Time constant τ = RC:
Z = (1 / R + jωC )−1 =
R(1 − jωτ)
1 + (ωτ)2
Semicircle in the impedance plot
Y = (R + 1 / jωC )−1 =
ωC (ωτ + j)
1 + (ωτ)2
Semicircle in the admittance plot
If an equivalent circuit has several time constants close to each other, the separation of circuit elements is difficult.
Cd,a
Cd,c
Simple model for a PEFC:
Rct,c
Rm
Rct,a
Often Rct,c » Rct,a ⇒ τc » τa and only one semi-circle appears in the impedance plot. Cathode and anode capacitances Cd,c
and Cd,a must usually be replaced by constant phase elements (depressed semi-circles); this will be discussed shortly.
Adsorption
Often electroactive species adsorb first on the electrode and electron transfer follows adsorption. Most often adsorption is
analyzed through a Langmuir type process. In a Langmuir type adsorption only one layer of an adsorbate is allowed and there
is no interaction between the adsorbates. The maximum amount of adsorbed molecules in a monolayer is denoted by Γmax
and the surface coverage by ϕ = Γ/Γmax where Γ is the amount of adsorbed amount in mol/cm2. A typical value of Γmax is
10−10… 10−9 mol/cm2. The kinetic equation of adsorption is
dϕ
= kac s (1 − ϕ) − kd ϕ
dt
where ka is the adsorption and kd desorption rate constant, and cs is the concentration (mol/cm3) of the adsorbate at the
electrode. From the above equation it is seen that the units of ka is cm3/(mol·s) and that of kd is 1/s. The first term on the
right hand side tells that adsorption can take place only on free surface, represented by the factor (1 − ϕ). At equilibrium
dϕ/dt is naturally zero and the surface concentration cs is equal to the bulk concentration cb. Hence:
b
kac (1 − ϕeq ) = kd ϕeq ⇔ ϕeq
ka
kac b
Kac b
K
=
=
;
=
a
kd
kac b + kd 1 + K ac b
The framed equation is the famous Langmuir adsorption isotherm; Ka is the adsorption equilibrium constant. At low
concentrations – which probably is the case in electroanalytical work – the linear form of the isotherm, i.e. the numerator
alone suffices: ϕeq ≈ Kacb. At steady-state, dϕ/dt = 0 but cs must be calculated from the steady-state transport equations.
Let’s start the analysis with an extreme case where a redox couple is immobilized on the surface. This can be done by grafting
the surface with, e.g. in the case of gold electrode with thiols, or in the case of carbon electrodes with diazonium salts. The
figure below depicts the situation. We assume that ΓR + ΓO = Γmax or ϕR + ϕO = 1, i.e. the surface is fully covered by R and O.
There is thus no materials exchange between the solution and the surface monolayer. Electric current
density is simply
i
dϕ dϕ
=− R = O
nF Γmax
dt
dt
Analogously to previous treatments, the linearized current-overpotential equation is
η=
ϕO 
ϕR
RT  i
−
+
nF  i 0 ϕR,eq ϕO,eq 
The specialty here is that adjusting the electrode potential to any value, after a transient phase, the surface coverages settle
according to the Nernst equation because the amount of reacted species is not replaced by transport from the solution:
(
)
ϕO ,eq
 nF

Eeq − E 0' 
= θ = exp 
ϕR ,eq
 RT

⇒
ϕO ,eq =
θ
1
; ϕR ,eq =
1+θ
1+θ
The “exchange current density” is analogously
i0 = nFk (ΓO,eq ) (ΓR,eq )
0
α
1− α
α
1− α
 θ   1 
= nFk Γmax 
 

+
θ
+
θ
1
1

 

0
θα
= nFk Γmax
1+θ
0
Note that the units of k0 is now 1/s and we have different i0 at each potential!
Straightforward analysis in Laplace domain proceeds as follows:
i (s)
= −(s ϕR (s) − ϕR ,eq ) = s ϕO (s) − ϕO ,eq
nF Γmax
ϕR (s) 1
ϕ (s) 1
i (s) 1
i (s) 1
= −
; O = +
ϕR,eq s nF ΓR,eq s
ϕO,eq s nAΓO,eq s
RT  i (s)
i (s)  1
1  1 
η(s) =
+
+
nF  i 0 nF Γmax  ϕR,eq ϕO,eq  s 
⇒
(
1+θ
1
1 + θ)2
RT
RT
= 2 2
;
Rct = 2 2
0
α
θ
C n F AΓmax
n F AΓmax k θ
Z (ω) =
 1
1  1
1
RT
RT

+ 2 2
+
R
=
+
ct
jωC
nFI0 n F AΓmax  ϕR,eq ϕO,eq  jω
8
R
6
k0
Taking = 1
the minimum charge transfer resistance is ca. 530
the resistance is very small with high surface areas.
s−1
Ω·cm2,
i.e.
Let’s consider next a more conventional case where a metal cation M+z is first
adsorbed on the electrode and then reduced:
i
0 (α −1) f (E −E 0 ' )
= −kred ΓM = −k e
ΓM
nF
'R ' =
0.3
1+θ
θα
0.2
C
In the plot aside, the potential dependent parts of Rct and C are plotted. Now
the capacitance has a maximum at E0’. Inserting Γmax = 10−9 mol/cm2 gives
Cmax ≈ 1 mF/cm2. Since the area of a porous electrode can be of the order of
1 m2/g capacitors of very high charge can be prepared (i.e. super capacitors).
4
0.1
2
-100
'C ' =
-50
0
θ
(1 + θ)2
50
0
100
(1)
We have thus ignored the anodic part of the Butler-Volmer equation since we are depositing metal (E « E0’), and current is
negative; also, z = n. The surface concentration of the cation, ΓM, can naturally be written as Γmaxϕ. The mass balance of the
adsorbed cation is, assuming Langmuir type adsorption
∂ϕ
i
i
=
+ kacMs (1 − ϕ) − kd ϕ =
+ kacMs − (kacMs + kd )ϕ
∂t nF Γmax
nF Γmax
consumption adsorption
in electrode
reaction (i < 0)
desorption
(2)
Let’s simplify the problem neglecting mass transfer, i.e. cMs = c b which means that adsorption is the rate determining step. At
steady-state (‘ss’) dϕ/dt = 0 and from the mass balance it is obtained:
iss / nF Γmax + kac b
ϕss =
kac b + kd
(3)
Linearizing eq. (1) and transforming it to Laplace domain gives
∆i (s)
= −kred [∆ ϕ(s) + (α − 1) fϕss ∆E (s)]
nF Γmax
(4)
where ∆’s denote deviation from steady-state due to the ac signal. Eq. (2) is already linear because ka or kd do not depend
on potential. Thus its Laplace transform is
(
)
kac b
∆i (s)
s∆ ϕ(s) =
+
− kac b + kd ∆ ϕ(s)
nF Γmax
s
⇒ ∆ ϕ(s) =
nF Γmax
kac b
∆i (s)
+
b
s + kac + kd s s + kac b + kd
(
) (
(5)
)
(6)
Now we apply for the first time the rule of finding the periodical components. The last term on the right is not periodical
because it contains neither ∆i (s) nor ∆E (s). We throw it away and insert the rest into eq. (4). After some algebra:
1
(1 − α) fϕss Z (s)
=
1
1
nFAΓmax
+
kred s + kac b + kd
where both sides have been divided by ∆i (s). After some rearrangements and replacing s with jω it is obtained:
Z=
RT
(1 − α)n2F 2 AΓmax ϕss
 1

1
+


b
k
ω
+
+
j
k
c
k
a
d
 red
The first term in the impedance does not depend on frequency: hence it is the charge transfer resistance. The second term is a
combination of two elements. Pondering for a while it is obvious that it is a parallel combination of a resistor and a capacitor.
Adding the always present electrode double layer capacitance and the solution resistance the impedance plot looks like this:
Combining eqs. (1) and (3), ϕss can be solved as
kac b
ϕss =
kac b + kd + kred
Let’s write down the explicit expressions of the faradaic impedance elements:
 kac b + kd
RT
1 +
Rct =
2 2
b
kred
(1 − α)n F AΓmax kac 
Rads =





RT
kred

1
+
(1 − α)n2F 2 AΓmax ka c b  kac0 + kd



(1 − α)n2F 2 A Γmax kac b
Cads =
RT kac b + kd + kred
(
)
These elements are obtained from a non-linear fit, and the values of the physical parameters are obtained from them:
Rct kac b + kd
=
Rads
kred
RadsCads =
1
kac b + kd
Rct Cads = 1 / kred
Impedance method thus gives the value of the electrochemical rate constant, kred, explicitly, but finding the values of ka or
kd requires measurements at varying solution concentrations cb.
Let’s consider next the case where a redox couple has adsorbed intermediates and adsorption is very fast, obeying Langmuir
isotherm. Adsorption thus is under mass transfer control. The generalized current boundary condition is
∂Γ
∂Γ
i
 ∂c 
 ∂c 
= DR  R 
− R = −DO  O 
+ O
nF
 ∂x  x = 0 ∂t
 ∂x  x = 0 ∂t
Linearized Langmuir isotherm:
Γk
K k cks
Kk
*
θk =
=
≈
θ
+
k
Γmax 1 + K k cks
1 + K k ckb
(
cRb
−
cR (x , s) = + A(s)e
s
s / DR x
[
)
(
s
c
k
2
− ckb
)
⇒
∂Γ k
ΓmaxK k
=
∂t
1 + K k ckb
(
)
 ∂cks 
 ∂cks 
 ∂cks 

 ≈ ΓmaxK k 



2  ∂t 
 ∂t  = Λ k  ∂t 






1 >> K k ckb
cRb
⇒ cR = + A(s)
s
s
]
(
)
and
cOs 1
= +
b
cO s nFcOb
i (s)
= − A(s) sDR − ΛR scRs − cRb = − A(s) sDR − ΛR sA(s) = − A(s) sDR + sΛR
nF
A(s) = −
nF
(
i (s)
sDR + sΛR
)
i (s)
cRs 1
⇒ b= −
cR s nFcRb sDR + sΛR
(
)
(
i (s)
sDO + sΛ O
)
Obs. extra terms sΛR,O
in the denominator!
Z=
RT
RT 
+ 2 2  b
nFI 0 n F A  cR
(
DR
1
+ b
jω + Λ R jω cO
)
(
DO
1
jω + Λ O jω
)




C
Rs
WR
WO
CR,ads
CO,ads
C k ,ads =
1
RT
n2F 2 Ackb Λ k jω
Rct
But this equivalent circuit is indistinguishable from the Randles’ circuit, demonstrating that more than one equivalent
circuits give similar impedance plots, and a researcher must study the system first with, e.g. cyclic voltammetry to see what
the system is like.
“If my tool is a hammer, I see all the problems as nails.”
Model for an electrode with an adsorption step
Rs = 1 Ω
Cd
Cd = 10 µF
Rs
Rct = 5 Ω
Cads = 50 µF
Cads
Rads = 20 Ω = adsorption kinetics resistance
σ = 10 Ωs–1/2
W
Rct
Rads
12
10
-Z''/Ω
8
6
4
2
0
0
5
10
15
20
Z'/Ω
25
30
35
Heterogeneity of the electrode surface ⇒ General element Q, the Constant Phase
Element, CPE
In reality, complete semi-circles are seldom achieved but they are flattened. This is treated mathematically by replacing a
capacitor with a constant phase element, the admittance of which is Y = Y0(jω)α. Below a simulation with R = 100 Ω, Y0 = 10–7
Ω–1 and α = 1, 0.9, 0.8: the lower the value of α, the flatter is the semi-circle.
50
40
30
Q
20
10
0
0
20
40
Zreal / Ω
60
80
100
Real measurement, human cadaver skin:
Features:
- elongation along the real axis
- flattening of the semi-circle
–Zimag / kΩ
Q
10
R1
Rs
best semi-circle
R2
measured fit
0
0
10
20
Zreal / kΩ
30
40
C
The lower RC branch corresponds to dipolar
relaxations taking place in the membrane
matrix with the characteristic relaxation time
τ = R2C. Skin lipids???
K. Kontturi, L. Murtomäki, Pharm. Res. 11(9), 1994, 1355-1357
J.R. Macdonald, ”Impedance Spectroscopy”, John Wiley & Sons, New York 1987, p. 49.
The origin of a CPE is in the dispersion of time constants. An RC circuit has only one
time constant τ = RC, but in reality, an electrode surface is heterogeneous with sites
of variable reactivity. Instead of a single RC circuit, the surface should be illustrated
as a network of RC circuits, forming a span of time constants over several decades.
The analysis of a CPE can also be done through fractal geometry:
α=
1
DF − 1
DF = fractal dimension
D F = 1.1
D = 1.3
F
Cross sections of two fractal surfaces with DF = 2.1 and 2.3, corresponding
to α = 0.91 and 0.77.
K. Kontturi, et al., Pharm. Res., 10(2), 1993, 381-385.
S.H. Liu, Phys. Rev. Lett. 55 (1985) 529
Also, in the Warburg impedance the surface heterogeneity is seen as the deviation of the phase angle from 45°. Warburg
impedance ~ (jω)–1/2 is replaced by CPE ~ (jω)–α with α < 0.5 for partially blocked surfaces and α > 0.5 for rough surfaces. In
this case, the relation to the fractal dimension is α = (DF – 1)/2 ⇔ DF = 2α + 1. In a potential step experiment I ~ t –α. These
relations have been proved correct with tailor-made electrodes with unambiguously defined fractal dimensions.
DF = 2.50
Impedance of a Koch electrode: T. Pajkossy, L. Nyikos, J.
Electrochem. Soc. 133 (1986) 2061-4
Potential step at a Sierpinski gasket: T. Pajkossy, L. Nyikos, Electrochim. Acta 34 (1989) 171-9
Cottrell behavior:
Sierpinski gasket. DF = 1.585