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
EIS &
My ‘where abouts’
Nano-El-Cat
Nov. ‘08.
E-mail: b.a.boukamp@utwente.nl
Address:
University of Twente
Dept. of Science and Technology
P.O.Box 217
7500 AE Enschede
The Netherlands
www.ims.tnw.utwente.nl
EIS &
Nano-El-Cat
Nov. ‘08.
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)
EIS &
Nano-El-Cat
Nov. ‘08.
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
EIS &
Nano-El-Cat
Nov. ‘08.
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
EIS &
Nano-El-Cat
Nov. ‘08.
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!
EIS &
Nano-El-Cat
Nov. ‘08.
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”
EIS &
Nano-El-Cat
Nov. ‘08.
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
EIS &
Nano-El-Cat
Nov. ‘08.
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
EIS &
Nano-El-Cat
Nov. ‘08.
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’ !
EIS &
Nano-El-Cat
Nov. ‘08.
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
EIS &
Combination of elements
Nano-El-Cat
Nov. ‘08.
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
EIS &
Nano-El-Cat
Nov. ‘08.
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
EIS &
Impedance plot (RC)
Nano-El-Cat
Nov. ‘08.
8.0E+04
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
EIS &
Limiting cases
Nano-El-Cat
Nov. ‘08.
What happens for ω << τ and for ω >> τ ?
ω << τ :
1 − jωτ
2
≈
−
ω
τ
≈
−
ω
Z (ω) = R
R
j
R
R
j
R
C
2 2
1+ 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
EIS &
Bode plot (Zre, Zim)
Nano-El-Cat
Nov. ‘08.
1.E+05
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
EIS &
Bode, abs(Z), phase
Nano-El-Cat
Nov. ‘08.
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
EIS &
Nano-El-Cat
Nov. ‘08.
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:
EIS &
Simple model
Nano-El-Cat
Nov. ‘08.
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])
EIS &
Nano-El-Cat
Nov. ‘08.
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.
EIS &
Nano-El-Cat
Nov. ‘08.
‘Debije’ model:
An ionic conductor between two blocking electrodes:
1
Y (ω) =
Z (ω)
Impedance representation
Admittance representation
EIS &
Nano-El-Cat
Nov. ‘08.
Other representations
Zimag
Zreal
‘Bode’ representation
Different Bode
representation
EIS &
Nano-El-Cat
Nov. ‘08.
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
EIS &
Nano-El-Cat
Nov. ‘08.
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
EIS &
Nano-El-Cat
Nov. ‘08.
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
⎞
⎟
⎟
⎠
EIS &
Nano-El-Cat
Nov. ‘08.
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
EIS &
Nano-El-Cat
Nov. ‘08.
Equivalent Circuit Concept
se
m
ic
ir
cl
e
ω
45°
EIS &
Nano-El-Cat
Nov. ‘08.
Instruments
Measurement methods
Bulk, conductivity:
• two electrodes
• pseudo-four electrodes
• true four electrodes
Electrode properties:
• three electrodes
EIS &
Nano-El-Cat
Nov. ‘08.
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)
EIS &
Nano-El-Cat
Nov. ‘08.
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
EIS &
Nano-El-Cat
Nov. ‘08.
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
EIS &
Nano-El-Cat
Nov. ‘08.
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.
EIS &
Nano-El-Cat
Nov. ‘08.
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
EIS &
Actual test
Nano-El-Cat
Nov. ‘08.
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
EIS &
Nano-El-Cat
Nov. ‘08.
Example ‘K-K check’
Impedance of a sample, not in
equilibrium with the ambient.
χ2KK = 0.9·10-4
χ2CNLS = 1.4 ·10-4
EIS &
Nano-El-Cat
Nov. ‘08.
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
EIS &
Nano-El-Cat
Nov. ‘08.
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!
EIS &
Nano-El-Cat
Nov. ‘08.
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 << a0:
Relation a ⇔ C from
‘titration curve’:
a
a0 + Δa
⎛ Δa ⎞ Δa
=
ln 0 = ln
ln
⎜1 + 0 ⎟ ≈ 0
0
a
a
⎝ a ⎠ a
da a0 dln a Δa
Δa
= 0
≈
=
dC C dln C ΔC Δc( x, t ) x=0
EIS &
Nano-El-Cat
Nov. ‘08.
Up to the Frequency Domain!
Laplace transformation of E(t) and I(t)
gives the complex impedance (with p=jω):
E(ω)
=
FSW Z (ω) =
I (ω)
Z0
E(ω)
Z (ω) =
=
I (ω)
Z0
FLW
Laplace space
solution of Fick-2:
jω
~
~ cothl D
jωD
with:
R T Vm ⎡ d ln a ⎤
=
Z0 = 2 2 ⎢
⎥
n F S ⎣ d ln c ⎦
Vm ⎡ d E ⎤
=
⎢ ⎥
nFS ⎣ d δ ⎦
jω
~
~ tanhl D
jωD
p
p
C( p) = A cosh x ~ + B sinh x ~
D
D
EIS &
Dispersions
Nano-El-Cat
Nov. ‘08.
High frequencies:
Vs nF ⎡ d E ⎤
Cint =
VM ⎢⎣ d δ ⎥⎦
−1
Z(ω) = Z0 (j ω)-1/2
= Warburg diffusion
Low frequency limit:
FSW = capacitive
FLW = dc-resistance
Impedance representation of
FSW and FLW.
EIS &
Nano-El-Cat
Nov. ‘08.
General finite length
diffusion
Generic finite
length diffusion:
Z (ω) =
Z0
jωD%⋅ coth l
jωD% k coth l
jω
D%
jω
D%
+k
+ jωD%
If k =0 then
blocking interface
⇒ FSW
If k = ∞ then ideal
passing interface
⇒ FLW
Å Plot for different
values of k.
EIS &
Nano-El-Cat
Nov. ‘08.
A Simple Example
H.J.M. Bouwmeester
AgxNbS2 is a layered structure,
consisting of two-dimensional NbS2
layers. Insertion and extraction of Ag+
ions goes in an ideal manner (see graph).
Isostatically pressed and sintered
sample. Some preferential orientation
(in the proper direction!) will occur.
Simple cell
design for EIS
measurements.
EIS &
Nano-El-Cat
Nov. ‘08.
Circuit Description Code
The Circuit Description Code presents an
unique way to define an equivalent circuit
in terms suitable for computer processing.
Elements: R, C, L, W
Finite length diffusion:
T = FSW = Tanhyp (Adm.)
Cothyp (Imp.)
O = FLW = Cothyp (Adm.)
Tanhyp (Imp.)
Constant Phase Element:
Q = CPE = Y0(j ω)n (Adm.)
Z0(j ω)-n (Imp.)
EIS &
The CPE
Nano-El-Cat
Nov. ‘08.
Constant Phase Element:
YCPE = Y0 ωn {cos(n π/2) + j sin(n π/2)}
n=1
Î
Capacitance: C = Y0
• n=½
Î
Warburg:
• n=0
Î
Resistance: R = 1/Y0
• n = -1
Î
Inductance: L = 1/Y0
σ = Y0
All other values, ‘fractal?’
‘Non-ideal capacitance’, n < 1 (between 0.8 and 1?)
EIS &
Non-ideal behaviour
Nano-El-Cat
Nov. ‘08.
General observations:
• Semicircle (RC ) ⇒ depressed
• vertical spur (C ) ⇒ inclined
• Warburg
⇒ less than 45°
Deviation from ‘ideal’ dispersion:
Constant Phase Element (CPE),
(symbol: Q )
YCPE
nπ ⎤
⎡ nπ
= Y0 ( jω) = Y0ω ⎢cos + j sin ⎥
2
2⎦
⎣
n
n
½,
nn == 1,1, ½,
0, -1,
-1, ??
0,
EIS &
Nano-El-Cat
Nov. ‘08.
The Fractal Concept
How to explain this non-ideal behaviour?
1980’s: ‘Fractal behaviour’ (Le Mehaut)
= fractal dimensionality
i.e.: ‘What is the length of the coast line of England?’
) Depends on the size of the measuring stick!
) Self similarity (
EIS &
Fractals
Nano-El-Cat
Nov. ‘08.
Fractal line
Self similarity!
similarity!
Self
‘Sierpinski carpet’
EIS &
Nano-El-Cat
Nov. ‘08.
‘Fractal electrode’
‘Cantor bar’
arrangement
Impedance of the network:
1a
a
Z⎛(ω)⎞= R +
a Z (ω)
22
Z⎜ ⎟ = R+
2
jω
CC+Z+(ω) + 2
jωaC
⎝a⎠
1
Z
(
)
ω
R+
aR
22
aC++ 2
jωC
aaR
R + ...
R
aR
a2R
a3R
EIS &
Nano-El-Cat
Nov. ‘08.
Arriving at the ‘CPE’
a Z (ω)
⎛ ω⎞
Frequency scaling relation: Z ⎜ ⎟ = R +
jωC Z (ω) + 2
⎝a⎠
In the low frequency
limit this reduces to:
Which is satisfied by
the formula:
⎛ ω⎞ a
Z ⎜ ⎟ = Z (ω)
⎝a⎠ 2
Z (ω) = A( jω)−n
with n = 1 – ln2/lna
Fractal dimension of Cantor bar, d = ln2/lna
Hence: n = 1 –d
S.H. Liu and T. Kaplan, Solid State Ionics 18 & 19 (1986) 65-71.
EIS &
Nano-El-Cat
Nov. ‘08.
CDC
CDC = ‘instruction string’ for response calculation
Uses brackets:
• [ … ] series combination, e.g.: [RC]
• ( … ) parallel combination, e.g. (RC)
Randles circuit: R(C[RW])
W
EIS &
Nano-El-Cat
Nov. ‘08.
Determining the CDC
(C[(Q[R(RQ)])(C[RQ])])
EIS &
CNLS data analysis
Nano-El-Cat
Nov. ‘08.
Model function, Z(ω,ak), or equivalent circuit.
Adjust circuit parameters, ak, to match data,
Minimise error function:
n
{
2
S = ∑ wi Zre,i − Zre (ωi ) + Zim,i − Zim (ωi )
i =1
with: wi = Zi
for k = 1 ..M
2
≈ Z (ωi , ak )
2
2
}
(weight factor)
d
S =0
dak
Non-linear, complex model function!
Effect of minimisation
EIS &
Nano-El-Cat
Nov. ‘08.
Non-linear systems
Function Y (a1..aM) is not linear in its parameters, e.g.:
[
Z (ω) = Z0 ⋅ k + ( jω)
]
β −α
= Z (ω, Z0 , k , α, β)
(‘Gerischer’)
Linearisation: Taylor development around ‘guess values’, ajo:
∂Y ( x, a1..aM )
Y ( x, a1..aM ) = Y ( x, a ..a ) + ∑
∂a j
j
ο
1
⋅ δa j + ....
ο
M
ο
a1ο..aM
Derivative of error sum with respect to δaj :
⎡
∂S
∂Y ( xi , a1..M ) ⎤ ∂Y ( xi , a1..M )
ο
= 0 = 2∑ wi ⎢ yi − Y ( xi , a1..M ) + ∑
δak ⎥
∂a j
∂ak
∂a j
i
k
⎣
⎦
EIS &
NLLS-fit
Nano-El-Cat
Nov. ‘08.
A set of M simultaneous equations, in matrix form:
α ⋅ δa = β
,
solution:
δa = α-1⋅ β = ε ⋅ β
∂Y ( xi , a1..M ) ∂Y ( xi , a1..M )
= ∑ wi
⋅
∂a j
∂ak
i
With:
α j,k
and:
∂Y ( xi , a1..M )
βk = ∑ wi [ yi − Y ( xi , a1..M )]
∂ak
i
Derivatives are taken in point ao1..M.
Iteration process yields new, improved values: a’j = aoj + δaj.
EIS &
Nano-El-Cat
Nov. ‘08.
Marquardt-Levenberg
Analytical search: fast and accurate near true minimum
slow far from minimum
(and often erroneous)
Gradient search or steepest descent (diagonal terms only):
fast far from minimum
slow near minimum
Hence, combination!
Multiply diagonal terms with (1+λ).
•
λ << 1, analytical search
•
λ >> 1, gradient search
Successfuliteration:
iteration:
Successful
Sold
new<<S
SSnew
old
Î
decrease
λ
(=λ/10).
λ/10).
Î decrease λ (=
Otherwiseincrease
increase
Otherwise
(=λ×10)
λ×10)
λλ(=
Bottom line: good starting parameter estimates are essential!
EIS &
Error estimates
Nano-El-Cat
Nov. ‘08.
For proper statistical analysis the weight factors, wi,
should be established from experiment.
Other (dangerous) method:
Step 1: set weight factors, wi = g ⋅ σi-2
Step 2: assume variances can be replaced by parent
distribution, hence χν2 ≈ 1
(with ν = N –M – 1)
Step 3:
χ 1 [ yi − Y ( xi , a1..M )] 1
1
S
χ = = ∑
= S∑
=
≅1
2
2
ν ν i
σi
ν i wi ⋅ σi g ⋅ ν
2
ν
2
2
Hence proportionality factor, g = S/ν.
EIS &
Nano-El-Cat
Nov. ‘08.
Error analysis NLLS-fit
Based on this assumption we can derive
the variances of the parameters: σ a2k = g ⋅ αk , k
[ ]
−1
= g ⋅ εk ,k
Error matrix, ε, also contains the covariance of the
parameters:
σ ⋅σ = g ⋅ ε
aj
ak
j ,k
g⋅
εj,k ≅ 0, no correlation between aj and ak.
g⋅
εj,k ≅ 1, strong correlation between aj and ak.
Only acceptable for many data points AND
random distribution of the ‘residuals’
EIS &
Nano-El-Cat
Nov. ‘08.
Weight factors and error
estimates
Errors in parameters:
• estimates from CNLS-fit procedure
• assumption: error distribution equal to ‘parent distribution’
• only valid for random errors,
• no systematic errors allowed!
Residuals graph:
Large error estimates:
strongly correlated
parameters (+ noise).
Option: modification of
weight factors.
Δre =
Zre,i − Zre (ωi )
Z (ωi )
, Δim =
Zim,i − Zim (ωi )
Z (ωi )
EIS &
Nano-El-Cat
Nov. ‘08.
Two different CNLS-fits
Example of correct
error estimates:
R(RC)(RC)
CDC: R(RQ)(RQ)
χ2 2.4⋅10-5
R1 999
R2 4000
Q3 1.03⋅10-9
-n3 0.898
R4 8020
Q5 1.03⋅10-7
-n5 0.697
0.8%
1.7%
7%
0.6%
0.9%
3.6%
0.7%
And of incorrect
error estimates:
CDC: R(RC)(RC)
Values seem O.K.
but look at the
residuals!
χ2 3.8⋅10-3
R1 1290
4%
R2 4650
2.7%
C3 2.38⋅10-103.8%
R4 6580
2.6%
C5 6.07⋅10-9 7.3%
EIS &
Nano-El-Cat
Nov. ‘08.
Residuals plot!
Systematic deviation,
‘Trace’, bad fit
Good fit (not bad for
a straight simulation!)
EIS &
Nano-El-Cat
Nov. ‘08.
‘Fingerprinting’
Classification of capacitance
source
approximate value
geometric
grain boundaries
double layer / space charge
surface charge /”adsorbed species”
(closed) pores
“pseudo capacitances”
“stoichiometry” changes
2-20 pF
(cm-1)
1-10 nF
(cm-1)
0.1-10 μF/cm2
0.2 mF/cm2
1-100 F/cm3
large !!!!
Modified after: Peter Holtappels, TMR symposium ‘Alternative anodes...’,
Jülich, March 2000.
EIS &
Nano-El-Cat
Nov. ‘08.
Gas phase capacitance
Capacitance of gas volume (e.g. O2):
PV=nRT
PV=nRT
dE
i dt
or: C =
dt
dE
O2 produced: i dt = 4F dn
RT P + dP RT dP (RT )2 dn
Nernst: dE =
ln
≈
=
4F
4F P
4F V
P
Capacitance: i = C
2
Combination: C = ⎛ 4F ⎞ V ⋅ P
ox
⎜
⎟
⎝ RT ⎠
Example:
air, 700°C, Vol. = 10 mm3
Cox = 0.456 F !
EIS &
Nano-El-Cat
Nov. ‘08.
Conclusions on ‘fitting’
Many parameter, complex systems modelling:
•
Use Marquardt-Levenberg when quality starting
values are available
•
Simplex (or Genetic Algorithm) for optimisation of
‘rough guess’ starting values, as input for M-L NLSF
•
Check residuals when calculating Error Estimates
•
Look for systematic error contributions, remove if
feasible.
•
Provide error estimates in publications!
It’s human to err, its dumb not to include
an error estimate with a number result
EIS &
Nano-El-Cat
Nov. ‘08.
‘Molecular Printboard’
ferrocenyl (Fc) decorated poly(propylene imine) dendrimer
β-cyclodextrin
Christian A. Nijhuis, B.A. Boukamp, B-J. Ravoo,
J. Huskens and D.N. Reinhoudt
J. Phys. Chem. C 111 (2007) 9799
EIS &
Nano-El-Cat
Nov. ‘08.
Electrochemical response
Impedance graphs of
an aqueous solution of 1
mM (in Fc
functionality) of G4PPI-(Fc)32-(β-CD)32
at a β-CD SAM.
(10 mM β-CD at pH = 2)
Potential: -0.15 V to 0.15 V
Frequency: 10 kHz to 10 mHz
KK-test:
< 10 × 10-6
EIS &
Nano-El-Cat
Nov. ‘08.
Subtraction procedure
• Partial CNLS-fit of recognizable structure
ƒ Semicircle
ƒ Straight line (CPE, Cap., Ind.)
• Subtract dispersion as series- or parallel component
• Repeat steps until ‘garbage’ is left
• Be aware of ‘errors’ due to consecutive subtractions
• Sometimes restart and do a partial fit of a larger
group of parameters
Nano-El-Cat
Nov. ‘08.
Impedance G-4 at 0.105 V
--Zimag,
Zimag,[ohm]
[ohm]
3.E+04
3.E+04
2.E+04
2.E+04
1.E+04
1.E+04
SAM cap.//resist.
EIS &
0.E+00
0.E+00
0.E+00
0.E+00
Randles response
1.E+04
1.E+04
2.E+04
2.E+04
3.E+04
3.E+04
Zreal,[ohm]
[ohm]
Zreal,
4.E+04
4.E+04
EIS &
Nano-El-Cat
Nov. ‘08.
Subtract Rel’lyte, CSAM
3.E+04
3.E+04
--Zimag,
Zimag,[ohm]
[ohm]
Fit of Randles circuit: R(C[RW])
2.E+04
2.E+04
1.E+04
1.E+04
0.E+00
0.E+00
0.E+00
0.E+00
1.E+04
1.E+04
2.E+04
2.E+04
3.E+04
3.E+04
Zreal,[ohm]
[ohm]
Zreal,
4.E+04
4.E+04
EIS &
First CNLS-result
Nano-El-Cat
Nov. ‘08.
-Z-ima g x 1e 5
EqCWIN analysis
Re a l,Ima g-e rror x 1e -2
4
Full circuit
3
2
1
0
-1
0.2
-2
-3
-4
1e -1
1
1e 1
1e 2
1e 3
Fre que ncy
Re a l,Ima g-e rror x 1e -2
4
0.1
3
2
Circuit without
Residuals
plot(R[RC])
1
0
-1
-2
-3
-4
1e -1
1
1e 1
1e 2
Fre que ncy
0
0
0.1
G4-ferrocene, 0.105 V
0.2
0.3
Z-re a l x 1e 5
0.4
1e 3
EIS &
Nano-El-Cat
Nov. ‘08.
Subtract Rel’lyte, CSAM
3.E+04
3.E+04
--Zimag,
Zimag,[ohm]
[ohm]
Fit of Randles circuit: R(C[RW])
2.E+04
2.E+04
1.E+04
1.E+04
0.E+00
0.E+00
0.E+00
0.E+00
1.E+04
1.E+04
2.E+04
2.E+04
3.E+04
3.E+04
Zreal,[ohm]
[ohm]
Zreal,
4.E+04
4.E+04
EIS &
--Zimag,
Zimag,[ohm]
[ohm]
Nano-El-Cat
Nov. ‘08.
Subtract Randles
100
100
00
3800
3800
3900
3900
Zreal,[ohm]
[ohm]
Zreal,
4000
4000
EIS &
Nano-El-Cat
Nov. ‘08.
Small difference, but …
-Z-ima g x 1e 5
EqCWIN analysis
Re a l,Ima g-e rror x 1e -2
4
Full circuit
3
2
1
0
-1
0.2
-2
-3
-4
1e -1
1
1e 1
1e 2
1e 3
Fre que ncy
Re a l,Ima g-e rror x 1e -2
4
0.1
Circuit without (R[RC])
3
2
1
0
-1
-2
-3
-4
1e -1
1
1e 1
1e 2
Fre que ncy
0
0
0.1
G4-ferrocene, 0.105 V
0.2
0.3
Z-re a l x 1e 5
0.4
1e 3
Nano-El-Cat
Nov. ‘08.
Equivalent Circuit
Au
EIS &
+
EIS &
Nano-El-Cat
Nov. ‘08.
Consistency of Circuit!
R2
R1
CSAM
C2
Tentative model
EIS &
Nano-El-Cat
Nov. ‘08.
Modelling of diffusion
Modelling diffusion:
Qdiff = Q0
1
⎛
DO ⎞
1 ⎞ ⎛
⎟⎟
⎜1+
⎟ + ⎜⎜ 1 + K θ
K
D
θ ⎠
R ⎠
⎝
⎝
n 2F 2 A 2 0
with: Q0 =
CFc ,tot DO
RT
for right hand side only.
Actually: Qdiff= Qconst + Qdiff(V)
nF
( η−η )
Modelling of doubleRT
e
Cdl = Cdl ,1θ + Cdl ,2 (1 − θ) , with: θ =
layer capacitance:
nF
0
1+ e
RT
( η−η0 )
EIS &
Nano-El-Cat
Nov. ‘08.
Diffusion & generation
σ (μF)
Generation 1 to 4 measured at the same βCD SAM:
■ : G1
▼: G2
100 ▲: G3
●: G4
⎛
1
1
RT
⎜
σ= 2 2
+ *
*
n F A 2 ⎜⎝ CO DO C R DR
10
-0.02
0.00
0.02
0.04
0.06
Polarization (V)
0.08
0.10
⎞
⎟
⎟
⎠
EIS &
Nano-El-Cat
Nov. ‘08.
Stokes-Einstein
Stokes-Einstein relation:
D=
kT
6ηπr
Qdiff ÷ √D
and r ÷ Generation nr.
Hence: Qdiff ÷ (Gx)-1/2
The Warburg admittance corrected for the
concentration of dendrimers and the number
of electrons involved per molecules plotted vs
the (square root of generation)-1.
EIS &
Nano-El-Cat
Nov. ‘08.
Reaction Pathways
Schematic of the potential-dependent surface coverage of the
dendrimers (left), and a scheme of adsorption and desorption kinetics
(right), Redsol = reduced dendrimers in solution, Redads = dendrimers
adsorbed at the βCD SAM, Oxsol = oxidized dendrimers in solution,
Oxads = oxidized dendrimers at the surface; kdr, kar, kdo and kao are
adsorption (a) and desorption (d) rates of oxidized (o) and reduced (r)
dendrimers; kb and kf are electrochemical rate constants.
EIS &
Nano-El-Cat
Nov. ‘08.
Praise of the time domain …
Intercalation cathode.
Change of potential = change
of aA at the interface, hence
A-diffusion:
dCA ( x, t )
%
J (t ) = −DA
dx x=0
Voltage-activity relation:
RT aA, x=0
ln 0
E(t ) =
nF
aA
Fick 1 & 2, boundary conditions
V (ω)
=
Z (ω) =
+ Laplace transform:
I (ω)
jω
coth l
D%
jωD%
Z0
EIS &
Nano-El-Cat
Nov. ‘08.
Real cathode: LixCoO2
10.0
10.0
Measurement
Measurement
Simulation
Simulation
e
RR
e
Z"
Z"[kΩ]
[kΩ]
7.5
7.5
20Hz
20Hz
5.0
5.0
2.5
2.5
sl
RR
sl
ct W
RR
ct W
sl
QQ
sl
dl
CC
dl
3.78V
3.78V
27Hz
27Hz
36Hz
36Hz
3.69V
3.69V
3.84V
3.84V
63Hz
63Hz
3.88V
3.88V
4.06V
4.06V
110Hz
110Hz
0.0
0.0
0.0
0.0
LiCoO2, RF film on silicon.
Peter J. Bouwman, Thesis,
U.Twente 2002.
2.5
2.5
5.0
5.0
7.5
10.0
7.5
10.0
[kΩ]
Z'Z'[kΩ]
12.5
12.5
15.0
15.0
IS of a RF-film electrode: (○) ‘fresh’;
(□) charged; (‘) intermediate SoC’s.
(+) CNLS-fit. Range: 0.01 Hz – 100
kHz.
Nano-El-Cat
Nov. ‘08.
Diffusive part?
The lithium diffusion
process is found at lower
frequencies!
Compare the potential-step
response time with lowest
frequency of EIS
experiment:
teq. >> 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] ]
EIS &
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.
EIS &
Nano-El-Cat
Nov. ‘08.
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
EIS &
Fourier Transform
Nano-El-Cat
Nov. ‘08.
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
EIS &
Simple exponential extension
Nano-El-Cat
Nov. ‘08.
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
EIS &
Nano-El-Cat
Nov. ‘08.
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 )
EIS &
Nano-El-Cat
Nov. ‘08.
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
EIS &
Nano-El-Cat
Nov. ‘08.
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 %
EIS &
Nano-El-Cat
Nov. ‘08.
Bode Graph
Double
logarithmic
display
almost
always
gives
excellent
result !
‘Bode plot’, Zreal and Zimag versus frequency in double log plot
EIS &
Nano-El-Cat
Nov. ‘08.
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!
EIS &
Nano-El-Cat
Nov. ‘08.
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!
EIS &
Nano-El-Cat
Nov. ‘08.
Last slide
EIS &
Nano-El-Cat
Nov. ‘08.
EIS &
Effect of truncation
Nano-El-Cat
Nov. ‘08.
10
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)
EIS &
More Fourier transform
Nano-El-Cat
Nov. ‘08.
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)
Nano-El-Cat
Nov. ‘08.
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]
EIS &
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!
⎣
⎦
EIS &
Nano-El-Cat
Nov. ‘08.
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!
⎦
}
EIS &
Nano-El-Cat
Nov. ‘08.
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