PhD School in Chemical Sciences and Technologies
Elettrochimica
per la caratterizzazione
di materiali funzionali avanzati
3. Spettroscopia d’impedenza elettrochimica
Electrochemistry
for the characterization
of advanced functional materials
3. Electrochemical Impedance Spectroscopy (EIS)
Prof. Patrizia R. Mussini
Dipartimento di Chimica Fisica ed Elettrochimica,Via Golgi 19, 20133 Milano
patrizia.mussini@unimi.it
Alternating voltage and current
IE
E = E0sin(ωt)
E0
I0
t
T
I = I0sin(ωt+ϕ)
E0= voltage amplitude
I0 = current amplitude
T = period
ν = frequency = 1/T
ω = pulsation = 2πν
ϕ = phase shift angle (between I and E)
Resistive effects to alternating voltage for the three basic elements of an electric circuit
Resistance R:
I E
E = E0sin(ωt)
Ohm Law
E=RI
I=GE
R = resistance
G = conductance
I in phase with E
ϕ=0
I = I0sin(ωt)
Inductance L:
E = XL I
t
I E
E = E0sin(ωt)
I = BL E
XL= inductive
BL = inductive
reactance
susceptance
= ωL
=1/ ωL
E is 90° ahead of I
t
I = I0sin(ωt − π/2)
ϕ = − π/2
Capacitance C:
E = XC I
XL = capacitive
reactance
= 1/ω
ωC
E = E0sin(ωt)
I = BC E
BC = inductive
susceptance
= ωC
I is 90° ahead of E
ϕ = π/2
I E
t
I = I0sin(ωt + π/2)
Impedance of a circuit including the three basic elements in series
The vector combination of the resistive effects of all the circuit elements
is the circuit impedance Z, and its reciprocal is admittance Y.
R
L
C
IXL
IZ
IX
ϕ
IR
IXC
In a SERIES circuit the current through each of the components is the same,
while the total voltage is the sum of the voltages across each component
(proportional to the corresponding resistance or reactance). Accordingly, in
order to find the global potential/current relationship it is convenient to perform
a vector analysis of the voltage contributions, taking as a reference the I vector.
X is the total reactance, i.e. the vector sum of inductive reactance and
capacitive reactance, perpendicular to ohmic resistance and pointing upwards
for prevailing inductive reactance, downwards for prevailing capacitive
reactance. For XL = XC total reactance is 0; therefore the circuit has purely ohmic
resistance and current is in phase with voltage, no matter how large XL and XC
(resonance condition).
The vectorial sum of R, ohmic resistance, and X, total reactance
(vectorial sum of capacitive and inductive reactances), is the
circuit Z impedance.
I
Dividing all terms by current I we obtain the “impedance triangle” of the circuit:
Z
|Z| = √(R2 + X2) =√(R2 + (ωL−(1/(ωC))2)
X
ϕ “phase shift angle”
or “power factor” (power is ÷ cosϕ; if ϕ = 0, cosϕ is maximum and = 1)
ϕ
R
R = Z cos ϕ
X = Z sin ϕ
X = R tg ϕ
Impedance of a circuit including the three basic elements in parallel
In a PARALLEL circuit, the voltage across each of the components is the same,
while the total current is the sum of the currents through each component (which
are proportional to the corresponding conductances or susceptances). Accordingly,
in order to find the global potential/current relationship it is convenient to perform
a vector analysis of the current contributions.
R
L
C
The rationale is analogous to the former one, performing the vector sum on the I
vectors (rather than the E ones) and therefore on susceptances (rather than
inductances), taking as a reference the E vector.
B is the total susceptance, vector sum of inductive susceptance and
capacitive susceptance, perpendicular to ohmic resistance and pointing
upwards for prevailing capacitive susceptance, or downwards for prevailing
inductive susceptance.
BLE
YE
BE
The vectorial sum of G, conductance, and B, total susceptance
(vector sum of capacitive and inductive susceptances) is the circuit
admittance Y, i.e. the reciprocal of impedance Z.
ϕ
GE
BCE
|Y| = √(G2 + B2) = √(1/R2 + (ωC −(1/(ωL))2)
tgϕ = B/G = R · (ωC −(1/(ωL))
E
Dividing all vectors by E:
For an RC parallel
R
(the most important case in electrochemistry)
Y
ϕ
G
B
|Y| = √(G2 + B2) = √(1/R2 + (ωC)2)
= 1/R · √(1+ ωRC)2
|Z| = R / √(1+ ωRC)2
e tgϕ = B/G = ω R C
C
A comparison among alternative impedance notations
Complex number
Vector
(real part = ohmic contributions
imaginary part = non-ohmic contributions)
For an RC series
Z = Z’+iZ”
Z”
Exponential
Z = Z e iϕ = Z (cos ϕ + i sin ϕ )
in particular, here
Z = R − i/ω
ωC = R +1/iω
ωC
R
XC
=1/(ω
ωC)
Z
Z’
For an RC parallel
R
C
BC =
ωC
Y
G
Y”
Y = Y’+iY”
in particular, here
Y = G + iω
ωC = 1/R + iω
ωC
Y’
The principle of Electrochemical Impedance Spectroscopy (EIS) measurements (I)
An electrochemical system opposes a number of obstacles to the current circulating under a
given potential, including:
•the resistance to charge transport in solution;
•the problem of mass transport to the electrode for the reactant species;
•the capacitive reactance of the electrical double layer at the electrode/solution interphase;
•the overpotential for the electron transfer between molecule and electrode,
and many possible others, according to the specific case.
Any of these steps/obstacles can be modeled as an electric circuit element or element
combination
Therefore analysis of the electrochemical system impedance can afford valuable
information concerning each of the above steps/obstacles.
The necessary condition is to be able to discriminate within the global impedance each
single contribution and assign it to the right step/obstacle, or identify a circuit model
corresponding to the electrochemical system and providing a simulated impedance
spectrum faithfully reproducing the experimental one.
To this aim it is necessary to repeat the impedance measurement at different frequencies
(i.e., on a whole frequency spectrum) of the alternating voltage; in fact all resistive terms
in the circuit excepting ohmic resistances have their own dependency on frequency; for
instance, capacitive reactances tends to zero at high frequencies.
The principle of Electrochemical Impedance Spectroscopy (EIS) measurements (II)
Electrochemical system,
with a given relationship
between potential and
current (see voltammetry)
The same determination is
repeated in a wide frequency
range, from very high ones
(MHz) to very low ones
(mHz)
(typically 100 kHz – 0.1 Hz).
(The necessary time for
obtaining the impedance
spectrum dramatically
increases with increasingly
lower frequency limits)
The resulting alternating
current is analyzed in
terms of amplitude and
phase angle
The system is polarized at the chosen potential and on this
fixed potential an alternating voltage is superimposed with a
convenient amplitude (a small one, usually within 10 mV, as
increasing amplitude results in increasing sensitivity, but
also in possible transgression of the required condition that
in the considered potential interval the I vs E relationship
can be considered constant).
ν
1. Nyquist diagram (−Z” vs Z’)
Each point represents the impedance at a given frequency,
as a complex number (the real part Z’ on the x axis and the
imaginary part Z” [with changed sign since in most cases
of capacitive origin and therefore negative] Frequency
increases from right to left, albeit it is not explicitly
accounted for.
Impedance
spectra, in
different
plotting modes
2. Bode modulus diagram
(ln|Z| vs lnν o lnω)
3. Bode phase diagram
(−ϕ vs lnν o lnω)
4. Complete 3D
diagram
(-Z” vs ν vs Z’)
5. Real time plot: Lissajous curves
resulting from real-time plotting of the alternating
current as a function of the alternating voltage.
They allow real time direct observation
of the phase shift between current and potential
pure R
pure C
Generic real
cell
Lissajous curves allow inter alia to detect anomalies
arising as a consequence of not meeting the required
linearity condition
RC series
In this example
Corresponds to an ideally polarizable electrode (i.e.
which potential can be varied virtually with no limits without
any faradaic reaction taking place)
Series → we consider the resistive terms in terms of complex number
(ohmic resistance as the real part, capacitive reactance as the (negative) imaginary one)
Z = R – i/(ωC) = R + 1/(iωC)
Nyquist:
Bode modulus:
Bode phase:
ν→0 Z→∞
|Z| =√(R2 +1/(ωC)2)
ϕ =1/(ωRC)
tgϕ
ν→∞ Z→R
ν→ 0 |Z| ∼ 1/(ωC)
→ ln|Z| / ln(ω
ω) ∼ −1
Nyquist admittance
ν→ ∞ ln|Z| ∼ lnR
ν→0
ϕ → 90°
ν→ ∞ ϕ → 0°
RC parallel
Corresponds to electron transfer at the interphase in the
absence of mass transfer control (like the Butler and Volmer
equation; “faradaic impedance”) also taking into account the
double layer capacitance (like the background in CV)
  αFη 
 − (1 − α) Fη 
i = i0 exp
 − exp

RT


  RT 
In this example with
Parallel → we consider the conductive terms in terms of complex number,
building admittance as the sum of ohmic conductance (real term) and capacitive
susceptance (imaginary positive term), then we invert it to achieve impedance.
1/Z = 1/R + i ωC
Z = 1/(1/R + i ωC)
We multiply
and divide by
(1/R − i ωC)
Z = R/(1 + ω2R2C2) − iωR2C /(1 + ω2R2C2)
Z’
Z’’
Nyquist:
ν→∞ Z’→ 0, Z’’→ 0
ν→0 Z’→ R, Z’’→ 0
dZ”/dω = 0 in ω = 1/RC (maximum)
In this point Z’ = R/2
Nyquist admittance
Bode modulus:
ν→ 0 ln|Z| ∼ lnR
ν→ ∞ |Z| ∼ 1/(ωC)
ln|Z| / ln(ω) ∼ −1
Bode phase:
ϕ = ωRC
tgϕ
ν → 0 ϕ → 0°
ν→ ∞ ϕ → 90°
Corresponds to electron transfer at the interphase in the
absence of mass transfer control (Butler and Volmer equation)
together with double layer capacitance, plus the solution
ohmic resistance (added in series to the RC parallel)
In this example, with
Z = RS + Rct/(1 + ω2 Rct 2Cdl2) − iωRct2Cdl /(1 + ω2Rct2Cdl2)
Nyquist:
Z’
ν→∞ Z’→ 0, Z’’→ 0
ν→0 Z’→ Rct+RS, Z’’→ 0
dZ”/dω = 0 in ω = 1/RctC (maximum)
at such point Z’ = RS + Rct/2
Z’’
Bode modulus:
ν→ 0 ln|Z| ∼ ln (RS + Rct)
ν→ ∞ ln|Z| ∼ ln (RS)
Bode phase:
tgϕ
ϕ = Z’’ /Z’
ν → 0 ϕ → 0°
ν→ ∞ ϕ → 0°
This circuit describes an electron transfer at the
interphase with both charge tranfer and mass
transfer control (therefore corresponding to the
complete I vs η equation obtained in the
Electrochemical Kinetics section)
Frequency increases
Rs
Parallel
Rct + Cdl
ZW
(charge
(diffusion of
transport in (polarization and
reagents and
solution) charge transfer at theproducts to /from
interphase)
the electrode)
With decreasing frequency
(stationary conditions, as in a
potentiostatic electrolysis) the
reagent diffusion to the electrode
becomes determining and is
accounted for by the “Warburg”
straight line.
On the contrary, with increasing
frequency the reactant diffusion
becomes negligible with respect to
charge transfer resistance and double
layer polarization (half circle); finally
at the highest frequencies also these
processes are excluded, and the
solution ohmic resistance can be
observed alone, as the small real
segment in proximity to the origin.
Semi-infinite diffusion: the Warburg element
To add a kinetically determining mass transfer from the solution bulk to the
electrode surface (semi-infinite diffusion) it is necessary to introduce in the circuit
a new element, the mass transfer impedance o semiinfinite Warburg
impedance ZW
ZW = σω-1/2 − i σω-1/2
with
Following such dependency on the square root of
frequency, the Warburg element cannot be
expressed as a combination of simple R e C
elements; as a “distributed element”, it can only be
approximated by an infinite series of simple
elements.
σ = σOx + σRed = RT/(n2F2√2) [(1/(DOx1/2COx*) +(1/(DRed1/2COx*)]
Therefore in Warburg impedance the real and imaginary components depend
on diffusion coefficients (in particular, they decrease with increasing diffusion
coefficients) and are identical in modulus; accordingly, the phase angle is
constant and = −45°. Thus:
•in the Nyquist diagram the Warburg element corresponds to a straight line
with 1 slope;
•in the Bode phase diagram, it corresponds to a plateau for ϕ = 45°;
•in the Bode modulus diagram, it corresponds to a straight line with 1/2 slope.
Slope ½ (Warburg)
Slope 1 (Capacitance)
Bode modulus
RCT (100 Ω)
Rs (10 Ω)
ϕ = 45°
Bode phase
ϕ=0
ϕ=0
It can be demonstrated that,
determining the charge
transfer resistance and the
Warburg impedance as a
function of the electrode
potential,
•the Warburg impedance ZW
has a minimum
corresponding to the
reversible half-wave potential
E1/2
•the charge transfer
resistance Rct has a minimum
corresponding to
Es = E1/2 + (RT/nF)ln(α/(1-α));
in particular, it correspondes
to E1/2 for α = 0.5 (symmetrical
charge transfer activation
barrier)
Modifications for electrode geometries other than planar
Semi infinite diffusion towards spherical surface of increasing curvature
(from the perspective of the molecule diffusing to the electrode; curvature is increasingly
perceived with decreasing radius and increasing diffusion coefficient)
Semi infinite diffusion towards
cylindrical electrode
Only the diffusive
component is here
represented, without
and with Rs
Semi infinite diffusion towards
disk electrode
Only the diffusive
component is here
represented, without
and with Rs
A practical example:
Modifications for diffusion within a finite boundary:
reflective and transmissive finite diffusion
A SEMI INFINITE: a “bare” electrode
a: semi infinite diffusion;
b: transmissive finite diffusion
c: reflective finite diffusion
Diffusion and Solution
resistance only
Adding charge transfer +
double layer polarization
involved in ET to/from an
electrochemically active molecule
diffusing to the electrode from the
solution bulk
c FINITE REFLECTIVE: the same
electrode covered by a conducting
polymer layer, in the absence of redox
active molecules in solution (at the
potential considered); In this case, the
RctCdl parallel corresponds to the
electrode/polymer interphase and Rct to the ET
between electrode and polymer, resulting in the
latter charging, with concurrent anion ingress
to preserve electroneutrality; this obviously
requires electrons and ions to diffuse in
opposite directions within the finite polymer
thickness (Warburg domain); however with
decreasing frequency capacitive saturation is
reached since the amount of sites available for
ET is limited and the charge cannot be
transmitted beyond the polymeric layer on
account of the absence of redox active
molecules in solution.
b: FINITE TRANSMISSIVE: the same
electrode covered by a conducting
polymer layer, in the presence of a
redox active molecule in solution in this
case, the charge can be transmitted beyond the
polymer layer, at the interphase between
polymer surface and solution.
Modification for adsorption of a reacting species
v1 = k10Γs a Ae
v2 = k Γ e
0
2 B
assuming adsorption to be represented by a
Langmuir isotherm
Y = A+
−
−
α 1 zF
RT
α 2 zF
RT
( E − E10 )
( E − E 20 )
− k10ΓB e
−k Γa e
0
2 s C
(1−α 1 ) zF
RT
(1−α 2 ) zF
RT
( E − E10 )
( E − E 20 )
Γs = concentration of free sites on the surface
ΓB = concentration of adsorbed B on the surface
B
iω + C
f = zF/(RT)
Θ = occupied sites/total sites
ififBB==00simple
simple
RC
semicircle
RC semicircle
B<0
B>0
Case of two adsorbed species
Effect of non ideality of solid electrodes: the phase constant element CPE
On solid electrodes deviations are often observed from ideal behaviour, particularly a
dispersive effect on frequencies and therefore on the time constants, as a consequence of (a)
microscopic roughness, resulting in coupling between solution resistance and surface capacity
and in (b) slow adsorption of ion and other molecules at the interphase.
For this reason the observed impedances cannot be represented as combination of simple R, C,
and L elements, but, as in the case of the Warburg element, by an infinite series of RC parallels
(“distributed element”) or by an ad hoc new element, which is called constant phase element
CPE, with equation:
Z CPE
1
=
T (iω)φ
that is
Z CPE
1
=
Tω φ
π
π 

cos(φ 2 ) − i sin (φ 2 )


where T is a constant in F cm-2 sφ-1 and φ is related to the α rotation angle in the complex
plane with respect to a purely capacitive characteristics:
α = (90°)×(1− φ)
It is worthwhile noticing that the equation can represent:
•a pure capacity for ϕ = 1
•a non ideal solid electrode for 1>ϕ > 0.5
•Warburg impedance for ϕ = 0.5
•a pure ohmic resistance for ϕ = 0
•a pure inductance for ϕ = -1
a) Case of an ideally polarizable
electrode: the straight line angle will
be no more 90° but a smaller one.
It is as in the case of a “leaking”
capacitor, i.e. with ohmic losses,
having a real component
besides the imaginary one
corresponding to pure capacity.
b) Case of an electron
transfer in parallel
with the double layer
capacitance: the half
circle appears
depressed (or, better,
partially rotated below
the first quadrant)
The extension of the CPE in the case of finite diffusion is the “Bounded Constant Phase”
element BCP
Z BCP =
[
1
φ
tanh RsT (iω)
φ
T (iω)
]
At high frequencies the equation becomes a simple CPE, reducing to Rs at low frequencies.
For φ = 0.5 it is shaped similarly to impedance in the finite diffusion case.
As for φ = 0.5 CPE represents semi infinite diffusion, for the same value BCP represents
finite diffusion.
Adsorption effects
The frequency dispersion resulting from surface irregularities is not found working on
monocrystalline Au(111) and Au(100) electrodes. However it is found even on such well
defined electrodes, in the presence of specifically adsorbable ions, such as halide anions, and
has been attributed to slow adsorption, diffusion and phase transitions at the surface
(“reconstructions”).
A fractal approach to surface roughness
Some researchers have elaborated impedance models for rough surfaces starting
from a fractal approach.
Zooming on a normal object will uncover finer, previously invisible,
new structure. When the same is done on a fractal object, however,
no new detail appears; nothing changes and the same pattern
repeats over and over. This feature is called “auto similarity”.
The fractal term has been coined in 1975 by Benoît Mandelbrot,
from Latin fractus (broken), like “fraction”; actually mathematics
considers fractal images as objects having fractional dimensions..
Porous electrodes
Porous electrodes are frequently adopted in electrocatalysis to increase the active surface.
In order to deal with pores they are usually assumed to have cylindrical shape with l length
and r radius.
Two cases can be considered:
a)
Porous electrodes in stationary conditions, without internal
diffusion. In other words, we can assume their internal
concentrations to be constant. The axially flowing dc
current, I, which enters the pore, flows towards the walls
and its value decreases with the distance x from the pore
orifice.
b)
Porous electrodes in the presence of axial diffusion (during
electrolysis, concentration changes in the pores).
Both the electric potential and the electroactive species
concentration are assumed to depend only on the distance
from the pore orifice; moreover, an excess of supporting
electrolyte is also assumed to be present (so that migration
can be neglected).
Pore model
1) Porous electrodes in
stationary conditions
a)
b)
c)
Intermediate general case.
Limiting case of shallow pores
in which current easily
reaches the pore base; in this
case the electrode behaves as
a flat electrode.
Limiting case of very deep
pores, in which the
penetration length is quite
shorter than the pore length.
Complex plane curves
calculated by De Levie at
different overpotentials
2) Porous electrodes in the presence of axial diffusion
In an electrolysis process
concentration in pores changes.
The presence of a concentration
gradient in pores results in two
semicircles modulated by the
overpotential.
The semicircle radius is also
modulated by concentration.
It is worthwhile noticing that at
high frequency a segment of
straight line with a 45° angle can
still be perceived.
Effect of the pore shape
Real case are much more complicated, involving a lot of pores of which morphology and
dimensional distribution are hardly known.
Other shapes than the cylindric one result in changes of the impedance spectrum:
Examples of applicative cases
Impedance of a protective coating in optimum conditions
Dealing with this case as with an ideal
capacity the fitting is not quite satisfactory.
This depends on the electrode roughness
However,
substituting C with
a CPE the fitting is
very good.
The presence of pores in the
protective layer changes the
situation drastically.
Impedance of a galvanic cell: two electrode
reactions (corresponding to two RC parallel
time constants)
Observation of a redox couple on a rotating
disk electrode at increasing speed (diffusion
in an increasingly shorter diffusion layer)
The RDE rotation
speed increases
The Fe(CN)63- / Fe(CN)64- couple on a
rotating disk electrode at increasing
speed: the thickness of the limiting
diffusion layer progressively
decreases.
Impedance of a membrane fuel cell as a function of
the gas fed to the cathode
Feeding hydrogen at the cathode (as well as at the anode) no
reduction can take place and therefore only the ohmic drop
corresponding to solution and membrane resistance is
measured.
Feeding oxygen at the cathode, then the cell reaction can take
place, and the corresponding faradaic (charge transfer)
resistance can be detected.
Feeding air rather than pure oxygen impedance increases on
account of oxygen “dilution”.
Moreover, feeding CO at the anode where hydrogen oxidation
must take place, we poison the catalytic electrode surface and
resistance increases; moreover, the EIS pattern is consistent
with a process involving adsorbed species.