### Impedance-based techniques

```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
-
- 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
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
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
- 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)
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
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