĖ = İ [RXC

2/(R2 + XC2) – jR2XC/(R2 + XC

2)]

Nyquist plots

02468

1012141618

0 50 100 150

Z Im

x 10

7

ZRe

w

0

10

20

30

40

50

60

0 20 40 60 80 100

Z Im

ZRe

w

103104105

Ė = İ (R – jXC) Ė = İ [RXC2/(R2 + XC

2) – jR2XC/(R2 + XC2)]

RC parallelRC series R = 100 W C = 1 mF

Equivalent circuit of cell

Zf

Cd

RW

ic + if

ic

if

Rct

ZwZf Rs Cs

= =

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)

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

Rep charge transfer between electrode-electrolyte

Characteristics of equiv circuit• 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

Zf

Cd

RW

ic + if

ic

if

Zf Rs Cs

=

• Assume Zf can be expressed as Rs, Cs in series𝐸=𝐸𝑅𝑠+𝐸𝐶𝑠 𝐸=𝑖 𝑅𝑠+

𝑞𝐶𝑠

𝑑𝐸𝑑𝑡 =𝑅𝑠

𝑑𝑖𝑑𝑡 +

𝑖𝐶𝑠

𝑖=𝐼 sin𝜔𝑡𝑑𝐸𝑑𝑡 =𝑅𝑠 𝐼𝜔 cos𝜔𝑡+

𝐼𝐶𝑠sin𝜔𝑡

Description of chemical systemO + ne R⇄ (O, R soluble)

E = E[i, CO(0,t), CR(0,t)]

𝑑𝐸𝑑𝑡 =( 𝜕𝐸𝜕𝑖 ) 𝑑𝑖𝑑𝑡 +[ 𝜕𝐸

𝜕𝐶𝑂(0 , 𝑡) ] 𝑑𝐶𝑂(0 , 𝑡)𝑑𝑡 +[ 𝜕𝐸𝜕𝐶𝑅 (0 , 𝑡) ] 𝑑𝐶𝑅 (0 ,𝑡)𝑑𝑡

𝑖=𝐼 sin𝜔𝑡𝑑𝐸𝑑𝑡 =𝑅𝑠 𝐼𝜔 cos𝜔𝑡+


𝑑𝐸𝑑𝑡 =𝑅𝑐𝑡

𝑑𝑖𝑑𝑡 +𝛽𝑂

𝑑𝐶𝑂(0 ,𝑡 )𝑑𝑡 +𝛽𝑅

𝑑𝐶𝑅(0 ,𝑡)𝑑𝑡

𝑑𝑖𝑑𝑡=𝐼 𝜔cos𝜔𝑡 Find

𝑑𝐶𝑂 (0 , 𝑡)𝑑𝑡 ,

𝑑𝐶𝑅(0 , 𝑡 )𝑑𝑡 by mass transfer considerations

𝐶𝑂 (0 ,𝑠)= 𝑖 (𝑠)

𝑛𝐹𝐴𝐷𝑂12 𝑠

12

+𝐶𝑂❑∗

𝑠 𝐶𝑅 (0 , 𝑠 )=− 𝑖 (𝑠)

𝑛𝐹𝐴𝐷𝑅12 𝑠

12

+𝐶𝑅❑∗

𝑠

Initial conditions: CO(x,0) = CO*, CR(x,0) = CR*

Recall from section 8.2.1: Notice the sign convention is opposite of usual

Because E is a function of 3 variables that depend on t, total differential is a combination of partial differentials

Determination of CO(0,t), CR(0,t)O + ne R⇄ (O, R soluble)

E = E[i, CO(0,t), CR(0,t)] 𝑖=𝐼 sin𝜔𝑡𝑑𝐸𝑑𝑡 =𝑅𝑠 𝐼𝜔 cos𝜔𝑡+


𝑑𝐸𝑑𝑡 =𝑅𝑐𝑡 𝐼 𝜔cos𝜔𝑡+ 𝛽𝑂

𝑑𝐶𝑂(0 ,𝑡)𝑑𝑡 +𝛽𝑅

𝑑𝐶𝑅 (0 , 𝑡 )𝑑𝑡

Find 𝑑𝐶𝑂 (0 , 𝑡 )𝑑𝑡 ,

𝑑𝐶𝑅(0 , 𝑡 )𝑑𝑡 by mass transfer considerations

𝐶𝑂 (0 ,𝑠)= 𝑖 (𝑠)

𝑛𝐹𝐴𝐷𝑂12 𝑠

12

+𝐶𝑂❑∗

𝑠 𝐶𝑅 (0 , 𝑠 )=− 𝑖 (𝑠 )

𝑛𝐹𝐴𝐷𝑅12 𝑠

12

+𝐶𝑅❑∗

𝑠

Initial conditions: CO(x,0) = CO*, CR(x,0) = CR*

Recall from section 8.2.1:

Recall Laplace Transform:

(s) 𝐿 {(𝜋𝑡 )− 1/2 }=∫0

∞

(𝜋𝑡 )−1/2𝑒− 𝑠𝑡 𝑑𝑡=𝑠−1 /2

𝐿 { 𝑓 (𝑠 )𝑔 (𝑠)}=F ( t )∗G(t )=∫0

𝑡

𝐹 (𝑡−𝜏 )𝐺 (𝜏 ) 𝑑𝜏 𝐿−1 {𝐹 (𝑠 )𝐺(𝑠)}=f ( t )∗g (t)=∫0

𝑡

𝑓 (𝑡−𝜏 )𝑔 (𝜏 )𝑑𝜏Convolution integral:

𝐶𝑂 (0 ,𝑡 )=𝐶𝑂❑∗+ 1

𝑛𝐹𝐴 𝜋12𝐷𝑂

12∫0

𝑡 𝑖 (𝑡−𝑢 )𝑢1/2

𝑑𝑢 𝐶𝑅 (0 , 𝑡 )=𝐶𝑅❑∗ − 1

𝑛𝐹𝐴 𝜋12𝐷𝑅

12∫0

𝑡 𝑖 (𝑡−𝑢 )𝑢1/2

𝑑𝑢

Evaluation of O + ne R⇄ (O, R soluble)






𝐶𝑂 (0 ,𝑡 )=𝐶𝑂❑∗+ 1


12∫0

𝑡 𝑖 (𝑡−𝑢 )𝑢1/2

𝑑𝑢 𝐶𝑅 (0 , 𝑡 )=𝐶𝑅❑∗ − 1

𝑛𝐹𝐴𝜋12𝐷𝑅

12∫0

𝑡 𝑖 (𝑡−𝑢 )𝑢1/2

𝑑𝑢

∫0

𝑡 𝑖 (𝑡−𝑢)𝑢1 /2

𝑑𝑢=∫0

𝑡 𝐼 sin𝜔 (𝑡−𝑢)𝑢1 /2

𝑑𝑢

Recall trig identity sin w(t – u) = sin wt cos wu – cos wt sin wuCan be derived from Euler identity ejx = cos x – j sin xAlso recall: sin x = (ejx – e–jx)/2j, cos x = (ejx + e–jx)/2

∫0

𝑡 𝐼 sin𝜔 (𝑡−𝑢 )𝑢1 /2

𝑑𝑢=𝐼 sin𝜔𝑡∫0

𝑡 cos𝜔𝑢𝑢1/2

𝑑𝑢− 𝐼 cos𝜔𝑡∫0

𝑡 sin𝜔𝑢𝑢1/2

𝑑𝑢

Evaluation of O + ne R⇄ (O, R soluble)






𝐶𝑂 (0 ,𝑡 )=𝐶𝑂❑∗+ 1


12∫0

𝑡 𝑖 (𝑡−𝑢 )𝑢1/2

𝑑𝑢 𝐶𝑅 (0 , 𝑡 )=𝐶𝑅❑∗ − 1


12∫0

𝑡 𝑖 (𝑡−𝑢 )𝑢1/2

𝑑𝑢

∫0

𝑡 𝐼 sin𝜔 (𝑡−𝑢 )𝑢1 /2


𝑡 cos𝜔𝑢𝑢1/2

𝑑𝑢− 𝐼 cos𝜔𝑡∫0

𝑡 sin𝜔𝑢𝑢1/2

𝑑𝑢

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 stateIntegrals rep transition from initial cond to steady stateBecause u–½ appears, integrands only significant at short timesObtain steady state by letting int limits go to

O + ne R⇄ (O, R soluble)






∫𝑠 𝑡𝑒𝑎𝑑𝑦𝑠𝑡𝑎𝑡𝑒

𝐼 sin𝜔 (𝑡−𝑢)𝑢1 /2


∞ 𝑒 𝑗𝜔𝑢+𝑒− 𝑗 𝜔𝑢

2𝑢1/2𝑑𝑢− 𝐼 cos𝜔𝑡∫

0

∞ 𝑒 𝑗 𝜔𝑢+𝑒− 𝑗𝜔𝑢

2 𝑗𝑢1 /2𝑑𝑢

Recall: sin x = (ejx – e–jx)/2j, cos x = (ejx + e–jx)/2

𝐿 {(𝜋𝑡 )− 1/2 }=∫0

∞

(𝜋𝑡 )−1/2𝑒− 𝑠𝑡 𝑑𝑡=𝑠−1 /2




∞ cos𝜔𝑢𝑢1/2

𝑑𝑢− 𝐼 cos𝜔 𝑡∫0

∞ sin𝜔𝑢𝑢1/2

𝑑𝑢

∫0

∞ 𝑒 𝑗 𝜔𝑢+𝑒− 𝑗 𝜔𝑢

2𝑢1 /2𝑑𝑢= 𝜋 1/2

2 (− 𝑗 )1 /2𝜔1/2+𝜋 1/2

2 ( 𝑗 )1 /2𝜔1/2=( 𝜋2𝜔 )1 /2

∫0

∞ 𝑒 𝑗 𝜔𝑢−𝑒− 𝑗 𝜔𝑢

2 𝑗𝑢1 /2𝑑𝑢= 𝜋 1 /2

2 𝑗 (− 𝑗 )1 /2𝜔1/2−𝜋 1/2

2 𝑗 ( 𝑗 )1 /2𝜔1/2=( 𝜋2𝜔 )1 /2

Can be derived from Euler identity ejx = cos x – j sin x

Evaluation of

Surface concentration expressionsO + ne R⇄ (O, R soluble)








𝑑𝑢=𝐼 ( 𝜋2𝜔 )1/2

sin𝜔𝑡− 𝐼 ( 𝜋2𝜔 )1 /2

cos𝜔𝑡

𝐶𝑂 (0 ,𝑡 )=𝐶𝑂❑∗+ 1

𝑛𝐹𝐴𝜋12𝐷𝑂

12∫0

𝑡 𝑖 (𝑡−𝑢 )𝑢1/2

𝑑𝑢 𝐶𝑅 (0 , 𝑡 )=𝐶𝑅❑∗ − 1


12∫0

𝑡 𝑖 (𝑡−𝑢 )𝑢1/2

𝑑𝑢

𝐶𝑂 (0 ,𝑡 )=𝐶𝑂❑∗+𝐼

𝑛𝐹𝐴 (2𝐷𝑂𝜔 )12

(sin𝜔𝑡− cos𝜔𝑡 ) 𝐶𝑅 (0 , 𝑡 )=𝐶𝑅❑∗ −𝐼

𝑛𝐹𝐴 (2𝐷𝑅𝜔 )12

(sin𝜔𝑡− cos𝜔𝑡 )

𝑑𝐶𝑂 (0 , 𝑡 )𝑑𝑡 = 𝐼

𝑛𝐹𝐴 ( 𝜔2𝐷𝑂 )12 (sin𝜔𝑡+cos𝜔𝑡 )

𝑑𝐶𝑂 (0 , 𝑡 )𝑑𝑡 =− 𝐼

𝑛𝐹𝐴 ( 𝜔2𝐷𝑂 )12 (sin𝜔𝑡+cos𝜔𝑡 )

Evaluation of Rs, CsO + ne R⇄ (O, R soluble)






𝑑𝐶𝑂 (0 , 𝑡 )𝑑𝑡 = 𝐼

𝑛𝐹𝐴 ( 𝜔2𝐷𝑂 )12 (sin𝜔𝑡+cos𝜔𝑡 ) 𝑑𝐶𝑅 (0 ,𝑡 )

𝑑𝑡 =− 𝐼𝑛𝐹𝐴 ( 𝜔2𝐷𝑅 )

12 (sin𝜔𝑡+cos𝜔𝑡 )

𝑑𝐸𝑑𝑡 =(𝑅𝑐𝑡+ 𝜎𝜔1/2 ) 𝐼 𝜔cos𝜔𝑡+𝐼 𝜎𝜔1/2 s∈𝜔𝑡 𝜎= 1

𝑛𝐹𝐴√2 ( 𝛽𝑂𝐷𝑂1/2 −𝛽𝑅𝐷𝑅

1 /2 )Finding Rs, Cs depends on finding Rct, bO, bR

Rct from heterogeneous charge-transfer kineticss/w1/2 and 1/sw1/2 from mass-transfer effects

𝒁 𝑓=𝑅𝑐𝑡+𝑅𝑊−𝑗

𝜔𝐶𝑊=𝑅𝑐𝑡+[𝜎𝜔−1/2− 𝑗 (𝜎𝜔− 1/2 ) ]

ZW

f dependent R

Pseudo C

Called pseudo C because energy is stored electrochemically (in rev faradaic redox reaction) rather than electrostatically (as in Cd)

Kinetic parameters from impedanceO + e R⇄ (O, R soluble) 𝑑𝐸

𝑑𝑡 =𝑅𝑐𝑡 𝐼 𝜔cos𝜔𝑡+ 𝛽𝑂𝑑𝐶𝑂(0 ,𝑡)𝑑𝑡 +𝛽𝑅


kf

kb

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)

𝜂=𝑅𝑇𝐹 [𝐶𝑂(0 , 𝑡)𝐶𝑂∗−𝐶𝑅(0 , 𝑡 )𝐶𝑅∗

+𝑖𝑖0 ] 𝑅𝑐𝑡=

𝑅𝑇𝐹 𝑖0

𝛽𝑂=𝑅𝑇𝐹 𝐶𝑂∗

𝛽𝑅=𝑅𝑇𝐹 𝐶𝑅∗

𝑑𝐸𝑑𝑡 =𝑅𝑠 𝐼𝜔 cos𝜔𝑡+

𝐼𝐶𝑠sin𝜔𝑡 𝑑𝐸

𝑑𝑡 =(𝑅𝑐𝑡+ 𝜎𝜔1/2 ) 𝐼 𝜔cos𝜔𝑡+𝐼 𝜎𝜔1/2 s∈𝜔𝑡

𝑅𝑠−1𝜔𝐶 𝑠

=𝑅𝑐𝑡=𝑅𝑇𝐹 𝑖0

k0 can be evaluated through i0 when Rs, Cs are known

𝜎= 𝑅𝑇𝐹 2𝐴√2 ( 1

𝐷𝑂1/2𝐶𝑂

∗+1

𝐷𝑅1 /2𝐶𝑅

∗)Rs

1/wCs

Rct Slope = s

w–1/2

R or

XC

Limiting case: reversible system, fast charge transfer

i0 , Rct 0, Rs s/w1/2


𝜔𝐶𝑊=𝑅𝑐𝑡+[𝜎𝜔−1/2− 𝑗 (𝜎𝜔− 1/2 ) ]



𝑍 𝑓=𝜎 ( 2𝜔 )1/2

ZW alone. Mass-transfer impedance (applies to any electrode reaction) minimum impedance.If kinetics are observable, Rct contributes and Zf is greater.

𝜎= 𝑅𝑇𝐹 2𝐴√2 ( 1

𝐷𝑂1/2𝐶𝑂

∗+1


∗)Large concentrations reduce mass-transfer impedanceConcentration ratio significantly different than one make s and Zf largeLarge transfer rates only achieved when concentrations are comparable (Zf minimal near E0’)Impedance measurements easiest near E0’

𝑅𝑠=𝑅𝑐𝑡+𝜎 /𝜔1/2 𝐶 𝑠=1

𝜎 𝜔1 /2

Limiting case: reversible system, fast charge transfer

i0 , Rct 0, Rs s/w1/2


𝜔𝐶𝑊=𝑅𝑐𝑡+[𝜎𝜔−1/2− 𝑗 (𝜎𝜔− 1/2 ) ]



𝑍 𝑓=𝜎 ( 2𝜔 )1/2

𝜎= 𝑅𝑇𝐹 2𝐴√2 ( 1

𝐷𝑂1/2𝐶𝑂

∗+1


∗) RW = s/w1/2

1/w

C s = s

/w1/

2

f < 45o

Rct

|Zf | > |Z

W |

|Zf | = |Z

W |

tan f = 1/wRsCs = (s/w1/2) / (Rct + s/w1/2)

0 ≤ f ≤ 45o, always a component of iac in-phase (0o) with Eac and can be measured with phase sensitive detector (lock-in amplifier) basis for discriminating against charging current in ac voltammetry

𝜎= 𝑅𝑇𝑛2𝐹2 𝐴√2 ( 1

𝐷𝑂1/2𝐶𝑂

∗+1

𝐷𝑅1/2𝐶𝑅

∗ )

Electrochemical impedance spectroscopy

Zf

Cd

RW

ic + if

ic

if

Rct

ZwZf Rs Cs

= =

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)

• 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 fExtract RW, Cd, Rs, and Cs

Eliminates need for separate measurements w/o redox speciesEliminates need to assume redox species has no effect on nonfaradaic impedance

Electrochemical impedance spectroscopy

Zf

Cd

RW

ic + if

ic

if

Zf Rs Cs

=

- 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)

Measured Z is expressed as series RB + CB

ZRe = RB , ZIm = 1/wCB

0

10

20

30

40

50

60

0 20 40 60 80 100

Z Im

ZRe

w

103104105

𝑍𝑅𝑒=𝑅𝐵=𝑅Ω+𝑅𝑠𝐴2+𝐵2

𝑍 𝐼𝑚=1𝜔𝐶𝐵

=

𝐵2𝜔𝐶𝑑

+𝐴𝜔𝐶 𝑠

𝐴2+𝐵2A = Cd/Cs , B = wRsCd

See Section 10.1.2Can be shown by E = ERW + ECd(ERs + ECs)/(ECd + ERs +ECs)ER = IR, EC = –j/wC

Variation of total impedance

Zf

Cd

RW

ic + if

ic

if

Zf Rs Cs

=

𝑍𝑅𝑒=𝑅𝐵=𝑅Ω+𝑅𝑠𝐴2+𝐵2 𝑍 𝐼𝑚=

1𝜔𝐶𝐵

=

𝐵2𝜔𝐶𝑑

+𝐴𝜔𝐶 𝑠

𝐴2+𝐵2

A = Cd/Cs , B = wRsCd

𝑅𝑠=𝑅𝑐𝑡+𝜎 /𝜔1/2 𝐶 𝑠=1

𝜎 𝜔1 /2

𝑍𝑅𝑒=𝑅Ω+𝑅𝑐𝑡+𝜎𝜔− 1/2

(𝐶𝑑𝜎𝜔1 /2+1)2+𝜔2𝐶𝑑2 (𝑅𝑐𝑡+𝜎 𝜔−1 /2)

2

𝑍 𝐼𝑚=𝜔𝐶𝑑 (𝑅𝑐𝑡+𝜎𝜔− 1/2 )

2+𝜎𝜔− 1/2 (𝐶𝑑𝜎𝜔1 /2+1 )

(𝐶𝑑𝜎𝜔1 /2+1 )2+𝜔2𝐶𝑑2 (𝑅𝑐𝑡+𝜎 𝜔−1 /2)

2Obtain chem info by plotting Zim vs. ZRe

Impedance: low-frequency limit𝑍𝑅𝑒=𝑅Ω+

𝑅𝑐𝑡+𝜎𝜔− 1/2

(𝐶𝑑𝜎𝜔1 /2+1)2+𝜔2𝐶𝑑2 (𝑅𝑐𝑡+𝜎 𝜔−1 /2)

2 𝑍 𝐼𝑚=𝜔𝐶𝑑 (𝑅𝑐𝑡+𝜎𝜔− 1/2 )

2+𝜎𝜔− 1/2 (𝐶𝑑𝜎𝜔1 /2+1 )

(𝐶𝑑𝜎𝜔1 /2+1 )2+𝜔2𝐶𝑑2 (𝑅𝑐𝑡+𝜎 𝜔−1 /2)

2

As w 0

𝑍𝑅𝑒=𝑅Ω+𝑅𝑐𝑡+𝜎𝜔−1/2 𝑍 𝐼𝑚=𝜎𝜔−1 /2+2𝜎2𝐶𝑑

𝑍 𝐼𝑚=𝑍𝑅𝑒−𝑅Ω−𝑅𝑐𝑡+2𝜎 2𝐶𝑑 - Linear w/ unit slope and extrapolated line intersects ZRe axis at

- 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

ZIm

ZRe

Slope = 1𝑅 Ω

+𝑅 𝑐𝑡−2𝜎

2 𝐶 𝑑

Impedance: high-frequency limitAs w

𝑍𝑅𝑒=𝑅Ω+𝑅𝑐𝑡

1+𝜔2𝐶𝑑2𝑅𝑐𝑡

2

(𝑍𝑅𝑒−𝑅Ω− 𝑅𝑐𝑡2 )2

+𝑍 𝐼𝑚2=(𝑅𝑐𝑡2 )

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

Zf

Cd

RW

Cd

RW Rct

𝒁=𝑅Ω− 𝑗 ( 𝑅𝑐𝑡𝑅𝑐𝑡𝐶𝑑𝜔− 𝑗 ) 𝑍 𝐼𝑚=

𝜔𝐶𝑑𝑅𝑐𝑡2

1+𝜔2𝐶𝑑2𝑅𝑐𝑡

2

ZIm

ZRe

w = 1/RctCd

RW RW + Rct

Impedance: applications to real systems

ZIm

ZRe

w = 1/RctCd

Kinetic control Mass-transfer control

ZIm

ZRe

RW 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.

w = 1/RctCdZIm

ZRe

RW RW + Rct

Limits to measurable k0 by faradaic impedanceUpper limit- Rct must make sig contribution to Rs

(Rct ≥ s/w1/2)- 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- Think aromatic species to cation/anion

radicals in aprotic solvents (k0 > 1 cm/s)- Cs ≥ Cd and Rs ≥ RW

𝜎= 𝑅𝑇𝐹 2𝐴√2 ( 1

𝐷𝑂1/2𝐶𝑂

∗ −1

𝐷𝑅1/2𝐶𝑅

∗ ) 𝑅𝑐𝑡=𝑅𝑇𝐹 𝑖0 i0 = FAk0C (Eqn 3.4.7)

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

w = 1/RctCd

Kinetic control Mass-transfer control

ZIm

ZRe

RW RW + Rct

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.

Impedance-based techniques

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