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
𝑗 = −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
𝑗 = −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
𝐸 = 𝐸𝑅𝑠 + 𝐸𝐶𝑠
𝑞
𝐸 = 𝑖𝑅𝑠 +
𝐶𝑠
𝑖 = 𝐼 sin 𝜔𝑡
𝑑𝐸
𝐼
= 𝑅𝑠 𝐼𝜔 cos 𝜔𝑡 +
sin 𝜔𝑡
𝑑𝑡
𝐶𝑠
𝑑𝐸
𝑑𝑖
𝑖
= 𝑅𝑠 +
𝑑𝑡
𝑑𝑡 𝐶𝑠
Description of chemical system
O + ne ⇄ R
(O, R soluble)
E = E[i, CO(0,t), CR(0,t)]
𝑑𝐸
𝐼
= 𝑅𝑠 𝐼𝜔 cos 𝜔𝑡 +
sin 𝜔𝑡
𝑑𝑡
𝐶𝑠
𝑖 = 𝐼 sin 𝜔𝑡
Because E is a function of 3 variables that depend on t, total differential is a combination of partial differentials
𝑑𝐸
𝜕𝐸 𝑑𝑖
𝜕𝐸
𝑑𝐶𝑂 (0, 𝑡)
𝜕𝐸
𝑑𝐶𝑅 (0, 𝑡)
=
+
+
𝑑𝑡
𝜕𝑖 𝑑𝑡
𝜕𝐶𝑂 (0, 𝑡)
𝑑𝑡
𝜕𝐶𝑅 (0, 𝑡)
𝑑𝑡
𝑑𝐸
𝑑𝑖
𝑑𝐶𝑂 (0, 𝑡)
𝑑𝐶𝑅 (0, 𝑡)
= 𝑅𝑐𝑡 + 𝛽𝑂
+ 𝛽𝑅
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝐶𝑂 (0, 𝑡) 𝑑𝐶𝑅 (0, 𝑡)
𝑑𝑖
by mass transfer considerations
Find
,
= 𝐼𝜔 cos 𝜔𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
Initial conditions: CO(x,0) = CO*, CR(x,0) = CR*
Recall from section 8.2.1:
𝐶𝑂 0, 𝑠 =
𝑖 𝑠
1
𝑛𝐹𝐴𝐷𝑂 2 𝑠
𝐶𝑂 ∗
1+ 𝑠
2
Notice the sign convention is opposite of usual
𝐶𝑅 0, 𝑠 = −
𝑖 𝑠
1
𝑛𝐹𝐴𝐷𝑅 2 𝑠
𝐶𝑅 ∗
1+ 𝑠
2
Determination of CO(0,t), CR(0,t)
O + ne ⇄ R
(O, R soluble)
𝑑𝐸
𝐼
= 𝑅𝑠 𝐼𝜔 cos 𝜔𝑡 +
sin 𝜔𝑡
𝑑𝑡
𝐶𝑠
𝑖 = 𝐼 sin 𝜔𝑡
E = E[i, CO(0,t), CR(0,t)]
𝑑𝐸
𝑑𝐶𝑂 (0, 𝑡)
𝑑𝐶𝑅 (0, 𝑡)
= 𝑅𝑐𝑡 𝐼𝜔 cos 𝜔𝑡 + 𝛽𝑂
+ 𝛽𝑅
𝑑𝑡
𝑑𝑡
𝑑𝑡
Find
𝑑𝐶𝑂 (0, 𝑡) 𝑑𝐶𝑅 (0, 𝑡)
by mass transfer considerations
,
𝑑𝑡
𝑑𝑡
Initial conditions: CO(x,0) = CO*, CR(x,0) = CR*
Recall from section 8.2.1:
𝐶𝑂 0, 𝑠 =
𝑖 𝑠
1
𝑛𝐹𝐴𝐷𝑂 2 𝑠
𝐶𝑂 ∗
1+ 𝑠
2
𝐶𝑅 0, 𝑠 = −
𝑖 𝑠
1
𝑛𝐹𝐴𝐷𝑅 2 𝑠
Recall Laplace Transform:
𝐿 𝐹(𝑡) =
∞ −𝑠𝑡
𝑒 𝐹
0
𝑡 𝑑𝑡 = 𝐹(s)
Convolution integral:
𝑡
𝐿 𝑓 𝑠 𝑔(𝑠) = F t ∗ G(t) =
𝐹 𝑡 − 𝜏 𝐺 𝜏 𝑑𝜏
𝐿 𝜋𝑡
−1/2
∞
=
𝜋𝑡
𝐶𝑅 ∗
1+ 𝑠
2
−1/2 −𝑠𝑡
𝑒
0
−1
𝐿
𝑡
𝐹 𝑠 𝐺(𝑠) = f t ∗ g(t) =
𝐶𝑂 0, 𝑡 = 𝐶𝑂
+
𝑡
1
1
𝑛𝐹𝐴𝜋 2
𝐷𝑂
1
2 0
𝑖 𝑡−𝑢
𝑑𝑢
𝑢1/2
𝑓 𝑡 − 𝜏 𝑔 𝜏 𝑑𝜏
0
0
∗
𝑑𝑡 = 𝑠 −1/2
𝐶𝑅 0, 𝑡 = 𝐶𝑅
∗
−
1
𝑡
1
1
2 0
2
𝑛𝐹𝐴𝜋 𝐷𝑅
𝑖 𝑡−𝑢
𝑑𝑢
𝑢1/2
Evaluation of
O + ne ⇄ R
𝑡 𝑖 𝑡−𝑢
0 𝑢1/2
(O, R soluble)
𝑑𝐸
𝐼
= 𝑅𝑠 𝐼𝜔 cos 𝜔𝑡 +
sin 𝜔𝑡
𝑑𝑡
𝐶𝑠
𝑖 = 𝐼 sin 𝜔𝑡
E = E[i, CO(0,t), CR(0,t)]
𝑑𝑢
𝑑𝐸
𝑑𝐶𝑂 (0, 𝑡)
𝑑𝐶𝑅 (0, 𝑡)
= 𝑅𝑐𝑡 𝐼𝜔 cos 𝜔𝑡 + 𝛽𝑂
+ 𝛽𝑅
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝐶𝑂 0, 𝑡 = 𝐶𝑂
𝑡
0
∗
+
𝑖 𝑡−𝑢
𝑑𝑢 =
𝑢1/2
𝑡
1
1
𝑛𝐹𝐴𝜋 2
𝑡
0
𝐷𝑂
1
2 0
𝑖 𝑡−𝑢
𝑑𝑢
𝑢1/2
𝐶𝑅 0, 𝑡 = 𝐶𝑅
∗
−
1
1
1
2 0
2
𝑛𝐹𝐴𝜋 𝐷𝑅
𝐼 sin 𝜔 𝑡 − 𝑢
𝑑𝑢
𝑢1/2
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
𝑡
0
𝐼 sin 𝜔 𝑡 − 𝑢
𝑑𝑢 = 𝐼 sin 𝜔𝑡
𝑢1/2
𝑡
0
cos 𝜔𝑢
𝑑𝑢 − 𝐼 cos 𝜔𝑡
𝑢1/2
𝑡
0
𝑡
sin 𝜔𝑢
𝑑𝑢
𝑢1/2
𝑖 𝑡−𝑢
𝑑𝑢
𝑢1/2
Evaluation of
O + ne ⇄ R
(O, R soluble)
𝑖 = 𝐼 sin 𝜔𝑡
E = E[i, CO(0,t), CR(0,t)]
𝑡 𝑖 𝑡−𝑢
0 𝑢1/2
𝑑𝑢
𝑑𝐸
𝐼
= 𝑅𝑠 𝐼𝜔 cos 𝜔𝑡 +
sin 𝜔𝑡
𝑑𝑡
𝐶𝑠
𝑑𝐸
𝑑𝐶𝑂 (0, 𝑡)
𝑑𝐶𝑅 (0, 𝑡)
= 𝑅𝑐𝑡 𝐼𝜔 cos 𝜔𝑡 + 𝛽𝑂
+ 𝛽𝑅
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝐶𝑂 0, 𝑡 = 𝐶𝑂
𝑡
0
∗
+
𝑡
1
1
𝑛𝐹𝐴𝜋 2
𝐷𝑂
1
2 0
𝑖 𝑡−𝑢
𝑑𝑢
𝑢1/2
𝐼 sin 𝜔 𝑡 − 𝑢
𝑑𝑢 = 𝐼 sin 𝜔𝑡
𝑢1/2
𝑡
0
𝐶𝑅 0, 𝑡 = 𝐶𝑅
cos 𝜔𝑢
𝑑𝑢 − 𝐼 cos 𝜔𝑡
𝑢1/2
∗
𝑡
0
−
1
𝑡
1
1
2 0
2
𝑛𝐹𝐴𝜋 𝐷𝑅
𝑖 𝑡−𝑢
𝑑𝑢
𝑢1/2
sin 𝜔𝑢
𝑑𝑢
𝑢1/2
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 
𝑡 𝑖 𝑡−𝑢
0 𝑢1/2
Evaluation of
O + ne ⇄ R
(O, R soluble)
𝑑𝐸
𝐼
= 𝑅𝑠 𝐼𝜔 cos 𝜔𝑡 +
sin 𝜔𝑡
𝑑𝑡
𝐶𝑠
𝑖 = 𝐼 sin 𝜔𝑡
E = E[i, CO(0,t), CR(0,t)]
𝑑𝑢
𝑑𝐸
𝑑𝐶𝑂 (0, 𝑡)
𝑑𝐶𝑅 (0, 𝑡)
= 𝑅𝑐𝑡 𝐼𝜔 cos 𝜔𝑡 + 𝛽𝑂
+ 𝛽𝑅
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝐼 sin 𝜔 𝑡 − 𝑢
𝑑𝑢 = 𝐼 sin 𝜔𝑡
𝑠𝑡𝑒𝑎𝑑𝑦
𝑢1/2
𝑠𝑡𝑎𝑡𝑒
∞
0
∞
cos 𝜔𝑢
𝑑𝑢 − 𝐼 cos 𝜔𝑡
𝑢1/2
0
sin 𝜔𝑢
𝑑𝑢
𝑢1/2
Recall: sin x = (ejx – e–jx)/2j, cos x = (ejx + e–jx)/2
𝐼 sin 𝜔 𝑡 − 𝑢
𝑑𝑢 = 𝐼 sin 𝜔𝑡
𝑠𝑡𝑒𝑎𝑑𝑦
𝑢1/2
𝑠𝑡𝑎𝑡𝑒
𝐿 𝜋𝑡
−1/2
∞ 𝑗𝜔𝑢
𝑒
0
∞
=
𝜋𝑡
+ 𝑒 −𝑗𝜔𝑢
𝑑𝑢 − 𝐼 cos 𝜔𝑡
2 𝑢1/2
−1/2 −𝑠𝑡
𝑒
∞ 𝑗𝜔𝑢
𝑒
0
+ 𝑒 −𝑗𝜔𝑢
𝑑𝑢
2𝑗 𝑢1/2
𝑑𝑡 = 𝑠 −1/2
0
∞ 𝑗𝜔𝑢
𝑒
0
+
2 𝑢1/2
∞ 𝑗𝜔𝑢
𝑒
0
𝑒 −𝑗𝜔𝑢
𝑑𝑢 =
𝜋 1/2
2 −𝑗
1/2 𝜔
+
1/2
𝜋 1/2
2 𝑗
1/2 𝜔1/2
=
𝜋
2𝜔
1/2
− 𝑒 −𝑗𝜔𝑢
𝜋 1/2
𝜋 1/2
𝜋
𝑑𝑢
=
−
=
2𝜔
2𝑗 𝑢1/2
2𝑗 −𝑗 1/2 𝜔1/2 2𝑗 𝑗 1/2 𝜔1/2
1/2
Can be derived from
Euler identity
ejx = cos x – j sin x
Surface concentration expressions
O + ne ⇄ R
(O, R soluble)
𝑑𝐸
𝐼
= 𝑅𝑠 𝐼𝜔 cos 𝜔𝑡 +
sin 𝜔𝑡
𝑑𝑡
𝐶𝑠
𝑖 = 𝐼 sin 𝜔𝑡
E = E[i, CO(0,t), CR(0,t)]
𝑑𝐸
𝑑𝐶𝑂 (0, 𝑡)
𝑑𝐶𝑅 (0, 𝑡)
= 𝑅𝑐𝑡 𝐼𝜔 cos 𝜔𝑡 + 𝛽𝑂
+ 𝛽𝑅
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝐶𝑂 0, 𝑡 = 𝐶𝑂
∗
𝑡
1
+
1
𝑛𝐹𝐴𝜋 2
𝐷𝑂
1
2 0
𝑖 𝑡−𝑢
𝑑𝑢
𝑢1/2
𝐼 sin 𝜔 𝑡 − 𝑢
𝜋
𝑑𝑢
=
𝐼
𝑠𝑡𝑒𝑎𝑑𝑦
2𝜔
𝑢1/2
1/2
𝑠𝑡𝑎𝑡𝑒
= 𝐶𝑂
𝜋
sin 𝜔𝑡 − 𝐼
2𝜔
1/2
+
𝐼
𝑛𝐹𝐴 2 𝐷𝑂 𝜔
𝑑𝐶𝑂 0, 𝑡
𝐼
𝜔
=
𝑑𝑡
𝑛𝐹𝐴 2 𝐷𝑂
1
2
1
2
sin 𝜔𝑡 − cos 𝜔𝑡
sin 𝜔𝑡 + cos 𝜔𝑡
−
𝑡
1
1
1
2 0
𝑛𝐹𝐴𝜋 2 𝐷𝑅
𝑖 𝑡−𝑢
𝑑𝑢
𝑢1/2
cos 𝜔𝑡
𝐶𝑅 0, 𝑡
𝐶𝑂 0, 𝑡
∗
𝐶𝑅 0, 𝑡 = 𝐶𝑅
∗
= 𝐶𝑅 ∗ −
𝐼
𝑛𝐹𝐴 2 𝐷𝑅 𝜔
𝑑𝐶𝑂 0, 𝑡
𝐼
𝜔
=−
𝑑𝑡
𝑛𝐹𝐴 2 𝐷𝑂
sin 𝜔𝑡 − cos 𝜔𝑡
1
2
1
2
sin 𝜔𝑡 + cos 𝜔𝑡
Evaluation of Rs, Cs
O + ne ⇄ R
(O, R soluble)
𝑑𝐸
𝐼
= 𝑅𝑠 𝐼𝜔 cos 𝜔𝑡 +
sin 𝜔𝑡
𝑑𝑡
𝐶𝑠
𝑖 = 𝐼 sin 𝜔𝑡
E = E[i, CO(0,t), CR(0,t)]
𝑑𝐸
𝑑𝐶𝑂 (0, 𝑡)
𝑑𝐶𝑅 (0, 𝑡)
= 𝑅𝑐𝑡 𝐼𝜔 cos 𝜔𝑡 + 𝛽𝑂
+ 𝛽𝑅
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝐶𝑂 0, 𝑡
𝐼
𝜔
=
𝑑𝑡
𝑛𝐹𝐴 2 𝐷𝑂
1
2
𝑑𝐶𝑅 0, 𝑡
𝐼
𝜔
=−
𝑑𝑡
𝑛𝐹𝐴 2 𝐷𝑅
sin 𝜔𝑡 + cos 𝜔𝑡
𝑑𝐸
𝜎
= 𝑅𝑐𝑡 + 1/2 𝐼𝜔 cos 𝜔𝑡 + 𝐼𝜎𝜔1/2 sin 𝜔𝑡
𝑑𝑡
𝜔
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
𝜎=
f dependent R
𝑗
𝒁𝑓 = 𝑅𝑐𝑡 + 𝑅𝑊 −
= 𝑅𝑐𝑡 + 𝜎𝜔 −1/2 − 𝑗 𝜎𝜔 −1/2
𝜔 𝐶𝑊
Pseudo C
ZW
1
𝛽𝑂
𝑛𝐹𝐴 2 𝐷𝑂
1
2
−
1/2
sin 𝜔𝑡 + cos 𝜔𝑡
𝛽𝑅
𝐷𝑅 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
𝑑𝐸
𝑑𝐶𝑂 (0, 𝑡)
𝑑𝐶𝑅 (0, 𝑡)
= 𝑅𝑐𝑡 𝐼𝜔 cos 𝜔𝑡 + 𝛽𝑂
+ 𝛽𝑅
𝑑𝑡
𝑑𝑡
𝑑𝑡
(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)
𝑅𝑇 𝐶𝑂 (0, 𝑡) 𝐶𝑅 (0, 𝑡) 𝑖
𝜂=
−
+
𝐹
𝐶𝑂 ∗
𝐶𝑅 ∗
𝑖0
𝑅𝑐𝑡 =
𝑑𝐸
𝐼
= 𝑅𝑠 𝐼𝜔 cos 𝜔𝑡 +
sin 𝜔𝑡
𝑑𝑡
𝐶𝑠
𝜎=
𝑅𝑇
1
𝑅𝑇
= 𝑅𝑐𝑡 =
𝜔 𝐶𝑠
𝐹𝑖0
1
𝐹 2 𝐴 2 𝐷𝑂 1/2 𝐶𝑂
+
∗
𝛽𝑂 =
𝑅𝑇
𝐹𝐶𝑂 ∗
𝛽𝑅 =
𝑅𝑇
𝐹𝐶𝑅 ∗
𝑑𝐸
𝜎
= 𝑅𝑐𝑡 + 1/2 𝐼𝜔 cos 𝜔𝑡 + 𝐼𝜎𝜔1/2 sin 𝜔𝑡
𝑑𝑡
𝜔
k0 can be evaluated through i0 when Rs, Cs are known
1
𝐷𝑅 1/2 𝐶𝑅 ∗
R or XC
𝑅𝑠 −
𝑅𝑇
𝐹𝑖0
Rct
Rs
1/wCs
Slope = s
w–1/2
Limiting case: reversible system, fast
charge transfer
i0  , Rct  0, Rs  s/w1/2
𝑅𝑠 −
𝑅𝑠 = 𝑅𝑐𝑡 + 𝜎
𝒁𝑓 = 𝑅𝑐𝑡 + 𝑅𝑊 −
2
𝑍𝑓 = 𝜎
𝜔
𝜎=
𝑅𝑇
1
𝑅𝑇
= 𝑅𝑐𝑡 =
𝜔 𝐶𝑠
𝐹𝑖0
𝜔1/2
𝑗
= 𝑅𝑐𝑡 + 𝜎𝜔 −1/2 − 𝑗 𝜎𝜔 −1/2
𝜔 𝐶𝑊
1/2
1
𝐹 2 𝐴 2 𝐷𝑂 1/2 𝐶𝑂 ∗
1
𝐶𝑠 =
𝜎𝜔1/2
ZW alone. Mass-transfer impedance (applies to any electrode
reaction)  minimum impedance.
If kinetics are observable, Rct contributes and Zf is greater.
+
1
𝐷𝑅 1/2 𝐶𝑅 ∗
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
𝜎=
𝑅𝑇
𝑗
= 𝑅𝑐𝑡 + 𝜎𝜔 −1/2 − 𝑗 𝜎𝜔 −1/2
𝜔 𝐶𝑊
1
𝐹 2 𝐴 2 𝐷𝑂
1/2
𝐶𝑂
∗
+
1
𝐷𝑅
1/2
𝐶𝑅
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
𝑅𝑇
1
𝑛2 𝐹 2 𝐴
1/2
2 𝐷𝑂
𝐶𝑂
∗
+
1
𝐷𝑅 1/2 𝐶𝑅 ∗
2
𝑍𝑓 = 𝜎
𝜔
RW = s/w1/2
∗
tan f = 1/wRsCs = (s/w1/2) / (Rct + s/w1/2)
𝜎=
1
𝑅𝑇
= 𝑅𝑐𝑡 =
𝜔 𝐶𝑠
𝐹𝑖0
f < 45o
1/wCs = s/w1/2
𝒁𝑓 = 𝑅𝑐𝑡 + 𝑅𝑊 −
𝑅𝑠 −
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
𝑅𝑠
= 𝑅𝐵 = 𝑅Ω + 2
𝐴 + 𝐵2
𝑍𝐼𝑚
1
=
=
𝜔𝐶𝐵
30
20
10
Measured Z is expressed as series RB + CB
ZRe = RB , ZIm = 1/wCB
𝑍𝑅𝑒
w

40
𝐵2
𝐴
+
𝜔𝐶𝑑
𝜔𝐶𝑠
𝐴2 + 𝐵 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
𝑍𝑅𝑒
Cd
RW
Rs
𝑍𝑅𝑒 = 𝑅Ω +
𝑍𝐼𝑚 =
𝑅𝑠 = 𝑅𝑐𝑡 + 𝜎 𝜔
if
=
𝑍𝐼𝑚
1
=
=
𝜔𝐶𝐵
𝐵2
𝐴
𝜔𝐶𝑑 + 𝜔𝐶𝑠
𝐴2 + 𝐵 2
A = Cd/Cs , B = wRsCd
Zf
ic + if
Zf
𝑅𝑠
= 𝑅𝐵 = 𝑅Ω + 2
𝐴 + 𝐵2
1
𝐶𝑠 =
𝜎𝜔1/2
1/2
Cs
𝑅𝑐𝑡 + 𝜎𝜔 −1/2
𝐶𝑑 𝜎𝜔1/2 + 1
𝜔𝐶𝑑 𝑅𝑐𝑡 + 𝜎𝜔 −1/2
𝐶𝑑 𝜎𝜔1/2 + 1
2
2
2
+ 𝜔 2 𝐶𝑑 2 𝑅𝑐𝑡 + 𝜎𝜔 −1/2
+ 𝜎𝜔 −1/2 𝐶𝑑 𝜎𝜔1/2 + 1
+ 𝜔 2 𝐶𝑑 2 𝑅𝑐𝑡 + 𝜎𝜔 −1/2
2
2
Obtain chem info by
plotting Zim vs. ZRe
Impedance: low-frequency limit
𝑍𝑅𝑒 = 𝑅Ω +
𝑅𝑐𝑡 + 𝜎𝜔 −1/2
𝐶𝑑 𝜎𝜔1/2 + 1
2
+ 𝜔 2 𝐶𝑑 2 𝑅𝑐𝑡 + 𝜎𝜔 −1/2
2
𝑍𝐼𝑚 =
𝜔𝐶𝑑 𝑅𝑐𝑡 + 𝜎𝜔 −1/2
𝐶𝑑 𝜎𝜔1/2 + 1
2
2
+ 𝜎𝜔 −1/2 𝐶𝑑 𝜎𝜔1/2 + 1
+ 𝜔 2 𝐶𝑑 2 𝑅𝑐𝑡 + 𝜎𝜔 −1/2
As w  0
𝑍𝑅𝑒 = 𝑅Ω + 𝑅𝑐𝑡 + 𝜎𝜔 −1/2
𝑍𝐼𝑚 = 𝑍𝑅𝑒 − 𝑅Ω − 𝑅𝑐𝑡 + 2𝜎 2 𝐶𝑑
ZIm
Slope = 1
ZRe
𝑍𝐼𝑚 = 𝜎𝜔 −1/2 + 2𝜎 2 𝐶𝑑
- Linear w/ unit slope and extrapolated line
intersects ZRe axis at 𝑅Ω + 𝑅𝑐𝑡 − 2𝜎 2 𝐶𝑑
- 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
𝑅𝑐𝑡
𝒁 = 𝑅Ω − 𝑗
𝑅𝑐𝑡 𝐶𝑑 𝜔 − 𝑗
𝑅𝑐𝑡
𝑍𝑅𝑒 − 𝑅Ω −
2
2
+ 𝑍𝐼𝑚
𝑍𝑅𝑒 = 𝑅Ω +
2
𝑅𝑐𝑡
=
2
2
w = 1/RctCd
ZIm
RW
ZRe
RW + Rct
RW
𝑅𝑐𝑡
1 + 𝜔 2 𝐶𝑑 2 𝑅𝑐𝑡 2
Rct
𝑍𝐼𝑚 =
𝜔𝐶𝑑 𝑅𝑐𝑡 2
1 + 𝜔 2 𝐶𝑑 2 𝑅𝑐𝑡 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
𝑅𝑇
𝑅𝑇
1
1
𝑅
=
𝜎=
− 1/2 ∗
𝑐𝑡
i0 = FAk0C (Eqn 3.4.7)
∗
1/2
2
𝐹𝑖
0
𝐹 𝐴 2 𝐷𝑂 𝐶𝑂
𝐷𝑅 𝐶𝑅
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.