mathematical modeling of electrochemical and photoelectrochemical...
TRANSCRIPT
Didrik René Småbråten
Mathematical Modeling of Electrochemicaland Photoelectrochemical Impedance
TMT4500 Materialteknologi, fordypningsprosjekt
Trondheim, Fall 2013
Supervisor: Svein Sunde
Norwegian University of Science and Technology
Faculty of Engineering Science and Technology
Department of Materials Science and Engineering
i
Acknowledgment
First of all I would like to thank my supervisor Professor Svein Sunde for brilliant guidance and dis-
cussions this fall. It has been a true honor to work with such a competent scientist. Ph.D. Morten
Tjelta and Ph.D. Lars-Erik Owe are acknowledged for deriving the analytical models for photoelectro-
chemical impedance spectra of semiconducting films modeled numerical throughout this project.
I would also like to thank my fellow classmates at the M.Sc. program in Chemical Engineering and
at the Department of Materials Science and Engineering, especially the "Pippi"-gang for countless
hours of academic and non-academic moments.
Ph.D. candidate Siri Marie Skaftun also needs to be thanked for company during most needed, and
well earned, lunch breaks this fall.
ii
Preface
This report gives a summary of the course TMT4500 Materialteknologi, fordypningsprosjekt during
my master thesis at the Department of Materials Science and Engineering at the Norwegian Univer-
sity of Science and Technology (NTNU), fall 2013.
The project is a numerical modeling study of electrochemical and photoelectrochemical impedance
spectra of semiconducting thin film materials. The analytical models have been developed by Profes-
sor Svein Sunde, Ph.D. Morten Tjelta, and Ph.D. Lars-Erik Owe, and verified throughout the project
work.
Didrik René Småbråten
Trondheim, December 17, 2013
iii
Summary
Newman’s BAND subroutine solves initial, boundary-condition coupled linear differential equations.
The subroutine is supplied coefficient matrices based on the differential equations and boundary
conditions to solve the coupled differential equations. Traditionally, the coefficient matrices sup-
plied are real numbers. In this project, we have verified that the subroutine is suitable for complex
numbers, thus the compatibility to calculate impedance spectra.
A mathematical model for the electrochemical and photoelectrochemical impedance for a thin
film electrode has been established. The analytical model is based on the dilute solution theory and
binary electrolyte speciation. The numerical method is finite difference method, and uses Newman’s
BAND subroutine to solve a modified diffusion equation, where a sink term corresponding to a non-
faradaic reaction across the electrode (e.g. recombination or trapping) and a source term due to
photon absorption have been added. The concentration profile from the solved diffusion equation
was used to calculate the impedances in the thin film electrode.
The sensitivity of the method was investigated, and two main contributors to deviation were iden-
tified. The first is related to the frequency region of the calculated concentrations, where a deviation
at high frequencies was found at low number of steps. This deviation was solved by choosing a higher
number of steps until proper convergence. The second contribution is related to the parameteri-
zation of the diffusion equation, where large deviations and non-convergence of the model were
observed. The BAND subroutine uses dimensionless lengths, and the parameterization to dimen-
sionless lengths caused large deviations at low frequencies. These deviations were solved by alter the
dimensionless length parameterization to obtain proper convergence.
The numerical model of the photoelectrochemical impedance at zero light intensity deviates from
the electrochemical model. Three suggestions to solve this deviation was suggested. First, under no
illumination, a steady-state concentration profile was observed for the photoelectrochemical model,
but assumed constant for the electrochemical model. One source of this concentration gradient could
be a non-faradaic reaction that may affect the steady-state concentration. Second, the expression of
the steady-state concentration gradient neglects the contribution of carriers under no faradaic cur-
rent from e.g. thermal excitation. An additive term to the given expression should be included. Third,
the faradaic steady-state current is dependent of the light intensity, and under no illumination this
should be zero. However, the suggested model disregard this dependency, and an expression to take
this into consideration has been suggested.
General mathematical models for the infinite, transmissive and reflective Warburg diffusion impedances,
and transmissive and reflective Gerischer impedances were also derived, to trace deviations of the
numerical models and explain the behavior of the mixed ionic thin film electrode.
TABLE OF CONTENTS iv
Table of contents
Acknowledgment i
Preface ii
Summary iii
Table of contents v
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The photoelectrochemical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Impedance spectroscopy and modeling with Newman’s BAND subroutine . . . . . . . . 5
2 Theory 7
2.1 Impedance and impedance spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Mathematical treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Analysis of impedance spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Photoelectrochemical film electrode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Numerical method 24
3.1 Finite difference method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Coefficients for the BAND routine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Results 32
4.1 Warburg diffusion impedance spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Gerischer impedance spectra for thin film electrodes . . . . . . . . . . . . . . . . . . . . . 39
4.3 Electrochemical impedance spectra for semiconducting thin films . . . . . . . . . . . . . 47
4.4 Photoelectrochemical impedance spectra for semiconducting thin films . . . . . . . . . 53
5 Discussion 62
5.1 Parameterization sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Impedance spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6 Conclusion 67
References 70
TABLE OF CONTENTS v
A Proof that the impedance can be derived by setting s = jω in the Laplace-transform t-space
equations 73
B Full derivation of Warburg impedance 76
B.1 Warburg infinite diffusion impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
B.2 Warburg transmissive diffusion impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
B.3 Warburg reflective diffusion impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
C Full derivation of Gerischer impedance 82
C.1 Transmissive Gerischer impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
C.2 Reflective Gerischer impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
D Electrochemical impedance for a semiconducting film electrode 88
D.1 The photoelectrochemical film electrode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
D.2 Electrochemical impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
E Finite difference method 103
E.1 Coupled, linear, difference equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
E.2 Solution of coupled, linear difference equations . . . . . . . . . . . . . . . . . . . . . . . . 104
F Program codes for the Matlab environment 106
F.1 Finite difference method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
F.2 Warburg impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
F.3 Gerischer impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
F.4 Electrochemical impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
F.5 Photoelectrochemical impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
1 INTRODUCTION 1
1 Introduction
1.1 Background
The earth’s population has since the 19th century grown exponentially from around one billion, to
over seven billions today. Prognoses say that this increase will continue until it flattens at 10 billions
[1]. This, combined with an increase in the standard of livings and industrialization of the "third
world" will lead to an increase in the worlds energy demand. The historical and predicted energy
demand is given in Figure 1.1.
Figure 1.1: The historical and predicted energy demand for different energy sources, from [2].
Researchers agree that the emission of greenhouse gases may contribute to global warming. Green-
house gases involve gases that absorb more heat than they emit, which results in an increase in the
temperature of the gas. CO2, methane and nitrous gases are amongst the greenhouse gases that have
the largest manmade concentration contribution. The largest source to CO2 emission is energy pro-
duced from fossil fuels, where the emission has increased with 145% the last 30 years [3]. The future
energy demand needs to be met by increasing the energy produced by emission-free, "green", energy
sources.
The Earth’s entire annual energy demand is supplied by solar irradiation in about 1.5 hours [4].
This energy is free and green, and solar energy is one promising energy source to meet the future
energy demand. However, utilizing and harvesting this energy is difficult, and there is a constant
research in this field. Solar cells, or photovoltaic cells, convert light energy to usable energy, in form
1 INTRODUCTION 2
of electricity, which can be fed directly to the energy mixture. Photovoltaics are also the most green
alternative to produce hydrogen used as fuel in hydrogen fuel cells.
1.2 The photoelectrochemical system
1.2.1 Photoelectrochemistry and electrochemical photovoltaic cells
If a semiconductor electrode is brought in contact with an electrolyte, a potential barrier may be
created at the surface of the electrode [5]. When the interface is illuminated and the semiconduct-
ing electrode is connected to a conducting counter-electrode, a photocurrent may flow between the
electrodes. The most important process of creating a photocurrent is the generation of electron-hole
pairs in the bulk of the semiconductor when the incident light has a greater energy than the semicon-
ductor band gap [5]. In the presence of a depletion layer, the majority carriers will be transported into
the interior of the electrode and minority carriers will be transported to the surface [6]. If transfer to
a redox couple in the electrolyte whose energy lies within the band-gap can take place, a photocur-
rent is seen. This process is described below for a TiO2 semiconducting electrode and a Pt electrode
immersed in an acidic Fe3+/Fe2+ electrolyte.
An n-type TiO2 semiconductor [5] is immersed in an acidic Fe3+/Fe2+ electrolyte, containing an
oxidized and reduced state of Fe, known as a redox couple. Under illumination, minority carrier holes
generated in the semiconductor move to the electrolyte interface where they are oxidized in a redox
reaction with the electrolyte
Fe2++h+ → Fe3+ (1.1)
When the semiconductor is connected to the Pt counter electrode immersed in the same electrolyte,
electrons are reduced at the counter electrode interface by
Fe3++e− → Fe2+ (1.2)
The redox potential of the electrolyte is 0.77 V, and the conduction band lies close to zero. Thus, the
fermi level of TiO2 under no illumination lies close to 0.77 V, and there will be zero photovoltage.
Under illumination exceeding an energy of 0.77 eV, a photocurrent will flow through the system, with
the semiconductor serving as the anode and the counter electrode as the cathode [5]. A schematic of
this cell is given in Figure 1.2.
The magnitude of the photocurrent will, in general, depend on the electrode material, incident
light energy, electrode potential and electrolyte composition [5]. Photoelectrochemical cells can be
used to produce electricity or to produce hydrogen, as described below.
1 INTRODUCTION 3
Figure 1.2: Example of a electrochemical photovoltaic cell consisting of a TiO2 semiconductor anode,a Pt counter electrode cathode, and an Fe3+/Fe2+ redox couple electrolyte.
1.2.2 Dye-sensitized solar cells (DSSC)
Commercially available photovoltaic solar cells up to now are based on inorganic materials, which
require high costs and highly energy consuming preparation methods. Several of the conventional
materials are toxic and have low natural abundance. The use of organic materials have been pro-
posed to solve these issues. However, conventional organic photovoltaic cells consisting of two elec-
trodes in a heterojunction have a significally lower efficiency compared to the commercial inorganic
photovoltaic cells [7]. The organic materials should have both good light harvesting properties and
good carriers transport properties which is difficult to achieve. Dye-sensitized solar cells (DSSC) are
promising organic-bases photoelectrochemical cells that may solve these problems for organic pho-
tovoltaics. DSSCs separates the two requirements as the light harvesting is done semiconductor-dye
interface and charge transport is done by the semiconductor and the electrolyte [7].
A schematic of a DSSC is given in Figure 1.3. A porous TiO2 semiconductor film is mounted on a
transparent conductive oxide (TCO), and a sensitizer material consisting of a ruthenium complex is
adsorbed onto the TiO2 particle surface. The semiconductor is in contact with an electrolyte consist-
ing of a triiodide/iodine redox couple. A Pt counter electrode is in contact with the electrolyte and
connected in an outer circuit with the semiconductor through the conducting glass [7].
Under illumination, the dye S absorbs photons to an excited sensitizer state S∗. S∗ injects an
1 INTRODUCTION 4
Figure 1.3: Schematic representation of the dye-sensitized solar cell. A ruthenium-complex sensi-tizer are adsorbed on the TiO2 semiconductor particles mounted on a transparent conducting ox-ide (TCO). The Pt counter electrode is connected to the semiconductor through an outer circuit andthrough the triiodide/iodine redox couple electrolyte.
electron in the conduction band of TiO2, leading to an oxidized sensitizer state S+. The electron is
transferred to the counter electrode, where it ejected by reducing the triiodide/iodine redox couple.
The reduced redox couple then regenerates the oxidized dye. These operating principles are summa-
rized in the chemical reactions [7]
S(ads) +hν→ S∗(ads) (1.3)
S∗(ads) → S+
(ads) +e−(inj, TiO2) (1.4)
I−3 +2e−(Pt) → 3I− (1.5)
S+(ads) +
3
2I− → S(ads) +
1
2I−3 (1.6)
1.2.3 Photoelectrochemical water splitting
Photoelectrochemical water splitting is a photoelectrolysis cell that utilizes illumination to produce
hydrogen by water electrolysis [6]. For an n-type semiconductor, an oxidation reaction occurs at the
semiconductor interface, and a reduction reaction occurs at the counter electrode. For the case of
1 INTRODUCTION 5
water splitting, the reaction at the semiconductor is [6]
H2O+2p+ → 1
2O2 +2H+ (1.7)
and at the counter electrode
2H++2e− → H2 (1.8)
which give the net process reaction
2H2O → 2H2 +O2 (1.9)
This net process reaction has an energy of 1.23 eV [8], which places a lower limit for the band gap
of the semiconductor. A schematic figure of a photoelectrochemical water splitting cell with a TiO2
semiconductor electrode, a Pt counter electrode and a KOH electrolyte [5] is given in Figure 1.4. Here,
H2O is the redox couple.
Figure 1.4: Example of a photoelectrochemical water splitting cell consisting of a TiO2 semiconductoranode, a Pt counter electrode cathode, and an KOH electrolyte.
1.3 Impedance spectroscopy and modeling with Newman’s BAND subroutine
Designing and improving the materials used in photovoltaic cells are essential to optimize the perfor-
mance of these cells. When the material is designed and synthesized, the material needs to be charac-
1 INTRODUCTION 6
terized to determine wether certain material property requirements are fulfilled. Experimental data
have to be compared to theoretical models to presume something about the behavior of the system,
and models are needed in order to describe these theoretical systems. Impedance spectroscopy is
one characterization technique to investigate electrical properties of (amongst others) electrochem-
ical and photoelectrochemical cells, and will be modeled in this project. In impedance measures,
a harmonically modulating electric stimulus is applied to the system, and the modulating response
is measured. Because of this harmonic oscillation, the impedance can be represented as a complex
number, described below. The mathematical treatment of impedance spectra involves solving partial
differential equations.
In general, there are two modeling approaches. The preferred approach is analytical modeling,
where different models, expressions and equations are combined to form one single expression for
the wanted property. However, in complex systems, the analytical approach can be difficult, or even
impossible. At this point, numerical modeling is required, where approximate numerical solutions
are used to calculate the wanted property for a specific system. By applying the numerical approach
to simple systems with well-known analytical expressions for the wanted property, the numerical re-
sults can be compared to the analytical results. Thus, the method may either be verified and used for
calculating the complex problem for proper convergence, or be falsified and disregarded for inappro-
priate convergence.
John S. Newman gives in "Electrochemical Systems" [9] a routine for solving a set of n coupled,
linear, second-order differential equations numerically. The routine is suitable for calculating initial,
boundary-value problems, which are common to define several electrochemical and material prop-
erties. A program code for a subroutine BAND(J) is given to solve these boundary-value problems,
where coefficient matrices defined by the differential equations and boundary-values are supplied to
the subroutine. Traditionally, this routine coefficients consisting real numbers. The purpose of this
study is to investigate if the BAND routine may be supplied complex numbers to solve differential
equations. Since impedance can be expressed as a complex number described below, we investigate
the suitability of Newman’s BAND subroutine to calculate impedance spectra for electrochemical and
photoelectrochemical systems. The investigation is done by first establishing an analytical expression
for the impedance spectrum of a electrochemical or photoelectrochemical system, then apply New-
man’s BAND subroutine to the system to calculate numerically the impedance spectrum, and finally
compare the numerical results with the analytical model.
2 THEORY 7
2 Theory
2.1 Impedance and impedance spectroscopy
2.1.1 Impedance spectroscopy and the importance of interfaces
The interface of a material is important in the study of material properties. Physical properties – crys-
tallographic, mechanical and electrical – change rapidly at an interface and polarization reduce the
conductivity of the system [10]. Each interface in the system will polarize differently when subjected
to an applied potential difference. The rate of polarization change when the applied voltage is re-
versed is characteristic for the interface type; slow for chemical reactions at the electrode–electrolyte
surface, substantially faster in the aqueous electrolyte [10].
Impedance spectroscopy (IS) is a characterization method where electrical properties and inter-
facial transfer processes in a system can be determined. In impedance spectroscopy measurements, a
harmonically modulated electrical stimulus is applied to the electrodes and the modulating response
is measured. For instance, a modulating voltage is applied to the electrochemical system, and the
resulting modulating current is measured. We define the impedance, Z , as the ratio of the stimulus
and the response. When a faradaic reaction is present, we define the faradaic impedance Z f by
Z f (t ) = V (t )
I (t )(2.1)
where V (t ) and I (t ) are the modulating voltage stimulus and alternating current response, respec-
tively.
The electrical response is a result of several microscopic processes throughout the system. Amongst
the processes, and usually the most important in the characterization of the electrical properties of
the system, are charge transport through the electrode and electrolyte phases, and charge transfer
across heterogeneous interfaces. For a semiconductor under illumination, a contribution from the
generation of photocurrent when electrons are excited gives a contribution to the electrical response.
Application of modulating potential forces these processes to oscillate with the applied frequency. If
a reversible couple is present at the electrode surface, the concentrations of both the reduced and
oxidized species will also oscillate not only at the surface but away of the electrode [5]. The motion of
charge across the system is thus affected by the ohmic resistance in the different phases, and the rate
of the charge transfer at the interfaces [10].
Impedance spectroscopy measurements are well suited to characterize the electrical behavior of
the system, including characterization of the motion of charge control and measuring material prop-
erties like diffusion coefficients, rate constants, mobilities, transport numbers and conductivities. It
may be used to investigate the dynamics of charge species of any kind of solid or liquid material:
2 THEORY 8
ionic, semiconducting, mixed ionic-electronic and so on [10].
2.1.2 Impedance spectroscopy measurements
There are several impedance spectroscopy techniques, where single-frequency impedance spectroscopy
is the most common. One single frequency of the electrical stimuli is chosen, and the phase shift and
amplitude of the electrical response is measured. This process is then repeated across a frequency
range, typically between 1 mHz to 1MHz [10], and a series of data is collected.
In electrical impedance spectroscopy (EIS) a harmonically modulated voltage V (t ) = V0 sin(ωt )
with angular frequencyω is applied, and an AC current I (t ) = I0 sin(ωt+φ) results. A phase difference
φ between the stimuli and response may be observed [11]. I0 and V0 are the steady state current and
voltage, respectively. The EIS signal is the electrical impedance
Z ( jω) = V (t )
I (t )(2.2)
The equivalent circuit for the electrical impedance spectroscopy setup is given in Figure 2.1,
where Φ0 is the incident light intensity. EIS measurements can be performed under any bias illu-
mination, either for one wave length, one sun, or several suns, depending on the wanted conditions
Figure 2.1: Equivalent sketch of an EIS measurement. Φ0 is the incident light intensity, ω the angularfrequency, φ the phase shift, and I0 and U0 are the steady state current and voltage, respectively.
The experimental setup of EIS at the department of materials science at NTNU is showed in Figure
2.2. Here, the ZAHNER IM6x electrochemical workstation potentiostat applies a harmonically mod-
2 THEORY 9
ulated stimulus, and the modulated response is measured. The light intensity source is connected to
a monochromator, where one single wavelength may be chosen if wanted.
Figure 2.2: EIS experimental setup at department of materials science at NTNU, located in Chemistrybuilding 2 (K2).
2.2 Mathematical treatment
2.2.1 The Laplace transform
The relation between the system response and system properties are usually very complex in the
time domain [10]. One method to greatly simplify the mathematical treatment of the system is to
use Laplace transform from the time domain to the frequency domain. It can be shown that the
impedance can be derived by setting s = jω in the Laplace-transformed time region equations, where
j =p−1. This proof is given in Appendix A, established by Professor Svein Sunde. Thus, the electrical
impedance in Eq. (2.1) can be expressed by the Laplace-transform of the voltage and the current by
Z ( jω) = LV
( jω)
L
I
( jω)(2.3)
where V and I are the modulated frequency-dependent voltage and current, respectively. This math-
ematical treatment of the impedance is used in this study.
2 THEORY 10
2.2.2 Representation of impedance spectra
The phase shift and amplitude of the impedance can be used to represent the impedance as a com-
plex number
Z = Re(Z )+ j Im(Z ) = |Z |cosθ+ j |Z |sinθ (2.4)
This relation can be represented in an impedance plane plot (often named Nyquist plot [5]), where
the impedances at different frequencies are plotted as a planar vector in the real and imaginary plane
given by Eq. (2.4). As an example, an impedance plane plot for a Ni/Ti-doped YSZ solid-oxide fuel cell
(SOFC) is given in Figure 2.3.
Figure 2.3: Impedance plane plot representation of the impedance for a Ni/Ti-doped YSZ SOFC as afunction of PH2 , from [12].
These plots give a good representation of the phase shift and amplitude of the impedance. How-
ever, it can be quite ambiguous to determine at which frequency each point in the plot is measured.
The impedance plane plot often does not give the frequency at each measured point explicit. This can
be shown explicit by plotting the magnitude of the impedance versus the frequency in a Bode plot [5,
p. 285]. A Bode plot of the electrical impedance spectrum of a TiO2 dye-sensitized solar cell (DSSC) is
given in Figure 2.4 [11].
2 THEORY 11
Figure 2.4: Bode representation of the electrical impedance spectrum for a DSSC with different con-centration of Pt at the counter electrode.
2.3 Analysis of impedance spectra
2.3.1 Electrode half-cell and the Warburg diffusion impedance
We consider a half-cell consisting of an electrode electrode immersed in an electrolyte with the gen-
eral electrochemical process at the surface
Ox+e− → Red (2.5)
The potential drop E across the interface will be a function of the current density flowing , i , and the
concentrations of Ox and Red at the surface, c sO and c s
R
E = E(i ,c sO ,c s
R ) (2.6)
We shall assume that the concentration of supporting electrolyte is sufficient to ensure that all the
potential drop at the interface is accommodated in the Helmholtz layer, so that the species transport
takes place only by diffusion [5]. The material balance is given by Fick’s second law of diffusion [9]
∂c
∂t= D
∂2c
∂x2 (2.7)
Here we have introduced the concentration c = cO
νO= cR
νRdescribed below.
We assume a harmonic perturbation is applied in the driving force for the diffusion for t > 0, so
2 THEORY 12
that the concentration may be written [13]
c(t ) = c(0)+ c(t ) (2.8)
where c(0) corresponds to the expected steady state concentration, and is time-independent [5].The
diffusion equation becomes∂c
∂t= D
(d 2c(0)
d x2 + d 2c
d x2
)(2.9)
Since the steady state concentration c(0) is time-independent, by doing the Laplace transform it
follows from Eq. (2.9) that L c(0)(s) = 0 [13]. The Laplace transformed diffusion equation becomes
L
∂c
∂t
(s) = D L
d 2c
d x2
(s) (2.10)
By using the definition of the Laplace transform of the n − th derivative of a function [14]
L
f (n) (s) = snL
f
(s)− sn−1 f (0)− ...− f (n−1)(0) (2.11)
we get
s L c (s) = Dd 2 L c (s)
∂x2 (2.12)
This frequency region diffusion equation has the general solution
L c (s) = A exp
(√s
Dx
)+B exp
(−
√s
Dx
)(2.13)
We assume a leaking interface at x = 0 with a transmissive boundary condition described in [10]
and by Glarum et. al [15], where the current at the interface is proportional to the derivative of the
modulating concentration with respect to the length. The diffusion layer is set to be infinitely long,
and the modulating concentration is zero in the bulk electrolyte. The boundary conditions are thus
Li
(s)
nF= D
d L c (s)
d x
∣∣∣∣∣x=0
; limx→∞L c (s) → 0 (2.14)
This gives the specific solution
L c (s) =−Li
(s)
nF
1pDs
exp
(−
√s
Dx
)(2.15)
The harmonically perturbed voltage applied is given by invoking the linearity conditions
E = E(0)+ E(t ) = E(0)+ dE
dcc(t ; x = 0) (2.16)
2 THEORY 13
By rearranging and doing the Laplace transform, we get
LE
(s) = dE
dcL c (s; x = 0) =−dE
dc
Li
(s)
nF
1pDs
(2.17)
We let s → jω as described in Section 2.2.1, and use the definition of the electrical impedance given in
Eq. (2.3). This gives the infinite Warburg diffusion impedance, ZW,inf( jω), at the electrode interface
ZW,inf( jω) = LE
( jω)
Li
( jω)=−dE
dc
1
nF
1√D · jω
(2.18)
By similar approach, it can be showed that by choosing a finite length for the diffusion layer, we
get the expression for the transmissive Warburg diffusion impedance
ZW,trans( jω) = LE
( jω)
Li
( jω)=−dE
dc
1
nF
1√D · jω
tanh
√jω
DL
(2.19)
We can also assume a blocking interface with a reflective boundary condition at the electrode
surface described by Ho et al. [16] and Franceschetti et al. [17]. The boundary conditions are given by
Li
(s)
nF= D
d L c (s)
d x
∣∣∣∣∣x=0
; 0 = d L c (s)
d x
∣∣∣∣x=L
(2.20)
which has the specific solution at the interface
L c (s) = Li
(s)
nF
1pDs
coth
√jω
DL
(2.21)
By assuming a harmonically perturbed voltage applied as described above, the reflective Warburg
diffusion impedance at the electrode interface becomes
ZW,refl( jω) =−dE
dc
1
nF
1√D · jω
coth
√jω
DL
(2.22)
A full derivation of this model is given in Appendix B
The current flow in the system is given by an electron-transfer resistance across the interface, RC T ,
and a Warburg impedance across the diffusion layer, ZW , in series. In practice, this is connected in
parallel with the double layer capacitance, CD , and in series with the electrolyte resistance, RE [5]. An
equivalent circuit of this half-cell system is given in Figure 2.5.
2 THEORY 14
Figure 2.5: Equivalent circuit for the electrode half-cell. RC T is the charge transfer resistance, ZW isthe Warburg impedance, CD is the double layer capacitance, and RE is the electrolyte resistance.
2.3.2 Diffusion limited process
The infinite Warburg diffusion impedance is only diffusion controlled. In an impedance plane plot,
the infinite Warburg diffusion impedance is a straight line for all frequencies. In general, a system is
diffusion controlled for a specific frequency region if the impedance plane plot is a straight line with
an inclination angle −φ=α= 45 for all frequencies within the region. Physically, the system shows a
infinite-Warburg-like diffusion impedance behavior in this frequency region.
2.3.3 Electron-transfer limited process
We assume that the Warburg diffusion impedance becomes sufficiently small compared to the charge
transfer so that the equivalent circuit in Figure 2.5 is reduced to that of Figure 2.6a [5]. In this limiting
case, the system is only electron-transfer limited. It can be shown [5] that the impedance plane plot
gives a perfect semi-circle. The length of the real part of the semi-circle equals the charge trans-
fer resistance, and the center of the semi-circle is located at a frequency of ω = 1/RC T CD . Thus,
the double-layer capacitance can be investigated for this plot. A schematic impedance plane plot
of an electron-transfer limited process is given in Figure 2.6b. The system effectively charges and
discharges the electrical double layer in this frequency region.
2.3.4 Gerischer impedance
A non-faradaic side reaction that affects the concentrations may occur simultaneously with the het-
erogeneous charge transfer [18, 12]. We introduce a sink term −kc to the material balance, where the
effective rate constant k also includes any linearization of the heterogeneous kinetics at the interface
2 THEORY 15
(a) (b)
Figure 2.6: (a) Equivalent circuit for a rate limiting process, and (b) impedance plane plot for a ratelimiting process. RC T is the charge transfer resistance, CD the double layer capacitance, and RE theelectrolyte resistance.
[9]. The modified Fick’s law in one dimension is written
∂c
∂t= D
∂2c
∂x2 −kc (2.23)
Laplace transforming from the time domain to the frequency domain gives
L
∂c
∂t
(s) =L
D∂2c
∂x2
(s)−L kc (s) (2.24)
By assuming a harmonic perturbation applied, we get the Laplace transform
s L c (s) = D∂2 L c (s)
∂x2 −k L c (s) (2.25)
where we again have used the relation c(0) = 0 [13].
This frequency region diffusion equation has the general solution
L c (s) = A exp
√s +k
Dx
+B exp
−√
s +k
Dx
(2.26)
For a leaking electrode, we assume a transmissive boundary condition [15, 10] at the inspected
interface, as described above. That is
Li
(s)
nF= D
d L dc (s)
d x
∣∣∣∣∣x=0
; limx→L
L c (s) → 0 (2.27)
2 THEORY 16
This gives the specific solution at the interface
L c (s) = −Li
(s)
nF
1pD(s +k)
tanh
√s +k
DL
(2.28)
We assume a harmonically perturbed voltage applied as described above, and the transmissive Gerischer
impedance becomes
ZG ,trans( jω) =−dE
dc
1
nF
1√D( jω+k)
tanh
√jω+k
DL
(2.29)
A blocking electrode with a reflective boundary condition [16, 17]
Li f
(s)
nF= D
d L c (s)
d x
∣∣∣∣∣x=0
; 0 = d c (s)
d x
∣∣∣∣∣x=L
(2.30)
has the specific solution at the interface
L c (s) = Li
(s)
nF
1pD(s +k)
coth
√s +k
DL
(2.31)
This gives the reflective Gerischer impedance with a harmonically perturbed voltage applied
ZG ,refl( jω) =−dE
dc
1
nF
1√D( jω+k)
coth
√jω+k
DL
(2.32)
A full derivation of this model is given in Appendix C.
2.4 Photoelectrochemical film electrode
We assume thin film electrode consisting mounted on a metal bracket support. A schematic of the
electrode system is given in Figure 2.7.
Figure 2.7: Schematics of the modeled thin film electrode system.
2 THEORY 17
The charge balance for the charge-neutral bulk is [6]
n −Nd − (p −Na) = 0 (2.33)
where n, p, Nd , and Na are electron, hole, donor, and acceptor concentrations, respectively.
We introduce the nomenclature
c+ = p −Na (2.34)
c− = n −Nd (2.35)
which gives the charge balance
z+ν+c++ z−ν−c− = 0 (2.36)
2.4.1 Solution of the diffusion equation for mixed conducting thin film electrode
We employ the dilute-solution approximation in which the flux density vector of species i in the film,
N i , is described by [9]
N i =−zi ui F ci∇Φ1 −Di∇ci (2.37)
where zi is the charge number, ui the mobility, ci the concentration, and Di the diffusion coefficient
of species i .
The material balance∂ci
∂t=−∇N i +Ri (2.38)
becomes, by combination of Eqs. (2.34), (2.35), and (2.36) [9]
∂c
∂t= D∇2c +Rc (2.39)
where
D = z+u+D−− z−u−D+z+u+− z−u−
, Rc = z+u+R−− z−u−R+z+u+− z−u−
(2.40)
and the concentration of the electrolyte
c = c+ν+
= c−ν−
(2.41)
ν+ and ν− are the numbers of cations and anions produced by the dissociation of one molecule of
electrolyte, respectively.
A harmonic perturbation is assumed applied in the driving force for current flow for t > 0. The
concentration may be written
c = c(r ,0)+ c(r , t ) (2.42)
2 THEORY 18
The Laplace transform of the time derivative can be written as [14]
L
∂c
∂t
= s L c−
=0︷︸︸︷c(0) = s L c = sc(s) (2.43)
Combination of Eqs. (2.43) and (2.39) gives, for a one-dimensional electrode system
sc(s) = D∂2c(s)
∂x2 +LRc
(2.44)
We introduce a sink term corresponding to recombination or trapping of charge carriers [15]. The
rate constant k includes all rate constants and any other pre-factors stemming from linearization of
R+ and R−. We also add a source term due to photon absorption [19]. The diffusion equation becomes
sc(s) = D∂2c(s)
∂x2 −kc(s)+L
I0αexp−αx (2.45)
The current in the electrolyte is given by [9]
i = F∑
izi Ni (2.46)
and the net flux density is ∑i
Ni = i
F∑
i zi(2.47)
We see that N i is the integrated form of Eq. (2.38). We get
− i
z+ν+F= (z+u+− z−u−)F c∇Φ1 + (D+−D−)∇c = N+ (2.48)
We assume that electronic species are blocked at the solution interface x = 0, in the present example
the positive species, and we get the species fluxes [9]
N+x = 0 =−z+u+Fν+c∂Φ
∂x−D+ν+
∂c
∂x(2.49)
N−x = ix
z−F=−z−u−Fν−c
∂Φ
∂x−D−ν−
∂c
∂x(2.50)
where Ni x is the flux of species i in x-direction. The flux of negative carriers in the x-direction is
related to the faradaic current as
i f =−z−F N−x (2.51)
because a vacancy flux in the x-direction represents oxidation of the oxide. Thus, ix =−i f . By elimi-
2 THEORY 19
nation of the potential in Eqs. (2.50) and (2.49), we get
i f
z−ν−F=−z−u−D+− z+u+D−
z+u+∂c
∂x(2.52)
We introduce the transport number of the positive species [9]
t+ = 1− t− = z+u+z+u+− z−u−
(2.53)
Laplace-transforming gives the boundary condition at the electrolyte interface x = 0
Li f
z−ν−F
= D
1− t−∂L c
∂x
∣∣∣∣x=0
(2.54)
For the substrate interface we get similar species fluxes [9]
N+x = ix
z+F=−z+u+ν+c
∂Φ
∂x−D+ν+
∂c
∂x(2.55)
N−x = 0 =−z−u−Fν−c∂Φ
∂x−D−ν−
∂c
∂x(2.56)
which give the boundary condition at x = L
Li f
z+ν+F
= D
1− t+∂L c
∂x
∣∣∣∣x=L
(2.57)
Boundary conditions (2.54) and (2.57) are used to solve the diffusion equation (2.45), which give the
specific solution
L c =− Li f
F
√D( jω+k)sinh
(√jω+k
D L
)
·K1(ω)cosh
√jω+k
D(x −L)
−K2(ω)cosh
√jω+k
Dx
− L
I0
α
Dα2 −k − jωe−αx
(2.58)
where
K1(ω) = 1− t−z−ν−
− F D L
I0α2
Li f
(Dα2 −k − jω)
(2.59)
K2(ω) = 1− t+z+ν+
− F D L
I0α2
Li f
(Dα2 −k − jω)
e−αL (2.60)
2 THEORY 20
2.4.2 Electrochemical impedance for a mixed conductivity thin film electrode
We assume no illumination. That is
L
I0
(s) = 0; I0(x,0) = 0 (2.61)
The faradaic current at the electrode–electrolyte interface, i f , is assumed to be a function of the
proton concentration in the oxide film and the potential difference between electrode and electrolyte,
Φ1 −Φ2, given by the expression
i f (t ) =(∂i f
∂c
)Φ1−Φ2, x=0
c(x = 0)+[
∂i f
∂(Φ1 −Φ2)
]c, x=0
[Φ1(0)−Φ2(0)
]= A0c(0)+B0
[Φ1(0)−Φ2(0)
] (2.62)
with A0 =(∂i f
∂c
)Φ1−Φ2, x=0
and B0 =[
∂i f
∂(Φ1 −Φ2)
]c, x=0
.
The local admittance at the electrode–electrolyte interface for the combined faradaic reaction and
diffusion, Y0, is found by combining Eqs. (2.58) and (2.62) evaluated at x = 0
Y0 =L
i f
L
Φ1(0)−Φ2(0)
= B0
1− A0Z ′D
(2.63)
with
Z ′D =− 1
F√
D( jω+k)sinh
(√jω+k
D L
)
·1− t−
z−ν−cosh
√jω+k
DL
− 1− t+z+ν+
(2.64)
By similar approach, the local admittance at the metal–oxide boundary, YL , is given by
YL = Li f
L
Φm(L)−Φ1(L)
= BL
1− AL Z ′′D
(2.65)
with
Z ′′D =− 1
F√
D( jω+k)sinh
(√jω+k
D L
)
·1− t−
z−ν−− 1− t+
z+ν+cosh
√jω+k
DL
(2.66)
whereΦM (L) is the potential of the metal support at x = L.
To relate the potential difference LΦ1(0)−Φ2(0)
to the potential measured with respect to a
2 THEORY 21
reference electrode, we assume the iridium oxide system and write the proton-transfer reaction as
HxH*)V′
H +H+(aq) (2.67)
where HxH is an intercalated hydrogen, and the interface to the electrode support connecting the elec-
trode
0*) h++e− (2.68)
The electrochemical potential of electrons in the collecting leads is given through (in the dilute solu-
tion limit [9])
µe =µh =µ0h +RT lnc(L)+FΦ1(L) (2.69)
when Eq. (2.68) is assumed to be in equilibrium. The measured potential is related to the electro-
chemical potential in a reference electrode as −FV =µe −µrefe .
We assume that µrefe can be measured by a reference electrode so that its value is representative of
Φ2(0) plus a constant. The amplitude and phase of the measured electrode potential is thus derived
by taking the time dependent part of the linearized Eq. (2.69)
F LV
(s) = RT
ceL c(L) (s)+F L
Φ1(L)
(s)−F L Φ(0)(s) (2.70)
The potential Φ1 can in turn be related to the concentration c and i f through [9]
Li f
(s)
z+ν+F= (z+u+− z−u−)F
(ce∂L
Φ1
(s)
∂x+ c
∂Φ1e
∂x
)+ (D+−D−)
∂L c (s)
∂x(2.71)
where "e" refers to steady state, and we have neglected terms beyond first order (c∇Φ1 ≈ ce∂Φ1∂x +
c∂Φ1e
∂x).
We expect the steady-state concentration ce to be independent of the position [20]. For zero cur-
rent the relation [9]
F∂Φ1e
∂x=− D+−D−
(z+u+− z−u−)
∂ lnce
∂x(2.72)
implies that ∂Φ1e /∂x = 0. Using this result in Eq. (2.71) and integrating from x = 0 through x = L gives
LΦ1(L)
(s) =L
Φ1(0)
(s)− D+−D−
F ce (z+u+− z−u−)[L c(L) (s)−L c(0)(s)]+ L
i f
L
κ(2.73)
with κ= F 2 ∑i z2
i ui ci being the film conductivity.
2 THEORY 22
Combination of Eqs (2.70) and (2.73) gives
F LV
(s) = RT
ceL c(L) (s)
+F LΦ1(0)
(s)− D+−D−
ce (z+u+− z−u−)[L c(L) (s)−L c(0)(s)]−FΦ2(0)+ F L
i f
(s)L
κ
= F LΦ1(0)−FΦ2(0)
(s)
+[
RT
ce− D+−D−
ce (z+u+z−u−)
]L c(L) (s)+ D+−D−
ce (z+u+− z−u−)L c(0)(s)+ F L
i f
(s)L
κ
= F LΦ1(0)−FΦ2(0)
(s)
+[
(z+−1)D+− (z−−1)D−ce (z+u+− z−u−)
]L c(L) (s)+ D+−D−
ce (z+u+− z−u−)L c(0)(s)+ F L
i f
(s)L
κ(2.74)
where we have used the Nernst-Einstein relation Di = RTui [9, p. 253]. Inserting the expressions for
L c(0)(s) and L c(L) (s) in Eq. (2.58), and that for the potential difference F LΦ1(0)−FΦ2(0)
(s)
implied by Eq. (2.63), the faradaic impedance for the electrode, Z f ( jω), may be written
Z f =L
V
L
i f
= Z0 +Zφ+ZΩ (2.75)
where Z0 = Y −10 from Eq. (2.63), ZΩ = L/κ and
Zφ = D+−D−F ce (z+u+− z−u−)
Z ′D +
[(z+−1)D+− (z−−1)D−
F ce (z+u+− z−u−)
]Z ′′
D (2.76)
A full derivation of the electrochemical impedance spectrum model is given in Appendix D.
2.4.3 Photoelectrochemical impedance for a mixed conductivity thin film electrode
We assume a constant bias illumination, thus no intensity modulated illumination. That is
L
I0
(s) = 0; I0(x,0) 6= 0 (2.77)
In this case it can no longer be assumed that the steady-state concentration ce is independent of x.
In fact, the steady-state concentration ce (x) is given by Eq. (2.58) with ω= 0 and all time-dependent
2 THEORY 23
quantities replaced by steady-state ones,
ce =− i f
Fp
Dk sinh
(√kD L
)
·K1 cosh
√k
D(x −L)
−K2 cosh
√k
Dx
− I0α
Dα2 −ke−αx
(2.78)
with
K1 = 1− t−z−ν−
− F D I0α2
i f (Dα2 −k)(2.79)
K2 = 1− t+z+ν+
− F D I0α2
i f (Dα2 −k)e−αL (2.80)
The steady-state equivalent of Eq. (2.72) with no linearization is
F∂Φ1e
∂x=− D+−D−
z+u+− z−u−∂ lnce
∂x+ i f
z+ν+F ce (z+u+− z−u−)(2.81)
and gives the steady-state potential gradient ∂Φ1e /∂x to be used in Eqs. (2.71) and (2.73)
F LΦ1(L)
(s) = F L
Φ1(0)
−∫ L
0
[ −Li f
(s)
ce z+ν+F (z+u+− z−u−)+F
L c (s)
ce
∂Φ1e
∂x+ D+−D−
ce (z+u+− z−u−)
∂L c (s)
∂x
]d x
(2.82)
or
F LΦ1(L)
(s) = F L
Φ1(0)
−∫ L
0
[ −Li f
(s)
ce z+ν+F (z+u+− z−u−)
(1− L c (s)
ce
)+
D+−D−ce (z+u+− z−u−)
(∂L c (s)
∂x− L c(s)
ce
∂ce
∂x
)]d x
(2.83)
Here we assume Li f
(s) to be independent of x due to charge conservation [5].
3 NUMERICAL METHOD 24
3 Numerical method
The impedance is derived from the partial differential equation of the harmonically modulating con-
centration gradient. It can be seen from Eq. (2.45) that the system consists of a set of coupled linear
differential equations (here only one differential equation for the iridium oxide system and binary
electrolyte speciation).
The system is a typical initial boundary value problem set – that is, with boundary conditions at
the extrema of x-values, x = 0 and x = L (or x =∞).
3.1 Finite difference method
In general, a set of n coupled linear, second-order differential equations can be represented by [9]
n∑k=1
ai ,k (x)d 2ck
d x2 +bi ,k (x)dck
d x+di ,k (x)ck = gi (x) (3.1)
where ck (x) are the n unknown functions, i denotes the equation number, and each of the equations
can involve all of the unknowns ck through the sum.
For central difference approximations of the derivatives with a mesh distance h,
d 2ck
d x2 = ck (x j +h)+ ck (x j −h)−2ck (x j )
h2 +O(h2) (3.2)
dck
d x= ck (x j +h)− ck (x j −h)
2h+O(h2) (3.3)
we obtain the difference equations
n∑k=1
Ai ,k ( j )Ck ( j −1)+Bi ,k ( j )Ck ( j )+Di ,kCk ( j +1) =Gi ( j ) (3.4)
where j denotes a certain point number in the mesh, and the coefficients are given by [9]
Ck ( j ) = ck (x) (3.5)
Ai ,k ( j ) = ai ,k (x j )− h
2bi ,k (x j ) (3.6)
Bi ,k ( j ) =−2ai ,k (x j )+h2di ,k (x j ) (3.7)
Di ,k ( j ) = ai ,k (x j )+ h
2bi ,k (x j ) (3.8)
Gi ( j ) = h2gi (x j ) (3.9)
3 NUMERICAL METHOD 25
At j = 1, the equations are
n∑k=1
Bi ,k (1)Ck (1)+Di ,k (1)Ck (2)+Xi ,kCk (3) =Gi (1) (3.10)
where the third term at j = 3 has been added to allow the treatment of complex boundary conditions
[9]. General boundary conditions for equation number 1 at x = 0 would read
n∑k=1
pi ,kdck
d x+ei ,k ck = fi (3.11)
A forward difference accurate to order h2 has the boundary coefficients in Eq. (E.10)
Xi ,k =−0.5pi ,k , Di ,k (1) = 2pi ,k ,
Bi ,k (1) = hei ,k −1.5pi ,k , Gi (1) = h fi
(3.12)
A central-difference form given in Eq. (E.3) introduces an image point at x = −h, outside the
domain of interest has the boundary coefficients in Eq. (E.10)
Xi ,k =−Bi ,k (1) = pi ,k
2, Di ,k (1) = hei ,k , Gi (1) = h fi (3.13)
where the coefficients Xi ,k allow the introduction of complex boundary conditions at x = 0.
For similar reasons, the difference equations at j = jmax are written
n∑k=1
Yi ,kCk ( j −2)+ Ai ,k ( j )Ck ( j −1)+Bi ,k ( j )Ck ( j ) =Gi ( j ) (3.14)
where the coefficients Yi ,k again allow the introduction of complex boundary conditions at x = L.
Newman gives a program code to solve these equations written for the Fortran environment, and
a translated program code to the Matlab environment is established by J. W. Evans and P. Kar at Dept.
of Materials Science and Engineering at University of California, Berkley. Suitable coefficients for
the system as defined above are supplied to the BAND routine, and the routine solves the diffusion
equation. A derivation of the solution of coupled, linear, difference equations is given in Appendix E.
The program code in the Matlab environment is given in Appendix F.
3 NUMERICAL METHOD 26
3.2 Simulation parameters
We assume Butler-Volmer type kinetics
i f = i0
exp
((1−β)F (Φ1 −Φ2 −U )
RT
)−exp
(−βF (Φ1 −Φ2 −U )
RT
) (3.15)
By assuming A0 = AL = A and B0 = BL = B in Eqs. (2.62) and (2.65), we get directly from Eq. (3.15)
A = i0F
RT
∂U
∂c, B = i0F
RT(3.16)
The diffusion coefficients for each species are calculated by using Nernst-Einstein relation [9]
Di = kB T
qiu′
i (3.17)
and the diffusion coefficient for the system is given by [9, p. 11]
D = z+u+D−− z−u−D+z+u+− z−u−
(3.18)
where kB is the Boltzmann constant, qi is the charge of the species, zi is the valence of the species
and ui is the mobility of the species. Here we have used a different version of the Nernst-Einstein
relation than the one given in the theory section, in accordance with ref. [13].
The conductivity of the electrode material is given by [9]
κ= F 2∑
iz2
i ui ci (3.19)
The transport numbers are given by [9]
t j =z2
j u j c j∑i z2
i ui ci(3.20)
In general, to calculate the impedance spectra numerically, we solve the modified diffusion equa-
tion in Eq. (2.45)
s L c (s) = D∂2 L c (s)
∂x2 −k L c (s)+L
I0
(s) ·αe−αx (3.21)
where s = jω. The calculated modulating concentration profiles are inserted in the respective impedance
equations described above. The BAND-routine calculates the differential equations across a dimen-
sionless length domain, while the electrode system is defined for a dimension length domain. The
diffusion equation is written as a function of dimensionless length by assuming a length y = x/L, and
3 NUMERICAL METHOD 27
proper L-factors are added in the coefficients supplied to the BAND-routine.
The material simulation parameters for the iridium oxide system are given in Table 3.1. The values
are given by Sunde et al. [13], if not otherwise stated. The calculated values are given in Table 3.2. The
modulating faradaic current is set to 1 for simplicity, since this term is cancelled when the impedance
is calculated.
Table 3.1: Simulation parameters for the iridium oxide system.
Parameter Value Unit Reference
c0 0.025 mol cm-3 –ce 0.004 mol cm-3 –z+ 1 – –z− -1 – –ν+ 1 – –ν− 1 – –u′+ 0.1 cm2 (V s)-1 –u′− 1.7×10−8 cm2 (V s)-1 –i0 0.69×10−3 A cm-2 –dE/dc -20.27 V cm3 mol-1 –k 1×10−2 s-1 –L 100 µm –T 353.15 K –α 0.25×10−5 cm-1 [19]F 96485 A s mol-1 –kB 8.618×10−5 eV K-1 –
L i f (s) = i f 1 A cm-2 –
Table 3.2: Calculated simulation parameters from Table 3.1.
Parameter Value Unit Reference
1/B 44.06 Ω cm2 –A -0.46 A cm mol-1 –D 1×10−9 cm2 s-1 –
3.3 Coefficients for the BAND routine
3.3.1 Warburg diffusion impedance
The coefficients supplied to the Newman’s BAND routine for a central difference method with mesh
distance H for calculating the transmissive Warburg diffusion impedance in Eq. (2.19) are given in
Table 3.3.
The coefficients supplied to the Newman’s BAND routine for a central difference method with
mesh distance H for calculating the reflective Warburg diffusion impedance in Eq. (2.22) are given in
Table 3.4.
3 NUMERICAL METHOD 28
Table 3.3: Coefficients supplied to the BAND-routine for calculating the transmissive Warburg diffu-sion impedance in Eq. 2.18.
Coefficient Differential equation Boundary at x = 0 Boundary at x = L
A1,1 D1
L2– –
B1,1 −2D1
L2 −H 2s −D
2
1
L1
D1,1 D1
L20 0
G1 0 Hi f
nF0
X1,1 –D
2
1
L–
Y1,1 – – 1
Table 3.4: Coefficients supplied to the BAND-routine for calculating the reflective Warburg diffusionimpedance in Eq. (2.22).
Coefficient Differential equation Boundary at x = 0 Boundary at x = L
A1,1 D1
L2– –
B1,1 −2D1
L2 −H 2s −D
2
1
L−1
2
D1,1 D1
L20 0
G1 0 Hi f
nF0
X1,1 –D
2
1
L–
Y1,1 – –1
2
3 NUMERICAL METHOD 29
3.3.2 Gerischer impedance
The coefficients supplied to the Newman’s BAND routine for a central difference method with mesh
distance H to calculate the transmissive Gerischer impedance in Eq. (2.29) are given in Table 3.5
Table 3.5: Coefficients supplied to the BAND-routine for calculating the transmissive Gerischerimpedance in Eq. (2.29).
Coefficient Differential equation Boundary at x = 0 Boundary at x = L
A1,1 D1
L2– –
B1,1 −2D1
L2 −H 2(s +k) −D
2
1
L1
D1,1 D1
L20 0
G1 0 Hi f
nF0
X1,1 –D
2
1
L–
Y1,1 – – 1
The coefficients supplied to the Newman’s BAND routine for a central difference method with
mesh distance H to calculate the reflective Gerischer impedance in Eq. (2.32) are given in Table 3.6
3.3.3 Electrochemical impedance of the mixed ionic conductor
The coefficients supplied to the Newman’s BAND routine for a central difference method with mesh
distance H to calculate the electrochemical impedance in Eq. (2.75) are given in Table 3.7.
3.3.4 Photoelectrochemical impedance of the mixed ionic conductor
The coefficients supplied to the Newman’s BAND routine for a central difference method with mesh
distance H to calculate the steady-state concentration under illumination in Eq. (2.78) are given in
Table 3.8. This steady state concentration profile is then used in combination with the numerically
solved modulating concentration in Eq. (2.58) to calculate the photoelectrochemical impedance with
L I0 = 0.
3 NUMERICAL METHOD 30
Table 3.6: Coefficients supplied to the BAND-routine for calculating the reflective Gerischerimpedance in Eq. (2.32).
Coefficient Differential equation Boundary at x = 0 Boundary at x = L
A1,1 D1
L2– –
B1,1 −2D1
L2 −H 2(s +k) −D
2
1
L−1
2
D1,1 D1
L20 0
G1 0 Hi f
nF0
X1,1 –D
2
1
L–
Y1,1 – –1
2
Table 3.7: Coefficients supplied to the BAND-routine for calculating the electrochemical impedancein Eq. (2.75).
Coefficient Differential equation Boundary at x = 0 Boundary at x = L
A1,1 D1
L2– –
B1,1 −2D1
L2 −H 2(s +k) −1
2
D
1− t−1
L−1
2
D
1− t+1
L
D1,1 D1
L20 0
G1 0 Hi f
z−ν−FH
i f
z+ν+F
X1,1 –1
2
D
1− t−1
L–
Y1,1 – –1
2
D
1− t+1
L
3 NUMERICAL METHOD 31
Table 3.8: Coefficients supplied to the BAND-routine for calculating the steady-state concentrationunder illumination in Eq. (2.78).
Coefficient Differential equation Boundary at x = 0 Boundary at x = L
A1,1 D1
L2– –
B1,1 −2D1
L2 −H 2(s +k) −1
2
D
1− t−1
L−1
2
D
1− t+1
L
D1,1 D1
L20 0
G1 −H 2L2I0αe−αx Hi f
z−ν−FH
i f
z+ν+F
X1,1 –1
2
D
1− t−1
L–
Y1,1 – –1
2
D
1− t+1
L
4 RESULTS 32
4 Results
In these simulations, the impedances are calculated for a frequency range between 10 mHz and 100
MHz. The diffusion coefficient dependency, and rate constant dependency when suitable, of the
process are investigated to show any process control changes.
4.1 Warburg diffusion impedance spectra
4.1.1 Warburg infinite diffusion impedance
The Warburg infinite diffusion impedance from Eq. (2.18) with a diffusion coefficient D = 10−9 cm2
s-1 is given in Figure 4.1, where the solid line represent the analytical solution, and the markers repre-
sent calculated numerical solutions. The impedance spectra are calculated with the boundary condi-
tions d L c(s)d x
∣∣∣x=0
= L i f (s)nF and limx→∞L c(s) = 0. The numerical approach is a central difference
accurate to the order H 2 for the boundary conditions, where H is the inverse of number of steps.
The numerical calculations are done for 10, 100, 1000 and 10000 steps, where the numerical so-
lution approaches the analytical by increasing number of steps. A quicker convergence for lower
frequencies is observed. The impedance plane plot is a straight line with an inclination angle α= 45
for all frequencies, as it should be for a diffusion-controlled process.
4 RESULTS 33
0 20 40 60 80 100 120 140 160 180 2000
20
40
60
80
100
120
140
160
180
200
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
← ω = 0.0062832
← ω = 0.0019869
← ω = 0.00099582
ZW
, Analytical
ZW
, Numerical. Steps = 10000
ZW
, Numerical. Steps = 1000
ZW
, Numerical. Steps = 100
ZW
, Numerical. Steps = 10
Figure 4.1: Impedance plane plot from Eq. (2.18) of the Warburg infinite diffusion impedance spec-trum for a diffusion coefficient D = 10−9 cm2 s-1 and n = 1. Calculated with a central differenceaccurate to the order of H 2, and plotted for a frequency range of 0.1 mHz to 100 MHz.
By doing the same calculation with D = 10−5 cm2 s-1 plotted in Figure 4.2, we see that the nu-
merical solution has a sufficient convergence for number of steps above 1000. Again, the impedance
plane plot is a straight line with an inclination angle α = 45 and the process is diffusion controlled,
as it should be.
4 RESULTS 34
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
← ω = 0.0062832
← ω = 0.0019869
← ω = 0.00099582
ZW
, Analytical
ZW
, Numerical. Steps = 10000
ZW
, Numerical. Steps = 1000
ZW
, Numerical. Steps = 100
ZW
, Numerical. Steps = 10
Figure 4.2: Impedance plane plot from Eq. (2.18) of the Warburg diffusion impedance spectrum for adiffusion coefficient D = 10−5 cm2 s-1 and n = 1. Calculated with a central difference accurate to theorder of H 2, and plotted for a frequency range of 0.1 mHz to 100 MHz.
The same calculations for a forward difference accurate to order H 2 of the boundary conditions
for the iridium oxide electrode is given in Figure 4.3 for 1000 and 10000 steps. We see that a forward
difference requires a higher number of steps, that is, smaller step sizes, to give proper convergence,
and that the magnitude of the deviation is larger than that of the central difference. Forward differ-
ence has been done for the impedance spectra described below, and all calculations have shown the
same trend. Thus, the forward difference method has been proven poor to solve the differential equa-
tion in this study, and will be neglected in the remainder of this report. All numerical solutions below
are done with a central difference with accuracy to order H 2.
4 RESULTS 35
0 50 100 150 200 2500
50
100
150
200
250
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
← ω = 0.0062832
← ω = 0.0019869
← ω = 0.00099582
ZW
, Analytical
ZW
, Numerical. Steps = 10000
ZW
, Numerical. Steps = 1000
Figure 4.3: Impedance plane plot from Eq. (2.18) of the Warburg diffusion impedance spectrum for adiffusion coefficient D = 10−9 cm2 s-1. Calculated with a forward difference accurate to the order ofH 2, and plotted for a frequency range of 0.1 mHz to 100 MHz.
4.1.2 Transmissive Warburg diffusion impedance
The transmissive Warburg diffusion impedance from Eq. (2.19) with a diffusion coefficient D = 10−9
cm2 s-1 is given in Figure 4.4, where the solid line represent the analytical solution, and the markers
represent calculated numerical solutions. The impedance spectra are calculated with the boundary
conditions d L c(s)d x
∣∣∣x=0
= L i f (s)nF and limx→L L c(s) = 0.
The numerical calculations are done for 10, 100 and 1000 steps. Deviations at both high and low
frequencies are observed, and by increasing number of steps the numerical solution approaches the
analytical. A proper convergence is observed for a calculation with 1000 steps. The impedance plane
plot is a straight line with an inclination angle α= 45 for all frequencies within the frequency range,
4 RESULTS 36
which implies a diffusion controlled process.
0 20 40 60 80 100 120 140 160 180 2000
20
40
60
80
100
120
140
160
180
200
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
← ω = 0.62832← ω = 0.19869
← ω = 0.062832
ZW
, Analytical
ZW
, Numerical. Steps = 1000
ZW
, Numerical. Steps = 100
ZW
, Numerical. Steps = 10
Figure 4.4: Impedance plane plot from Eq. (2.19) of the transmissive Warburg diffusion impedancespectrum for a diffusion coefficient D = 10−9 cm2 s-1. Plotted for a frequency range of 0.1 mHz to 100MHz.
The same calculations for a system with a diffusion coefficient D = 10−5 cm2 s-1 are given in Figure
4.5. The diffusion coefficient is chosen to be larger than the previous to show the diffusion coefficient
dependency of the impedance spectrum. We observe a straight line with an inclination angle α =45 for high frequencies and a dome shape for lower frequencies. This implies a mixed control for
the system, where the system is rate limiting at lower frequencies and diffusion controlled at higher
frequencies. Again, a proper convergence is observed for a calculation process with 1000 steps.
4 RESULTS 37
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
← ω = 0.62832
← ω = 0.19869
← ω = 0.062832
ZW
, Analytical
ZW
, Numerical. Steps = 1000
ZW
, Numerical. Steps = 100
ZW
, Numerical. Steps = 10
Figure 4.5: Impedance plane plot from Eq. (2.19) of the transmissive Warburg diffusion impedancespectrum for a diffusion coefficient D = 10−5 cm2 s-1. Plotted for a frequency range of 0.1 mHz to 100MHz.
4.1.3 Reflective Warburg diffusion impedance
The reflective Warburg finite diffusion impedance spectrum from Eq. (2.22) with a diffusion coeffi-
cient D = 10−9 cm2 s-1 is given in Figure 4.6, where the solid line represents the analytical solution and
the markers represent calculated numerical solutions. The impedance spectra are calculated with the
boundary conditions d L c(s)d x
∣∣∣x=0
= L i f (s)nF and d L c(s)
d x
∣∣∣x=L
= 0.
The numerical calculations are done for 10, 100 and 1000 steps. The numerical solution ap-
proaches the analytical solution with increasing number of steps, and a quicker convergence for lower
frequencies than higher is observed. A proper convergence is observed for 1000 steps. The impedance
plane plot is a straight line with an inclination angle α= 45 for all frequencies in the range, and the
4 RESULTS 38
system is diffusion controlled.
0 20 40 60 80 100 120 140 160 180 2000
20
40
60
80
100
120
140
160
180
200
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
← ω = 0.0062832
← ω = 0.0019869
← ω = 0.00099582
ZW
, Analytical
ZW
, Numerical. Steps = 1000
ZW
, Numerical. Steps = 100
ZW
, Numerical. Steps = 10
Figure 4.6: Impedance plane plot from Eq. (2.22) for the reflective Warburg diffusion impedancespectrum for a diffusion coefficient D = 10−9 cm2 s-1. Plotted for a frequency range of 0.1 mHz to 100MHz.
The same calculations done for a system with a diffusion coefficient D = 10−7 cm2 s-1 are given
in Figure 4.7. We see that by increasing the diffusion coefficient, we observe a mixed reaction control
of the system. A divergence to infinite impedance is observed for the lower frequency region, and a
straight line with an inclination angle α= 45 is observed in the higher frequency region.
4 RESULTS 39
0 5 10 15 20 25 300
5
10
15
20
25
30
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
← ω = 0.0062832
← ω = 0.0019869
← ω = 0.00099582
ZW
, Analytical
ZW
, Numerical. Steps = 1000
ZW
, Numerical. Steps = 100
ZW
, Numerical. Steps = 10
Figure 4.7: Impedance plane plot from Eq. (2.22) of the reflective Warburg diffusion impedance spec-trum for a diffusion coefficient D = 10−7 cm2 s-1. Plotted for a frequency range of 0.1 mHz to 100MHz.
4.2 Gerischer impedance spectra for thin film electrodes
From the previous calculated impedance spectra, proper convergence is obtained for number of steps
above 1000. For simplicity, a calculation process of 1000 steps is chosen in this section.
4.2.1 Transmissive Gerischer impedance
The transmissive Gerischer impedance from Eq. (2.29) for an IrO2 electrode with a diffusion coeffi-
cient D = 10−9 cm2 s-1 is given in Figure 4.8, where the solid line represents the analytical solution and
the markers the numerical solution. Here we have chosen an effective rate constant k = 1×10−2 s-1.
4 RESULTS 40
For the higher frequency region a straight line with an inclination angle α = 45 is observed, which
implies a diffusion controlled process within this region. A dome shape is observed for lower frequen-
cies, corresponding to a mixed rate control between the heterogeneous transfer and a non-faradaic
reaction in the bulk.
0 10 20 30 40 50 60 700
10
20
30
40
50
60
70
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
← ω = 0.62832
← ω = 0.19869
← ω = 0.062832
ZG
, Analytical
ZG
, Numerical. Steps = 1000
Figure 4.8: Impedance plane plot from Eq. (2.29) of the transmissive Gerischer impedance spectrumfor a semiconducting thin film with a diffusion coefficient D = 10−9 cm2 s-1 and an effective rateconstant k = 1×10−2 s-1. Plotted for a frequency range of 0.1 mHz to 100 MHz.
We reduce the rate constant to k = 0 s-1 to show the rate constant dependency of the impedance
for the IrO2 system. This is given in Figure 4.9, where we observe a straight line with an inclination
angle α = 45 for all frequencies within the frequency range. The system is controlled by a diffusion
process.
4 RESULTS 41
0 10 20 30 40 50 60 700
10
20
30
40
50
60
70
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
← ω = 0.62832
← ω = 0.19869
← ω = 0.062832
ZG
, Analytical
ZG
, Numerical. Steps = 1000
Figure 4.9: Impedance plane plot from Eq. (2.29) of the transmissive Gerischer impedance spectrumfor a semiconducting thin film with a diffusion coefficient D = 10−9 cm2 s-1 and an effective rateconstant k = 0 s-1. Plotted for a frequency range of 0.1 mHz to 100 MHz.
To show the diffusion coefficient dependency of the transmissive Gerischer impedance, a hypo-
thetical electrode with a diffusion coefficient D = 10−5 cm2 s-1 is chosen. Calculations are done with
an effective rate constant k = 1×10−2 s-1 in Figure 4.10, and k = 0 s-1 in Figure 4.11. Both of the sys-
tems show a mixed controlled process for all frequencies within the frequency range, compared to
the IrO2 system above. A reduction in the effective rate constant results in a larger radius of the dome
shape in the rate controlled region. That is, we observe a higher magnitude of the impedance in the
rate controlled region for a lower effective rate constant.
4 RESULTS 42
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
← ω = 0.62832
← ω = 0.19869
← ω = 0.062832
ZG
, Analytical
ZG
, Numerical. Steps = 1000
Figure 4.10: Impedance plane plot from Eq. (2.29) of the transmissive Gerischer impedance spectrumfor a semiconducting thin film with a diffusion coefficient D = 10−5 cm2 s-1 and an effective rateconstant k = 1×10−2 s-1. Plotted for a frequency range of 0.1 mHz to 100 MHz.
4 RESULTS 43
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
← ω = 0.62832
← ω = 0.19869
← ω = 0.062832
ZG
, Analytical
ZG
, Numerical. Steps = 1000
Figure 4.11: Impedance plane plot from Eq. (2.29) of the transmissive Gerischer impedance spectrumfor a semiconducting thin film with a diffusion coefficient D = 10−5 cm2 s-1 and an effective rateconstant k = 0 s-1. Plotted for a frequency range of 0.1 mHz to 100 MHz.
4.2.2 Reflective Gerischer impedance
The reflective Gerischer impedance from Eq. (2.32) for an IrO2 electrode with a diffusion coefficient
D = 10−9 cm2 s-1 is given in Figure 4.12, where the solid line represents the analytical solution and
the markers the numerical solution. We have chosen an effective rate constant k = 1×10−2 s-1. We
observe a straight line with an inclination angle α = 45 for the high frequencies, and a dome shape
for lower frequencies. This implies a diffusion controlled process for high frequencies and a mixed
rate control between the heterogeneous transfer and a non-faradaic reaction for low frequencies.
4 RESULTS 44
0 10 20 30 40 50 60 700
10
20
30
40
50
60
70
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
← ω = 0.62832
← ω = 0.19869
← ω = 0.062832
ZG
, Analytical
ZG
, Numerical. Steps = 1000
Figure 4.12: Impedance plane plot from Eq. (2.32) of the reflective Gerischer impedance spectrum fora semiconducting thin film with a diffusion coefficient D = 10−9 cm2 s-1 and an effective rate constantk = 1×10−2 s-1. Plotted for a frequency range of 0.1 mHz to 100 MHz.
We reduce the effective rate constant to k = 0 s-1, and the results are given in Figure 4.13. In this
we observe a diffusion controlled system for all frequencies within the frequency range.
4 RESULTS 45
0 10 20 30 40 50 60 700
10
20
30
40
50
60
70
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
← ω = 0.62832
← ω = 0.19869
← ω = 0.062832
ZG
, Analytical
ZG
, Numerical. Steps = 1000
Figure 4.13: Impedance plane plot from Eq. (2.32) of the reflective Gerischer impedance spectrum fora semiconducting thin film with a diffusion coefficient D = 10−9 cm2 s-1 and an effective rate constantk = 0 s-1. Plotted for a frequency range of 0.1 mHz to 100 MHz.
To investigate the diffusion coefficient dependency of the reflective Gerischer impedance, we
choose a hypothetical electrode with a diffusion coefficient of D = 10−6 cm2 s-1. Calculation for an
effective rate constant k = 1× 10−2 s-1 is given in Figure 4.14, and k = 0 s-1 is given in Figure 4.15.
We observe a mixed controlled system for both rate constants, compared to the IrO2 system shown
above.
In Figure 4.15 we observe a similar trend as that observed for the reflective Warburg diffusion
impedance in Figure 4.7, where the impedance plane plot is a straight line for high frequencies and
diverges to infinity for lower frequencies. This should be expected, since the choice of k = 0 reduces
the process to a Warburg-like process. In Figure 4.14 a small bend after the diffusion controlled region
4 RESULTS 46
occurs, similar to the previous plot. However, for lower frequencies the impedance plane plot bends
towards zero in a dome shape rather than diverge to infinity compared to Figure 4.15. The dome
observed correspond to the mixed reaction controlled process with an effective rate constant k =1×10−2 s-1.
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
← ω = 0.62832
← ω = 0.19869
← ω = 0.062832
ZG
, Analytical
ZG
, Numerical. Steps = 1000
Figure 4.14: Impedance plane plot from Eq. (2.32) of the reflective Gerischer impedance spectrum fora semiconducting thin film with a diffusion coefficient D = 10−6 cm2 s-1 and an effective rate constantk = 1×10−2 s-1. Plotted for a frequency range of 0.1 mHz to 100 MHz.
4 RESULTS 47
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
← ω = 0.62832
← ω = 0.19869
← ω = 0.062832
ZG
, Analytical
ZG
, Numerical. Steps = 1000
Figure 4.15: Impedance plane plot from Eq. (2.32) of the reflective Gerischer impedance spectrumfor a semiconducting thin film electrode with a diffusion coefficient D = 10−6 cm2 s-1 and an effectiverate constant k = 0 s-1. Plotted for a frequency range of 0.1 mHz to 100 MHz.
From these results, we see that the blocking electrode appears as a leaking electrode – that is, the
reflective Gerischer impedance is similar to the transmissive Gerischer impedance – for low diffusion
coefficients (cf. Figure 4.8 and Figure 4.9 for comparison). It can also be seen that the rate control
process given by the effective rate constant k is more significant for a system with a larger diffusion
coefficient.
4.3 Electrochemical impedance spectra for semiconducting thin films
The analytical and numerical solution of the electrochemical impedance spectrum from Eq. (2.75) for
an iridium oxide thin film electrode with a diffusion coefficient D = 10−9 cm2 s-1 and an effective rate
4 RESULTS 48
constant k = 1×10−2 s-1 are given in Figure 4.16. We observe a straight line with an inclination angle
α= 45 for high frequencies, and a dome for low frequencies. The process is thus diffusion controlled
for higher frequencies and reaction controlled for lower frequencies.
40 50 60 70 80 90 100 1100
10
20
30
40
50
60
70
← ω = 0.62832
← ω = 0.19869
← ω = 0.062832
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
Zf, Analytical
Zf, Numerical
Figure 4.16: Impedance plane plot from Eq. (2.75) of the electrochemical impedance spectrum for aniridium oxide thin film electrode with a diffusion coefficient D = 10−9 cm2 s-1 and an effective rateconstant k = 1×10−2 s-1, calculated with 1000 steps. Plotted for a frequency range of 0.1 mHz to 100MHz.
To investigate the effective rate constant dependency, the impedance spectrum for the same irid-
ium oxide system for an effective rate constant k = 0 s-1 is given in Figure 4.17. Now, we observe a
diffusion controlled process for all frequencies.
4 RESULTS 49
40 50 60 70 80 90 100 1100
10
20
30
40
50
60
70
← ω = 0.62832
← ω = 0.19869
← ω = 0.062832
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
Zf, Analytical
Zf, Numerical
Figure 4.17: Impedance plane plot from Eq. (2.75) of the electrochemical impedance spectrum for aniridium oxide thin film electrode with a diffusion coefficient D = 10−9 cm2 s-1 and an effective rateconstant k = 0 s-1, numerical solution calculated with 1000 steps. Plotted for a frequency range of 0.1mHz to 100 MHz.
We see that by reducing the effective rate constant, the frequency range for diffusion control in-
creases. An effective rate constant analysis of the iridium oxide electrode with D = 10−9 cm2 s-1 is
given in Figure 4.18. Here, the electrochemical impedance is calculated for a range of effective rate
constants between k = 0.1 and k = 10−5 s-1. We see that by increasing the effective rate constant, the
impedance bends towards zero for lower frequencies, corresponding to the mixed rate control of the
heterogeneous transfer and the non-faradaic reaction.
The impedance spectra are similar to the transmissive gerischer impedance spectra (cf. Figure
4.12 and Figure 4.13), we observe a leaking electrode system for the iridium oxide electrode.
4 RESULTS 50
40 45 50 55 60 65 70 75 800
5
10
15
20
25
30
35
40
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
Zf, Numerical. k = 1e−05
Zf, Numerical. k = 0.0001
Zf, Numerical. k = 0.001
Zf, Numerical. k = 0.01
Zf, Numerical. k = 0.1
Figure 4.18: Rate constant analysis of the electrochemical impedance spectrum from Eq. (2.75) foran iridium oxide thin film electrode with a diffusion coefficient D = 10−9 cm2 s-1, numerical solutioncalculated with 1000 steps. Plotted for a frequency range of 0.1 mHz to 100 MHz.
The electrochemical impedance spectrum from Eq. (2.75) for a hypothetical semiconducting thin
film electrode with a diffusion coefficient D = 10−6 cm2 s-1 and an effective rate constant k = 1×10−2
s-1 is given in Figure 4.19. Again, we observe a straight line corresponding to a diffusion controlled
process for higher frequencies and a dome shape corresponding to a reaction controlled process for
lower frequencies, as for the iridium oxide electrode. However, compared to Figure 4.16, the diffusion
controlled region is smaller. A slight bend in the diffusion region located between ω = 0.062832 and
ω= 0.19869 similar to the mixed reflective gerischer impedance spectrum described in Figure 4.14 is
observed.
4 RESULTS 51
44 44.5 45 45.5 46 46.5 470
0.5
1
1.5
2
2.5
3
← ω = 0.62832
← ω = 0.19869
← ω = 0.062832
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
Zf, Analytical
Zf, Numerical
Figure 4.19: Impedance plane plot from Eq. (2.75) of the electrochemical impedance spectrum for asemiconducting thin film electrode with a diffusion coefficient D = 10−6 cm2 s-1 and an effective rateconstant k = 1×10−2 s-1, numerical solution calculated with 1000 steps. Plotted for a frequency rangeof 0.1 mHz to 100 MHz.
The same hypothetical electrode with D = 10−6 cm2 s-1 and an effective rate constant k = 0 s-1 is
given in Figure 4.20. Here, the bend of the impedance caused by the rate controlled process described
by the effective rate constant is not present, and the impedance diverges to infinity. This is similar to
the reflective gerischer impedance described in Figure 4.14
4 RESULTS 52
44 44.5 45 45.5 46 46.5 470
0.5
1
1.5
2
2.5
3
← ω = 0.62832
← ω = 0.19869
← ω = 0.062832
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
Zf, Analytical
Zf, Numerical
Figure 4.20: Impedance plane plot from Eq. (2.75) of the electrochemical impedance spectrum for asemiconducting electrode with a diffusion coefficient D = 10−6 cm2 s-1 and an effective rate constantk = 0 s-1, numerical solution calculated with 1000. Plotted for a frequency range of 0.1 mHz to 100MHz.
An effective rate constant analysis of this electrode with D = 10−6 cm2 s-1 is given in Figure 4.21.
The impedance spectrum is calculated for range of effective rate constants between k = 0.1 and k =10−5 s-1. This figure shows how the bending of the impedance spectrum caused by the effective rate
constant is reduced by reducing the rate constant. From these results we see that the impedance
spectrum is similar to the reflective gerischer impedance spectrum for high diffusion coefficients, we
observe a blocking electrode system for the hypothetical electrode.
4 RESULTS 53
44 46 48 50 52 54 56 58 600
2
4
6
8
10
12
14
16
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
Zf, Numerical. k = 1e−05
Zf, Numerical. k = 0.0001
Zf, Numerical. k = 0.001
Zf, Numerical. k = 0.01
Zf, Numerical. k = 0.1
Figure 4.21: Effective rate constant analysis of the electrochemical impedance spectrum from Eq.(2.75) for a semiconducting thin film electrode with a diffusion coefficient D = 10−6 cm2 s-1, numeri-cal solution calculated with 1000 steps. Plotted for a frequency range of 0.1 mHz to 100 MHz.
4.4 Photoelectrochemical impedance spectra for semiconducting thin films
For the electrochemical impedance, the steady state concentration is assumed constant across the
electrode. However, under illumination, this assumption does not hold. The steady-state concentra-
tion profile from Eq. (2.78) across a 100 µm thick iridium oxide electrode electrode with a diffusion
coefficient D = 10−9 cm2 s-1, an effective rate constant k = 1×10−2 s-1, a reciprocal absorption length
α = 0.25× 10−5 cm-1, and a light intensity of I0 = 1000 mol cm-2 s-1 is given in Figure 4.22. We ob-
serve reduction in the steady-state concentration across the electrode, with a steep decay close to
the illuminated surface. The concentration quickly reaches a constant value, which remains across
4 RESULTS 54
the rest of the electrode length. The numerical results are calculated for 1000 steps and show proper
convergence. A calculation process of 1000 steps is used in the remainder of this section.
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
0.5
1
1.5
2
2.5
3
3.5
Length [cm]
Co
nce
ntr
atio
n [
mo
l cm
−3]
Ce, analytical. I
0 = 1000
Ce, numerical. I
0 = 1000
Figure 4.22: The steady state concentration from Eq. (2.78) of a photoelectrochemical iridium ox-ide thin film electrode as a function of the length through the electrode. Calculations done with aneffective rate constant of k = 1×10−2 s-1, 1000 steps, α= 0.25×10−5 cm-1, and I0 = 1000 mol cm-2 s-1
.
A light intensity analysis of the same system is given in Figure 4.23, where we see an increase in the
concentration by increasing the intensity. The profile shape, however, shows no significant change,
it is only shifted when changing the light intensity. In Figure 4.24 an effective rate constant analysis
with an illumination I0 = 1000 mol cm-2 s-1 is given. We observe a decreases in the steady state con-
centration with an increase in the reaction rate, and an increases in the steady state concentration
difference across the electrode with decreasing effective rate constant.
4 RESULTS 55
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01−0.5
0
0.5
1
1.5
2
2.5
3
3.5
Length [cm]
Concentr
ation [m
ol cm
−3]
Ce, numerical. I
0 = 1000
Ce, numerical. I
0 = 1
Ce, numerical. I
0 = 0.001
Ce, numerical. I
0 = 1e−05
Ce, numerical. I
0 = 0
Figure 4.23: Light intensity analysis for the steady state concentration from Eq. (2.78), D = 10−9 cm2
s-1, k = 1×10−2 s-1, and α= 0.25×10−5 cm-1.
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
2
4
6
8
10
12
14
Length [cm]
Concentr
ation [m
ol cm
−3]
Ce, numerical. k = 1
Ce, numerical. k = 0.1
Ce, numerical. k = 0.01
Ce, numerical. k = 0.001
Figure 4.24: Rate constant analysis for the steady state concentration from Eq. (2.78), D = 10−9 cm2
s-1, α= 0.25×10−5 cm-1, and I0 = 1000 mol cm-2 s-1.
The photoelectrochemical impedance spectrum from Eqs. (2.78), (2.83) and (2.70) for the iridium
oxide electrode with an effective rate constant k = 1× 10−2 s-1 and a reciprocal absorption length
4 RESULTS 56
α = 0.25× 10−5 cm-1 is given in Figure 4.25 for different light intensities. No significant change in
the impedance for different illuminations is observed. The system shows mixed control for all light
intensities, diffusion controlled for higher frequencies and rate controlled for lower frequencies.
40 50 60 70 80 90 100 1100
10
20
30
40
50
60
70
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
Zf, numerical. I
0 = 100000
Zf, numerical. I
0 = 1000
Zf, numerical. I
0 = 1
Zf, numerical. I
0 = 0.001
Zf, numerical. I
0 = 1e−05
Zf, numerical. I
0 = 0
Figure 4.25: Impedance plane plot from Eqs. (2.78), (2.83) and (2.70) of the photelectrochemicalimpedance spectrum as a function of light intensity for an iridium oxide thin film electrode with adiffusion coefficient D = 10−9 cm2 s-1, an effective rate constant k = 10−2 s-1, and a reciprocal absorp-tion length α= 0.25×10−5 cm-1, calculated with 1000 steps. Plotted for a frequency range of 0.1 mHzto 100 MHz.
The photoelectrochemical spectrum from Eqs. (2.78), (2.83) and (2.70) for the same electrode with
an effective rate constant k = 0 s-1 is given in Figure 4.26, where the impedance shows no significant
dependency of light intensity. The plot is a straight line for all frequencies within the range, and is
diffusion controlled.
4 RESULTS 57
40 60 80 100 120 140 160 180 200 220 2400
20
40
60
80
100
120
140
160
180
200
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
Zf, numerical. I
0 = 100000
Zf, numerical. I
0 = 1000
Zf, numerical. I
0 = 1
Zf, numerical. I
0 = 0.001
Zf, numerical. I
0 = 1e−05
Zf, numerical. I
0 = 0
Figure 4.26: Impedance plane plot from Eqs. (2.78), (2.83) and (2.70) of the photelectrochemicalimpedance spectrum as a function of light intensity for an iridium oxide thin film electrode with adiffusion coefficient D = 10−9 cm2 s-1, an effective rate constant k = 0 s-1, and a reciprocal absorptionlength α = 0.25×10−5 cm-1, calculated with 1000 steps. Plotted for a frequency range of 0.1 mHz to100 MHz.
The effective rate constant is increased to k = 1 s-1 to investigate wether the light intensity is sig-
nificant for even higher effective rate constants. This is given Figure 4.27, where we observe a small
increase in the impedance in the lower frequency region when reducing the light intensity. However,
when the intensity approaches zero the numerical approach breaks down, and this plot is neglected
in this figure. We observe one dome shape at high frequencies going to a straight line when lowering
the frequency, and a larger dome shape at the lower frequencies for all light intensities. This implies
that the system is rate controlled in the highest frequency, diffusion controlled at intermediate fre-
quencies, and rate controlled for lower frequencies.
4 RESULTS 58
44 45 46 47 48 49 50 510
1
2
3
4
5
6
7
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
Zf, numerical. I
0 = 100000
Zf, numerical. I
0 = 1000
Zf, numerical. I
0 = 1
Zf, numerical. I
0 = 0.001
Zf, numerical. I
0 = 1e−05
Figure 4.27: Impedance plane plot from Eqs. (2.78), (2.83) and (2.70) of the photelectrochemicalimpedance spectrum as a function of light intensity for an iridium oxide thin film electrode with adiffusion coefficient D = 10−9 cm2 s-1, an effective rate constant k = 1 s-1, and a reciprocal absorptionlength α = 0.25×10−5 cm-1, calculated with 1000 steps. Plotted for a frequency range of 0.1 mHz to100 MHz.
By increasing the effective rate constant, we observe a larger light intensity dependency of the
photoelectrochemical impedance. We assume a hypothetical electrode with a diffusion coefficient
to D = 10−9 cm2 s-1 to investigate this dependency, since the electrode is diffusion controlled over
a smaller frequency range for higher diffusion coefficients. In Figure 4.28, the photoelectrochemical
impedance spectrum from Eqs. (2.78), (2.83) and (2.70) for this electrode is given with an effective
rate constant k = 10−2 s-1 and a reciprocal absorption length α = 0.25× 10−5 cm-1. An increase in
the impedance is observed in the lower frequency region, and the light intensity dependency is more
4 RESULTS 59
significant for this electrode than for the iridium system above.
44 44.5 45 45.5 46 46.5 470
0.5
1
1.5
2
2.5
3
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
Zf, numerical. I
0 = 100000
Zf, numerical. I
0 = 1000
Zf, numerical. I
0 = 1
Zf, numerical. I
0 = 0.001
Zf, numerical. I
0 = 1e−05
Zf, numerical. I
0 = 0
Figure 4.28: Impedance plane plot from Eqs. (2.78), (2.83) and (2.70) of the photelectrochemicalimpedance spectrum as a function of light intensity for a semiconducting thin film electrode witha diffusion coefficient D = 10−6 cm2 s-1, an effective rate constant k = 10−2 s-1, and a reciprocal ab-sorption length α= 0.25×10−5 cm-1, calculated with 1000 steps. Plotted for a frequency range of 0.1mHz to 100 MHz.
The photoelectrochemical impedance spectrum from Eqs. (2.78), (2.83) and (2.70) for the same
electrode with an effective rate constant k = 0.5 s-1 for different light intensities is given in Figure
4.29. The increase in the impedance for higher frequencies is even more significant for this system.
For high light intensities, we observe a system with diffusion control at high frequencies and rate
controlled at lower frequencies. A second dome in the lower frequency regions at light intensities
close to zero is observed, which disappears by increasing the light intensity.
4 RESULTS 60
44.05 44.1 44.15 44.2 44.25 44.3 44.350
0.05
0.1
0.15
0.2
0.25
0.3
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
Zf, numerical. I
0 = 100000
Zf, numerical. I
0 = 1000
Zf, numerical. I
0 = 1
Zf, numerical. I
0 = 0.001
Zf, numerical. I
0 = 1e−05
Zf, numerical. I
0 = 0
Figure 4.29: Impedance plane plot from Eqs. (2.78), (2.83) and (2.70) of the photelectrochemicalimpedance spectrum as a function of light intensity for a semiconducting thin film electrode witha diffusion coefficient D = 10−6 cm2 s-1 and an effective rate constant k = 0.5 s-1, calculated with 1000steps. Plotted for a frequency range of 0.1 mHz to 100 MHz.
The effective rate constant is increased to k = 1 s-1, the results are given in Figure 4.30. For high
light intensities we observe a mixed controlled system which is diffusion controlled for high frequen-
cies and rate controlled at low frequencies. A second dome at lower frequencies appears at lower light
intensities. The numerical approach breaks down when the light intensity approach zero, just like the
iridium oxide system with k = 1 s-1 described above.
4 RESULTS 61
44.05 44.1 44.15 44.2 44.25 44.3 44.35−0.05
0
0.05
0.1
0.15
0.2
0.25
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
Zf, numerical. I
0 = 100000
Zf, numerical. I
0 = 1000
Zf, numerical. I
0 = 1
Zf, numerical. I
0 = 0.001
Zf, numerical. I
0 = 1e−05
Zf, numerical. I
0 = 0
Figure 4.30: Impedance plane plot from Eqs. (2.78), (2.83) and (2.70) of the photelectrochemicalimpedance spectrum as a function of light intensity for a semiconducting thin film electrode witha diffusion coefficient D = 10−6 cm2 s-1, an effective rate constant k = 1 s-1, and a reciprocal absorp-tion length α= 0.25×10−5 cm-1, calculated with 1000 steps. Plotted for a frequency range of 0.1 mHzto 100 MHz.
5 DISCUSSION 62
5 Discussion
In this project, we want to verify if Newman’s BAND routine is compatible for complex coefficients,
thus suitable for impedance spectra calculations. The largest focus and most important in this dis-
cussion will be how, or if, the numerical solution approaches the expected analytical solution, what
measures have been done to obtain proper convergences, and origins to any deviations or non-
convergence.
5.1 Parameterization sensitivity
In general, we assume that the numerical solutions converge to the analytical solutions for infinitely
small step sizes H . That is
limH→0
(Numerical approach
)→ (Analytical approach
)(5.1)
By increasing the number of steps, thus reducing the step size H , the numerical solution should ap-
proach the analytical solution.
We have seen that to solve the diffusion equation
Dd 2 L c ( jω)
d x2 − (k + jω)L c ( jω) =−L
I0
( jω) ·αe−αx (5.2)
with the Newman’s BAND routine, we supply the coefficients
A1,1 = D1
L2
B1,1 =−2D1
L2 −H 2 · ( jω+k)
D1,1 = D1
L2
G1,1 =−H 2L
I0
( jω) ·αe−αx
(5.3)
We have two contributions that influences the step size, H 2 · ( jω+k) and H 2
I0
( jω)αe−αx . From
Figure 4.2, we see that the numerical solution for the infinite Warburg diffusion impedance converges
quickly for low frequencies with respect to number of steps, but more slow in the high frequency
region. This general trend is found in all of the calculations, and can be explained from the H 2·( jω+k)
term, where the low frequency range is not as sensitive to the step size H than for the high frequency
region. Deviations for high frequencies were solved by reducing the step size.
As mentioned above, the numerical solution converges in general more quickly for lower frequen-
cies. However, an increase in deviation with decreasing step size in the lower frequency region of the
5 DISCUSSION 63
faradaic impedance from Eq. (2.75) for the semiconducting thin film was observed. This is shown in
Figure 5.1a for a hypothetical thin film electrode with a diffusion coefficient D = 10−5 cm2 s-1 and rate
constant k = 10−2 s-1. The 1/L2-factor that originates from going from finite lengths to dimension-
less lengths effectively increases the a1,1 coefficient in the differential equation solved when L < 1,
and may alter the step size dependency. By multiplying the differential equation with the L2-factor,
proper convergence was observed. One explanation for this result could be that this multiplication
effectively reduces the step size, and should lead to an even quicker convergence. The same hypo-
thetical system solved with the parameterization where the differential equation solved is multiplied
with L2 is given in Figure 5.1b. A proper convergence trend with respect to step size is observed.
42 42.5 43 43.5 44 44.5 45 45.5 46 46.50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
Zf, Numerical. Steps = 10
Zf, Numerical. Steps = 100
Zf, Numerical. Steps = 1000
Zf, Numerical. Steps = 10000
(a)
44 44.5 45 45.5 46 46.50
0.5
1
1.5
2
2.5
Re(Z) [Ω cm2]
−Im
(Z)
[Ω c
m2]
Zf, Numerical. Steps = 10
Zf, Numerical. Steps = 100
Zf, Numerical. Steps = 1000
Zf, Numerical. Steps = 10000
(b)
Figure 5.1: (a) The faradaic impedance plane plot from Eq. (2.75) for a semiconducting thin filmwith diffusion coefficient D = 10−5 cm2 s-1 and rate constant k = 10−2 s-1 that shows an increase indeviation with decreasing step size. (b) The same semiconducting thin system that shows properconvergence dependency of the step size by multiplying the differential equation with L2.
5.2 Impedance spectra
5.2.1 Diffusion control at high frequencies and low diffusion coefficients
From the analytical expressions, the impedances are typically an expression with a tanh
(√jω+k
DL
)
or a coth
(√jω+k
DL
)term. Since we have the relations
lim√jω+k
D L→∞tanh
√jω+k
DL
→ 1 (5.4)
5 DISCUSSION 64
and
lim√jω+k
D L→∞coth
√jω+k
DL
→ 1 (5.5)
we see that
lim√jω+k
D L→∞Zfinite → Zinfinite (5.6)
Since the infinite Warburg diffusion impedance is diffusion controlled, all the processes described
above will be diffusion controlled when√
jω+kD L is substantially high. The largest contributions to
obtain this are a large frequency and a low diffusion coefficient. This can be seen from the results
above, where all processes are diffusion controlled at high frequencies, and the diffusion controlled
frequency region increases with decreasing diffusion coefficient. The magnitude of the rate constant
will also affect the process control. However, the rate constant only give contribution at lower fre-
quency regions when ω ≤ k. The rate constant will bend the impedance plane plot towards zero for
the blocking electrode, and more quickly towards zero for the leaking electrode.
5.2.2 Warburg diffusion impedance spectra
The derived numerical and analytical Warburg diffusion models above are in accordance with previ-
ous models, see refs. [16], [17] and [21] for comparison. Impedance measurements of tungsten oxide
(WO3) thin films deposited on tin oxide where carried out by Ho et al. [16], which showed a reflective
Warburg-like behavior. Transmissive Warburg-like behavior was observed by Sistat et al. [22] for an
AMX homogeneous anion-exchange membrane (described in ref. [23]).
The numerical model established may thus be fitted to Warburg-like impedance spectra to char-
acterize the electrode investigated.
5.2.3 Gerischer impedance spectra
The the Gerischer impedance models are in accordance with existing models, see refs. [18], [12]
and [24] for comparison. We see similar plots in this study for both the analytical model and, most
important, the numerical model. Thus, Newman’s BAND routine may be used for calculating the
Gerischer impedance for an electrochemical system. Observations strongly supporting the existence
of a finite-length Gerischer impedance responses in solid-oxide fuel cells (SOFC) have been reported
by Boukamp et. al [18, 12, 25].
Thus, the established numerical model may be fitted to these measurements to characterize the
electrical properties of these SOFC systems.
5 DISCUSSION 65
5.2.4 Electrochemical impedance for a mixed conductivity thin film electrode
The thin film electrode is, as expected, diffusion controlled in the higher frequency region for all
processes described above. The increase of the effective rate constant bends the impedance more
quicker toward zero, because of the increased reaction rate of the charge transfer across the interface
and reaction to neutral species. From the boundary conditions chosen, we expect the electrochem-
ical impedance to behave as a reflective Gerischer-type impedance. Thus, we expect to observe a
transmissive-like behavior for low diffusion coefficients, and blocking-like behavior for higher diffu-
sion coefficients. The impedance plane plots above are in accordance with the expected behavior,
and the numerical model shows proper trends for the mixed conducting thin film electrode.
5.2.5 Photoelectrochemical impedance spectra for a mixed conductivity thin film electrode
We observe that for a diffusion controlled process the photoelectrochemical impedance is nearly in-
dependent of the light intensity. This is shown in Figure 4.26 for the diffusion controlled iridium
oxide electrode with an effective rate constant k = 0 s-1. A small increase in the impedance for low
light intensities and low frequencies is observed for high effective rate constants, which indicates
that the rate controlled process is more light intensity dependent than the diffusion controlled pro-
cess. However, the photoelectrochemical impedance spectrum iridium oxide electrode appears close
to independent of light intensity, regardless of the effective rate constant. By increasing the diffu-
sion coefficient we observe a more significant light intensity dependence of the photoelectrochemical
impedance for high rate constants and low frequencies. Thus, the light intensity source term affects
the rate controlled process, but not the diffusion controlled process.
A second dome is observed for a diffusion coefficient D = 10−6 cm2 s-1 and an effective rate con-
stant k = 0.5 s-1. The at higher frequency dome correspond to the heterogeneous diffusion rate, and
the lower frequency dome correspond to the recombination process. One explanation for this could
be by reducing the light intensity, the recombination in the semiconductor is more significant, lead-
ing to a more significant contribution in the impedance. This dome is however not present for the
calculations electrochemical impedance spectra. From the steady-state concentration profiles, we
see that under no illumination there is a significant concentration profile close to the electrolyte in-
terface. This rapid change in concentration is caused by the non-faradaic trapping process described
above. Thus, the electrochemical impedances calculated neglects the non-faradaic reaction depen-
dency of the steady-state concentration, and should be included in the numerical model.
The steady-state concentration equation (2.78) neglects contributions to ce from e.g. thermal
excitation of carriers in a semiconductor. By setting the faradaic current and the light intensity in Eq.
(2.78) to zero leads to a zero steady state concentration, which is not the case for real cases where
5 DISCUSSION 66
thermal excitation e.g. are included. A constant additive term on the right of the equation should be
added to the right of Eq. (2.78) to include these contributions at zero current and light intensity. This
may significantly alter the calculated impedance spectra, and needs to be taken into consideration
for further work.
The faradaic photocurrent would be expected to be proportional to I0 by
i f = nF D∂ce
∂x(5.7)
Combination with the expression of the steady state concentration in Eq. (2.78) we get
i f =I0α
2e−αx
Dα2 −k
1− n
sinh
(√kD L
) ·K1 sinh
(√k
D(x −L)
)−K2 sinh
(√k
Dx
) (5.8)
with
K1 = 1− t−z−ν−
− F D I0α2
i f (Dα2 −k)(5.9)
K2 = 1− t+z+ν+
− F D I0α2
i f (Dα2 −k)e−αL (5.10)
which should be included in this model.
6 CONCLUSION 67
6 Conclusion
A numerical model to calculate impedance using Newman’s BAND subroutine has been established.
The numerical model show proper convergence, and is not time consuming. Thus, Newman’s BAND
subroutine has been proven valid to supply complex matrix coefficients to solve differential equa-
tions, hence it’s compatibility with impedance calculations.
An electrochemical impedance model for a mixed conducting thin film electrode based on di-
lute solution theory and a binary electrolyte has been established. The mixed conducting thin film
electrode behaves as a blocking electrode, and the impedance spectra are similar to the reflective
Gerischer impedance spectrum model.
A numerical model of the steady state concentration profile of a semiconducting thin film elec-
trode under illumination has been suggested. This model was used to describe the photoelectro-
chemical impedance under of a mixed conducting semiconductor. A second dome for a charge trans-
fer limiting process at low frequencies was observed under no illumination in this model, which was
not present for the previous electrochemical model. One suggested cause of this dome is an observed
steady-state concentration gradient under no illumination, which was assumed to be constant in the
first model.
The suggested steady-state model disregard the faradaic steady-state current dependency of the
light intensity, which should to be taken into consideration. One expression to include this was sug-
gested. Also, the model disregard steady-state concentration contribution under no illumination by
e.g. thermal excitation, which should to be included.
LIST OF SYMBOLS 68
List of symbols
A ∂i f /∂c, A mol-1 cm
B ∂i f /∂(Φ1 −Φ2),Ω−1 cm-2
c concentration of neutral hole-vacancy pair, mol cm-3
ce equilibrium concentration of neutral hole-vacancy pairs, mol cm-3
ci concentration of species i , mol cm-3
D diffusion coefficient, cm2 s-1
Di diffusion coefficient of species i , cm2 s-1
F Faraday constant, 96485 C mol-1
H xH intercalant hydrogen in the oxide lattice of effective zero charge
i current density, A cm-2
I0 light intensity, mol cm-2 s-1
I (t ) total time-dependent current, A
Im( ) imaginary part of complex number in parenthesis
j complex number,p−1
k effective rate constant, s-1
kB Boltzmann constant, 1.38×10−23 J K-1
L electrode thickness, cm
Na acceptor dopant concentration, cm-3
Nd donor dopant concentration, cm-3
Ni flux of species i , mol cm-2 s-1
qi charge of species i , C cm-2
R gas constant, 8.314 J K-1 mol-1
RC T charge transfer resistance,Ω cm2
RE electrolyte resistance,Ω cm2
Re( ) real part of complex number in parenthesis
s Laplace variable
ti transference number of species i
T temperature, K
u′i mobility of species i , cm2 (Vs)-1
V (t ) total time-dependent voltage, V
V ′H intercalant-hydrogen vacancy in the oxide lattice of effective negative charge
x coordinate normal to the electrode, cm
y dimensionless coordinate, x/L
LIST OF SYMBOLS 69
zi charge number of species i
Z impedance,Ω cm2
Z ′D , Z ′′
D diffusion impedances,Ω cm2
Greek
α reciprocal diffusion length, cm-1
∂ partial derivative
κ electrode conductivity,Ω−1 cm-1
∇ del-operator, x∂/∂x + y∂/∂y + z∂/∂z
µe electrochemical potential of electrons in electrode backing, J mol-1
µh electrochemical potential of holes in electrode, J mol-1
ν+ number of holes per electron extracted
ν− number of proton vacancies per electron extracted
σ electrolyte conductivity,Ω−1 cm-1
θ phase shift
Φ1,Φ2,ΦM electrode, electrolyte and metal support potentials, V
ω angular frequency, s-1
Subscripts
i species number
1 electrode phase in mixed conducting system
2 electrolyte phase in mixed conducting system
f faradaic
G Gerischer
M metal support
W Warburg
Superscripts
0 reference value
Overline
∼ time-dependent part of quantity
∧ unit vector
Others
L Laplace transform
REFERENCES 70
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Appendix A 73
A Proof that the impedance can be derived by setting s = jω in the Laplace-
transform t-space equations
Laplace-transform:
L
f (t )
(s) =∫ ∞
0f (t )exp(−st )d t = F (s) (A.1)
where s =σ+ jω ( j =p−1). Inverse transform:
−1L
f (s)
= 1
2π
∫ γ+ j∞
γ− j∞f (s)exp(st )d s = f (t ) (A.2)
Some simple rules and examples:
L t = 1
s2 (A.3)
L t 2 = 2
s2 (A.4)
L sinβt = β
s2 +β2 (A.5)
Applications in differential equations: Transform problem, solve transformed equations algebraically,
and then transform back.
For complex equations, the back-transformation is usually quite difficult, and only limiting forms
can (at best) be found. Numerical inversion is possible, but unstable. Some algorithms exist, see e. g.
Schittkowski [26].
Response to a harmonic driving signal
u(t ) = u0 sinωt : t > 0
0 : t < 0(A.6)
Driving signal:
u(s) = u0ω
s2 +ω2 (A.7)
Response when the transfer function is h(s):
y(s) = h(s)u0ω
s2 +ω2 (A.8)
In t-space
y(t ) = −1L
y(s)
= 1
2π j
∫ γ+ j∞
γ− j∞h(s)
u0ω
s2 +ω2 exp(st )d s (A.9)
Appendix A 74
Residue theorem: ∮C
f (z) = 2π jN∑
k=1Res fz=zk (z) (A.10)
If simple poles only:
Res fz=zk (z) = limξ→z0
(z − z0) f (z) (A.11)
If the transfer function h(s) has N poles ( uoωs2+ω2 = u0ω
(s+ jω2)(s− jω2) ) has two poles in s =± jω
y(t ) = 1
2π
∫ γ+ j∞
γ− j∞h(s)
u0ω
s2 +ω2 exp(st )d s
= h( jω)u0ωe jωt
2 jω+ h(− jω)u0ωe jωt
−2 jωN∑
k=1
(s − sk )h(s)u0ωe st
s2 +ω2
(A.12)
If we are primarily interested in the steady-state response, then we inspect the expression in Eq. (A.12)
when t →∞
• If all the poles are in the left half of the complex plane, Re(sk ) < 0, and the last term (the sum)
dies out exponentially.
• If there are any poles in the right half-plane, the system is unstable and will not be interesting
from an impedance point of view anyway.
• We therefore assume that h(s) only has poles in the left half-plane.
limt→∞ y(t ) = u0
[h( jω)
e jωt
2 j−h(− jω)
e− jωt
2 j
](A.13)
Setting (physical systems always have poles symmetrically about the real axis):
h( jω) = ∣∣h( jω)∣∣eφ(− jω) = ∣∣h( jω)
∣∣e−φ (A.14)
we get
limt→∞ y(t ) = u0
∣∣h( jω)∣∣sin(ω+φ) (A.15)
The steady-state response of the system to a sine stimulus is thus also a sine. The amplitude is scaled
by the absolute value of the transfer function. The phase is shifted by the phase of the transfer func-
tion.
We now assume that u(t ) takes the role of a voltage and y(t ) that of a current (for example). For
electrical and electrochemical systems, the transfer function as written with the argument s = jω is
defined as the admittance of the system:
Appendix A 75
The admittance of an electrochemical system is the ratio of the Laplace-transformed current to the
Laplace-transformed voltage. The impedance is the inverse of the admittance.
Note that we assumed that there is response linearly connected to the stimulus in the definition of
the transfer function. If the experimental arrangement arrangement does not comply with this re-
quirement, the analysis breaks down. In electrochemistry this is usually achieved by applying a small
amplitude.
Appendix B 76
B Full derivation of Warburg impedance
B.1 Warburg infinite diffusion impedance
Diffusion to an electrode in one dimension is given by
∂c(x, t )
∂t= D
∂2c(x, t )
∂x2 (B.1)
where D is the diffusion coefficient and c(x, t ) is the time dependent concentration.
The general solution of The Laplace transformation of a derivative is given by [14]
L
f (n)(t )= sn
L
f− sn−1 f ′(0)−·· ·− f n−1 f (0) (B.2)
where n denotes the n-th derivative of f .
Combination of Eq. (B.1) and (B.2) gives
L
∂c(t )
∂t
= s L c(t )− c(0) =L
D∂2c(t )
∂x2
(B.3)
where c(x, t ) is denoted as c(t ) for simplicity. c(x, t ) can be written as a linear combination c(t ) =c(0)+ c(t ). For the stationary case, Eq. (B.3) becomes
s
=s−1c(0)︷ ︸︸ ︷L c(0)+s L c(t )− c(0) = D
=0︷ ︸︸ ︷L
∂2c(0)
∂2x
+D L
∂2c(t )
∂x2
c(0)+ sc(s)− c(0) = D
∂2c(s)
∂2x
sc(s) = Dd 2c(s)
d x2 (B.4)
where c(s) =L c(t ). This differential equation has the general solution
c(s) = A exp
(√s
Dx
)+B exp
(−
√s
Dx
)(B.5)
Boundary conditions at x = 0 and x →∞ are given by the terms
i (s)
nF= D
dc(s)
d x
∣∣∣∣x=0
(B.6)
limx→∞ c(s) → 0 (B.7)
Appendix B 77
Combination of the general solution in Eq. (B.5) and the boundary condition in Eq. (B.6) gives
i (s)
nF= D
[√s
DA exp
(√s
D0
)−
√s
DB exp
(−
√s
D0
)]i (s)
nF= D
[√s
DA−
√s
DB
]i (s)
nF
1pDs
= A−B (B.8)
The boundary condition at x →∞ results in
limx→∞ c(s) → 0 = A exp
(√s
D∞
)+B
exp(−∞)→0︷ ︸︸ ︷exp
(−
√s
D∞
)0 = A (B.9)
Thus, the constants A and B can be found by combining Eqs. (B.8) and (B.9)
B =− i (s)
nF
1pDs
; A = 0 (B.10)
The expression for c(s) is given from Eqs. (B.5) and (B.10)
c(s) =− i (s)
nF
1pDs
exp
(−
√s
Dx
)(B.11)
The time dependent potential can be found by invoking the linearity condition and expanding
E(t ) = E(0)+ E(t ) (B.12)
s L
dE
dc
= s L
˜E(t )
−E(0)
s
=0︷ ︸︸ ︷L
dE(0)
dc
+s L
dE(t )
dc
= s
=s−1E(0)︷ ︸︸ ︷L E(0)+s L
E(t )
−E(0)
L
dE(t )
dc
= E(s)
dE
dcc(s) = E(s) (B.13)
The Laplace constant is related to the angular frequency by s → jω. The local impedance, Z0, at x = 0
becomes
Z0( jω) = E( jω)
i ( jω)
∣∣∣∣x=0
= dE
dc
c( jω)
i ( jω)=−dE
dc
1
nF
1√D · jω
(B.14)
Appendix B 78
B.2 Warburg transmissive diffusion impedance
Diffusion to an electrode in one dimension is given by
∂c(x, t )
∂t= D
∂2c(x, t )
∂x2 (B.15)
where D is the diffusion coefficient and c(x, t ) is the time dependent concentration.
The general solution of The Laplace transformation of a derivative is given by [14]
L
f (n)(t )= sn
L
f− sn−1 f ′(0)−·· ·− f n−1 f (0) (B.16)
where n denotes the n-th derivative of f .
Combination of Eqs. (B.15) and (B.16) gives
L
∂c(t )
∂t
= s L c(t )− c(0) =L
D∂2c(t )
∂x2
(B.17)
where c(x, t ) is denoted as c(t ) for simplicity. c(x, t ) can be written as a linear combination c(t ) =c(0)+ c(t ). For the stationary case, Eq. (B.17) becomes
s
=s−1c(0)︷ ︸︸ ︷L c(0)+s L c(t )− c(0) = D
=0︷ ︸︸ ︷L
∂2c(0)
∂2x
+D L
∂2c(t )
∂x2
c(0)+ sc(s)− c(0) = D
∂2c(s)
∂2x
sc(s) = Dd 2c(s)
d x2 (B.18)
where c(s) =L c(t ). This differential equation has the general solution
c(s) = A exp
(√s
Dx
)+B exp
(−
√s
Dx
)(B.19)
We assume a blocking electrode with a reflective boundary condition. Boundary conditions at
x = 0 and x = L are given by the terms
i (s)
nF= D
dc(s)
d x
∣∣∣∣x=0
(B.20)
limx→L
c(s) → 0 (B.21)
Appendix B 79
Combination of Eq. (B.19) and boundary condition at x = L from Eq. (B.21) gives
limx→L
c(s) = 0 = A exp
(√s
DL
)+B exp
(−
√s
DL
)
A =−Bexp
(−
√sD L
)exp
(√sD L
) (B.22)
Boundary condition at x = 0 from Eq. (B.20) with the result in Eq. (B.22) gives B
i (s)
nF= D
[√s
DA exp
(√s
D0
)−
√s
DB exp
(−
√s
D0
)]i (s)
nF
1pDs
=−B
exp(−
√sD L
)exp
(√sD L
) +1
i (s)
nF
1pDs
=−B
exp(−
√sD L
)+exp
(√sD L
)exp
(√sD L
)
i (s)
nF
1pDs
=−B
2cosh
(√s
DL
)exp
(√sD L
)
−B = i (s)
nF
1pDs
exp(√
sD L
)2cosh
(√s
DL
) (B.23)
Combination of Eqs. (B.22) and (B.23) gives
A = i (s)
nF
1pDs
exp(√
sD L
)2cosh
(√s
DL
) exp(−
√sD L
)exp
(√sD L
)= i (s)
nF
1pDs
1
2cosh
(√s
DL
) exp
(−
√s
DL
)(B.24)
Thus, the specific solution of Eq. (B.19) at the interface x = 0 for the transmissive boundary conditions
is
c(s) =− i (s)
nF
1pDs
2sinh(√
sD L
)2cosh
(√sD L
) =− i (s)
nF
1pDs
tanh
(√s
D
)(B.25)
The time dependent potential can be found by invoking the linearity condition above, and the trans-
missive Warburg diffusion impedance becomes
ZW,trans( jω) = LE
( jω)
Li
( jω)=−dE
dc
1
nF
1pDs
tanh
√jω
DL
(B.26)
Appendix B 80
B.3 Warburg reflective diffusion impedance
Diffusion to an electrode in one dimension is given by
∂c(x, t )
∂t= D
∂2c(x, t )
∂x2 (B.27)
where D is the diffusion coefficient and c(x, t ) is the time dependent concentration.
The general solution of The Laplace transformation of a derivative is given by [14]
L
f (n)(t )= sn
L
f− sn−1 f ′(0)−·· ·− f n−1 f (0) (B.28)
where n denotes the n-th derivative of f .
Combination of Eqs. (B.27) and (B.28) gives
L
∂c(t )
∂t
= s L c(t )− c(0) =L
D∂2c(t )
∂x2
(B.29)
where c(x, t ) is denoted as c(t ) for simplicity. c(x, t ) can be written as a linear combination c(t ) =c(0)+ c(t ). For the stationary case, Eq. (B.29) becomes
s
=s−1c(0)︷ ︸︸ ︷L c(0)+s L c(t )− c(0) = D
=0︷ ︸︸ ︷L
∂2c(0)
∂2x
+D L
∂2c(t )
∂x2
c(0)+ sc(s)− c(0) = D
∂2c(s)
∂2x
sc(s) = Dd 2c(s)
d x2 (B.30)
where c(s) =L c(t ). This differential equation has the general solution
c(s) = A exp
(√s
Dx
)+B exp
(−
√s
Dx
)(B.31)
We assume a blocking electrode with a reflective boundary condition. Boundary conditions at
x = 0 and x = L are given by the terms
i (s)
nF= D
∂dc(s)
∂x
∣∣∣∣x=0
(B.32)
0 = ∂dc(s)
∂x
∣∣∣∣x=L
(B.33)
Appendix B 81
Combination of Eq. (B.31) and boundary condition at x = L from Eq. (B.33) gives
∂dc(s)
∂x
∣∣∣∣x=L
= 0 =√
s
D
[A exp
(√s
DL
)+B exp
(−
√s
DL
)]
A = Bexp
(−
√sD L
)exp
(√sD L
) (B.34)
Boundary condition at x = 0 from Eq. (B.32) with the result in Eq. (B.34) gives B
i (s)
nF= D
[√s
DA exp
(√s
D0
)−
√s
DB exp
(−
√s
D0
)]i (s)
nF
1pDs
= B
exp(−
√sD L
)exp
(√sD L
) −1
i (s)
nF
1pDs
= B
exp(−
√sD L
)−exp
(√sD L
)exp
(√sD L
)
i (s)
nF
1pDs
= B
−2sinh
(√s
DL
)exp
(√sD L
)
B =− i (s)
nF
1pDs
exp(√
sD L
)2sinh
(√s
DL
) (B.35)
Combination of Eqs. (B.34) and (B.35) gives
A =− i (s)
nF
1pDs
exp(√
sD L
)2sinh
(√s
DL
) exp(−
√sD L
)exp
(√sD L
)=− i (s)
nF
1pDs
1
2sinh
(√s
DL
) exp
(−
√s
DL
)(B.36)
Thus, the specific solution of Eq. (B.31) at the interface x = 0 for the transmissive boundary conditions
is
c(s) =− i (s)
nF
1pDs
2cosh(√
sD L
)2sinh
(√sD L
) =− i (s)
nF
1pDs
coth
(√s
D
)(B.37)
The time dependent potential can be found by invoking the linearity condition above, and the reflec-
tive Warburg diffusion impedance becomes
ZW,refl( jω) = LE
( jω)
Li
( jω)=−dE
dc
1
nF
1√D · jω
coth
√jω
DL
(B.38)
Appendix C 82
C Full derivation of Gerischer impedance
C.1 Transmissive Gerischer impedance
Diffusion to an electrode one dimension is given by
∂c(x, t )
∂t= D
∂2c(x, t )
∂x2 (C.1)
where D is the diffusion coefficient and c(x, t ) is the time dependent concentration.
The general solution of The Laplace transformation of a derivative is given by [14]
L
f (n)(t )= sn
L
f− sn−1 f ′(0)−·· ·− f n−1 f (0) (C.2)
where n denotes the n-th derivative of f .
Combination of Eqs. (C.14) and (C.15) gives
L
∂c(t )
∂t
= s L c(t )− c(0) =L
D∂2c(t )
∂x2
(C.3)
where c(x, t ) is denoted as c(t ) for simplicity. c(x, t ) can be written as a linear combination c(t ) =c(0)+ c(t ). For the stationary case, Eq. (C.16) becomes
s
=s−1c(0)︷ ︸︸ ︷L c(0)+s L c(t )− c(0) = D
=0︷ ︸︸ ︷L
∂2c(0)
∂2x
+D L
∂2c(t )
∂x2
c(0)+ sc(s)− c(0) = D
∂2c(s)
∂2x
sc(s) = Dd 2c(s)
d x2 (C.4)
where c(s) = L c(t ). We introduce a sink term with an effective rate term that includes a non-
faradaic reaction in the bulk, and the modified Laplace transformed diffusion equation becomes
sc(s) = Dd 2c(s)
d x2 −kc(s) (C.5)
This differential equation has the general solution
c(s) = A exp
√s +k
Dx
+B exp
−√
s +k
Dx
(C.6)
We assume a leaking electrode with a transmissive boundary condition. Boundary conditions at
Appendix C 83
x = 0 and x = L are given by the terms
i (s)
nF= D
dc(s)
d x
∣∣∣∣x=0
(C.7)
limx→L
c(s) → 0 (C.8)
Combination of Eq. (C.6) and boundary condition at x = L from Eq. (C.8) gives
limx→L
c(s) = 0 = A exp
√s +k
DL
+B exp
−√
s +k
DL
A =−Bexp
(−
√s+k
D L
)exp
(√s+k
D L
) (C.9)
Boundary condition at x = 0 from Eq. (C.7) with the result in Eq. (C.9) gives B
i (s)
nF= D
√s +k
DA exp
√s +k
D0
−√
s +k
DB exp
−√
s +k
D0
i (s)
nF
1pD(s +k)
=−B
exp
(−
√s+k
D L
)exp
(√s+k
D L
) +1
i (s)
nF
1pD(s +k)
=−B
exp
(−
√s+k
D L
)+exp
(√s+k
D L
)exp
(√s+k
D L
)
i (s)
nF
1pD(s +k)
=−B
2cosh
(√s +k
DL
)
exp
(√s+k
D L
)
−B = i (s)
nF
1pD(s +k)
exp
(√s+k
D L
)2cosh
(√s +k
DL
) (C.10)
Combination of Eqs. (C.9) and (C.10) gives
A = i (s)
nF
1pD(s +k)
exp
(√s+k
D L
)2cosh
(√s +k
DL
) exp
(−
√s+k
D L
)exp
(√s+k
D L
)
Appendix C 84
= i (s)
nF
1pD(s +k)
1
2cosh
(√s +k
DL
) exp
−√
s +k
DL
(C.11)
Thus, the specific solution of Eq. (C.6) at the interface x = 0 for the transmissive boundary conditions
is
c(s) =− i (s)
nF
1pD(s +k)
2sinh
(√s+k
D L
)2cosh
(√s+k
D L
) =− i (s)
nF
1pD(s +k)
tanh
√s +k
D
(C.12)
The time dependent potential can be found by invoking the linearity condition above, and the trans-
missive Gerishcer impedance becomes
ZG ,trans( jω) = LE
( jω)
Li
( jω)=−dE
dc
1
nF
1√D( jω+k)
tanh
√jω+k
DL
(C.13)
Appendix C 85
C.2 Reflective Gerischer impedance
Diffusion to an electrode one dimension is given by
∂c(x, t )
∂t= D
∂2c(x, t )
∂x2 (C.14)
where D is the diffusion coefficient and c(x, t ) is the time dependent concentration.
The general solution of The Laplace transformation of a derivative is given by [14]
L
f (n)(t )= sn
L
f− sn−1 f ′(0)−·· ·− f n−1 f (0) (C.15)
where n denotes the n-th derivative of f .
Combination of Eqs. (C.14) and (C.15) gives
L
∂c(t )
∂t
= s L c(t )− c(0) =L
D∂2c(t )
∂x2
(C.16)
where c(x, t ) is denoted as c(t ) for simplicity. c(x, t ) can be written as a linear combination c(t ) =c(0)+ c(t ). For the stationary case, Eq. (C.16) becomes
s
=s−1c(0)︷ ︸︸ ︷L c(0)+s L c(t )− c(0) = D
=0︷ ︸︸ ︷L
∂2c(0)
∂2x
+D L
∂2c(t )
∂x2
c(0)+ sc(s)− c(0) = D
∂2c(s)
∂2x
sc(s) = Dd 2c(s)
d x2 (C.17)
where c(s) = L c(t ). We introduce a sink term with an effective rate term that includes a non-
faradaic reaction in the bulk, and the modified Laplace transformed diffusion equation becomes
sc(s) = Dd 2c(s)
d x2 −kc(s) (C.18)
This differential equation has the general solution
c(s) = A exp
√s +k
Dx
+B exp
−√
s +k
Dx
(C.19)
We assume a blocking electrode with a reflective boundary condition. Boundary conditions at
Appendix C 86
x = 0 and x = L are given by the terms
i (s)
nF= D
dc(s)
d x
∣∣∣∣x=0
(C.20)
0 = dc(s)
d x
∣∣∣∣x=L
(C.21)
Combination of Eq. (C.19) and boundary condition at x = L from Eq. (C.21) gives
dc(s)
d x
∣∣∣∣x=L
= 0 =√
s +k
D
A exp
√s +k
DL
+B exp
−√
s +k
DL
A = Bexp
(−
√s+k
D L
)exp
(√s+k
D L
) (C.22)
Boundary condition at x = 0 from Eq. (C.20) with the result in Eq. (C.22) gives B
i (s)
nF= D
√s +k
DA exp
√s +k
D0
−√
s +k
DB exp
−√
s +k
D0
i (s)
nF
1pD(s +k)
= B
exp
(−
√s+k
D L
)exp
(√s+k
D L
) −1
i (s)
nF
1pD(s +k)
= B
exp
(−
√s+k
D L
)−exp
(√s+k
D L
)exp
(√s+k
D L
)
i (s)
nF
1pD(s +k)
= B
−2sinh
(√s +k
DL
)
exp
(√s+k
D L
)
B =− i (s)
nF
1pD(s +k)
exp
(√s+k
D L
)2sinh
(√s +k
DL
) (C.23)
Combination of Eqs. (C.22) and (C.23) gives
A =− i (s)
nF
1pD(s +k)
exp
(√s+k
D L
)2sinh
(√s +k
DL
) exp
(−
√s+k
D L
)exp
(√s+k
D L
)
Appendix C 87
=− i (s)
nF
1pD(s +k)
1
2sinh
(√s +k
DL
) exp
−√
s +k
DL
(C.24)
Thus, the specific solution of Eq. (C.19) at the interface x = 0 for the transmissive boundary conditions
is
c(s) =− i (s)
nF
1pD(s +k)
2cosh
(√s+k
D L
)2sinh
(√s+k
D L
) =− i (s)
nF
1pD(s +k)
coth
√s +k
D
(C.25)
The time dependent potential can be found by invoking the linearity condition above, and the reflec-
tive Gerischer impedance becomes
ZG ,refl( jω) = LE
( jω)
Li
( jω)=−dE
dc
1
nF
1√D( jω+k)
coth
√jω+k
DL
(C.26)
Appendix D 88
D Electrochemical impedance for a semiconducting film electrode
D.1 The photoelectrochemical film electrode
The charge balance for the charge-neutral bulk is
n −Nd − (p −Na) = 0 (D.1)
where n, p, Nd , and Na are electron, hole, donor, and acceptor concentrations, respectively.
We introduce the nomenclature
c+ = p −Na (D.2)
c− = n −Nd (D.3)
which gives the charge balance from (2.33) as
z+ν+c++ z−ν−c− = 0 (D.4)
We employ the dilute-solution approximation in which the flux density vector of species i in the film,
N i , is described by
N i =−zi ui F ci∇Φ1 −Di∇ci (D.5)
where zi is the charge number, ui the mobility, ci the concentration, and Di the diffusion coefficient
of species i .
The material balance∂ci
∂t=−∇N i +Ri (D.6)
becomes, by combination of (2.34), (2.35), and (2.36)
∂c+∂t
= z+u+F c+∂2Φ1 +D+∇c+ (D.7)
∂c−∂t
= z−u−F c−∂2Φ1 +D−∇c− (D.8)
From (D.4) with z+ =−z− we get
c+ν+ = c−ν− (D.9)
where ν+ and ν− are the numbers of cations and anions produced by the dissociation of one molecule
of electrolyte, respectively. The concentration of the electrolyte, c, is then defined by
c = c+ν+
= c−ν−
(D.10)
Appendix D 89
Combined with the material balances (D.7) and (D.8) gives
∂c
∂t= z+u+F∇(c∇Φ1)+D+∇2c +R+ (D.11)
∂c
∂t= z−u−F∇(c∇Φ1)+D−∇2c +R− (D.12)
The potential gradient is eliminated by subtracting (D.12) from (D.12)
0 = (z+u+− z−u−)F∇(c∇Φ1)+ (D+−D−)∇2c + (R+−R−) (D.13)
F∇(c∇Φ1) = D−−D+z+u+− z−u−
∇2c + R−−R+z+u+− z−u−
(D.14)
Inserting (D.14) in (D.11) gives
∂c
∂t= z+u+
(D−−D+
z+u+− z−u−∇2c + R−−R+
z+u+− z−u−
)+D+∇2c +R+
∂c
∂t= z+u+D−+ z+u−D+
z+u+− z−u−∇2c + z+u+R−− z+u+R+
z+u+− z−u−+D+∇2c +R+
∂c
∂t= z+u+D−− z+u+D++ (z+u+D+− z−u−D+)
z+u+− z−u−∇2c + z+u+R−− z+u+R++ (z+u+R+− z−u−R+)
z+u+− z−u−∂c
∂t= z+u+D−− z−u−D+
z+u+− z−u−∇2c + z+u+R−− z −−u−R+
z+u+− z−u−∂c
∂t= D∇2c +Rc (D.15)
where
D = z+u+D−− z−u−D+z+u+− z−u−
, Rc = z+u+R−− z−u−R+z+u+− z−u−
(D.16)
A harmonic perturbation is assumed applied in the driving force for intercalation for t > 0. The con-
centration may be written
c = c(r ,0)+ c(r , t ) (D.17)
The Laplace transform of the time derivative can be written as
L
∂c
∂t
= s L c−
=0︷︸︸︷c(0) = s L c = sc(s) (D.18)
Combination of (D.18) and (D.15) gives in an one-dimensional electrode system
sc(s) = D∂2c(s)
∂x2 +LRc
(D.19)
where s → jω.
We introduce sink term corresponding to recombination or trapping, and a source term due to
Appendix D 90
photon absorption. The diffusion equation becomes
sc(s) = D∂2c(s)
∂x2 −kc(s)+L
I0αexp−αx (D.20)
where the rate constant k includes all rate constants and any other pre-factors stemming from lin-
earization of R+ and R−.
The current in the electrolyte is given by
i = F∑
izi Ni (D.21)
Thus, the net flux density is ∑i
Ni = i
F∑
i zi(D.22)
From (D.5) we see that Ni can be found by integration of (D.13). Thus, we get
− i
z+ν+F= (z+u+− z−u−)F c∇Φ1 + (D+−D−)∇c = N+ (D.23)
where Ri from (D.5) cancels with that of (D.13).
We assume an electrode in which electronic species are blocked at the solution interface x = 0
(here holes, h+), and we get the boundary conditions
N+x = 0 =−z+u+Fν+c∂Φ
∂x−D+ν+
∂c
∂x(D.24)
N−x = ix
z−F=−z−u−Fν−c
∂Φ
∂x−D−ν−
∂c
∂x(D.25)
where Ni x is the flux of species i in x-direction.
The flux of negative carriers in the x-direction is related to the faradaic current as
i f =−z−F N−x (D.26)
because a vacancy flux in the x-direction represents oxidation of the oxide. Thus, ix =−i f . We elimi-
nate the potential gradient by subtracting (D.25) from (D.24)
− ix
z−F=−(z+u+Fν+− z−u−Fν−)c
∂Φ
∂x− (D+ν+−D−ν−)
∂c
∂x
− ix
z−F=−F c
∂Φ
∂x(z+u+ν+− z−u−ν−)− (D+ν+−D−ν−)
∂c
∂x
− ix
z−F+ (D+ν+−D−ν−)
∂c
∂x=−F c
∂Φ
∂x(z+u+ν+− z−u−ν−)
Appendix D 91
−F c∂Φ
∂x=
[− ix
z−F+ (D+ν+−D−ν−)
∂c
∂x
](z+u+ν+− z−u−ν−)−1 (D.27)
Insertion of (D.27) into (D.24) gives
0 = z+u+[− ix
z−F+ (D+ν+−D−ν−)
∂c
∂x
](z+u+ν+− z−u−ν−)−D+
∂c
∂x
D+(z+u+ν+− z−u−ν−)∂c
∂x=−z+u+ix
z−F+ z+u+(D+ν+−D−ν−)
∂c
∂x
(D+z+u+ν+−D+z−u−ν−− z+u+D+ν++ z+u+D−ν−)∂c
∂x=−z+u+ix
z−F
(z+u+D−ν−−D+z−u−ν−)∂c
∂x= z+u+
ix
z−ν−F
− ix
z−ν−F= z+u+D−−D+z−u−
z+u+∂c
∂xi f
z−ν−F=−z−u−D+− z+u+D−
z+u+∂c
∂x(D.28)
We introduce the transport number
t+ = 1− t− = z+u+z+u+− z−u−
(D.29)
Thus, (D.28) becomesi f
z−ν−F= D
1− t−∂c
∂x(D.30)
We then do the Laplace transformation
L
i f
z−ν−F
=L
D
1− t−∂c
∂x
Li f
z−ν−F
= D
1− t−∂L c
∂x
∣∣∣∣x=0
(D.31)
For the substrate interface we get similar boundary conditions
N+x = ix
z+F=−z+u+ν+c
∂Φ
∂x−D+ν+
∂c
∂x(D.32)
N−x = 0 =−z−u−Fν−c∂Φ
∂x−D−ν−
∂c
∂x(D.33)
which gives the boundary condition at x = L, by similar approach above, as
Li f
z+ν+F
= D
1− t+∂L c
∂x
∣∣∣∣x=L
(D.34)
Boundary conditions (D.31) and (D.34) are then used to solve the diffusion equation (D.20). This is
Appendix D 92
solved in with the Maple software, with the solution
e
√s+k
D x(L I0DFα2ν+ν−z+z−e−
√s+k
D L −L I0DFα2ν+ν−z+z−e−αL
+Dα2L i f ν+t−z+e−
√s+k
D L −Dα2L i f ν+z+e−
√s+k
D L −Dα2L i f ν−t+z−
+Dα2L i f ν−z−−L i kν+t−z+e−
√s+k
D L −L i f ν+st−z+e−√
s+kD L +L i f kν+z+e−
√s+k
D L
+L i ν+sz+e−√
s+kD L +L i f kν−u−z−+L i f ν−su−z−−L i f kν−z−−L i f ν−sz−
)/
F√
D(s +k)ν+ν−z+z−(Dα2e
√s+k
D L −Dα2e−√
s+kD L −ke−
√s+k
D L − se−√
s+kD
+ke−√
s+kD L + se−
√s+k
D L)
+e−
√s+k
D x(L I0DFα2ν+ν−z+z−e
√s+k
D L −L I0DFα2ν+ν−z+z−e−αL
+Dα2L i f ν+t−z+e
√s+k
D L −Dα2L i f ν+z+e
√s+k
D L −Dα2L i f ν−t+z−
+Dα2L i f ν−z−−L i kν+t−z+e
√s+k
D L −L i f ν+st−z+e
√s+k
D L +L i f kν+z+e
√s+k
D L
+L i ν+sz+e
√s+k
D L +L i f kν−u−z−+L i f ν−su−z−−L i f kν−z−−L i f ν−sz−)/
F
√D(s +k)(Dα2 −k − s)ν+ν−z+z−
(e
√s+k
D L −e−√
s+kD L
)− L I0α2
Dα2 −k − se−αx
To find the appropriate expression, we write this as
L c = numerator 1
denominator 1+ numerator 2
denominator 2− L I0α2
Dα2 −k − se−αx (D.35)
and solve each numerator and denominator individually.
Denominator 1
From the maple solution, denominator 1 in (D.35) is
F√
D(s +k)ν+ν−z+z−(Dα2e
√s+k
D L −Dα2e−√
s+kD L
−ke−√
s+kD L +ke−
√s+k
D L − se−√
s+kD + se−
√s+k
D L)
Rearranging gives
F√
D(s +k)ν+ν−z+z−[
Dα2(e
√s+k
D L −e−√
s+kD L
)−k
(e
√s+k
D L −e−√
s+kD L
)−s
(e
√s+k
D L −e−√
s+kD L
)]
Appendix D 93
By using the definition 2sinh x = ex −e−x we get the denominator
F√
D(s +k)z+z−ν+ν−
2Dα2 sinh
√s +k
DL
−2k sin2sinh
√s +k
DL
−2s sin2sinh
√s +k
DL
2F√
D(s +k)z+z−ν+ν−(Dα2 −k − s)sinh
√s +k
DL
(D.36)
Denominator 2
From the maple solution, denominator 2 in (D.35) is
F√
D(s +k)(Dα2 −k − s)z+z−ν+ν−(e
√s+k
D L −e−√
s+kD L
)
We use the definition of sinh, and get
2F√
D(s +k)z+z−ν+ν−(Dα2 −k − s)sinh
√s +k
DL
(D.37)
Numerators
From (D.36) and (D.37) we see that the denominators are equal. Thus, we get one fraction where
numerator 1 and numerator 2 are added. By using the relation
e
√s+k
x e−√
s+kD L = e
√s+k
D (x−L) (D.38)
and the definition 2cosh x = ex +e−x we get the numerator
2L I0DFα2ν+ν−z+z− cosh
√s +k
D(x −L)
−2L I −0DFe−αLα2ν+ν−z+z− cosh
√s +k
Dx
+2Dα2
L i ν+t−z+ cosh
√s +k
D(x −L)
−2Dα2
L i f ν+z+ cosh
√s +k
D(x −L)
Appendix D 94
−2Dα2L i f ν−t+z− cosh
√s +k
Dx
+2Dα2
L i f ν−z− cosh
√s +k
Dx
−2L i f kν+t−z+ cosh
√s +k
D(x −L)
−2L i f ν+st−z+ cosh
√s +k
D(x −L)
+2L i f kν+z+ cosh
√s +k
D(x −L)
+2L i f ν+sz+ cosh
√s +k
D(x −L)
+i kν−t+z− cosh
√s +k
Dx
+L i f ν−st+z− cosh
√s +k
Dx
−L i f kν−z− cosh
√s +k
Dx
+L i f ν−sz− cosh
√s +k
Dx
Rearranging gives
2cosh
√s +k
D(x −L)
L I0DFα2ν+ν−z+z−+Dα2
L i f ν+t−z+−Dα2L i f
ν+z+−L i f kν+t−z+−L i f st−z++L i f kν+z+
+L i ν+sz+
+2cosh
√s +k
Dx
−L I0DFα2ν+ν−z+z−e−αL −Dα2L i f ν−t+z−
+Dα2L i f ν−z−+L i kν−t+z−+L i f ν−st+z−
−L i f kν−z−−L i f ν−sz−
This can be written as
2A cosh
√s +k
D(x −L)
+2B cosh
√s +k
Dx
(D.39)
and we calculate each constant, A and B , separately.
For constant A, by rearranging, we get
A =L i f ν+z+
(F Dα2 L I0ν−z−
L i f +Dα2t−−Dα2 −kt−− st−+k + s
)(D.40)
Appendix D 95
For constant A, by rearranging, we get
B =L i f ν−z−
(−F Dα2 L I0ν+z+
L i f e−αL −Dα2t++Dα2 +kt++ st+−k − s
)(D.41)
Solution of diffusion equation
From these expressions we see that (D.35) can be written as
L c =2A cosh
(√s+k
D (x −L)
)denominator
+2B cosh
(√s+k
D x
)denominator
− L I0α2
Dα2 −k − se−αx (D.42)
which gives the result
L c =L i f ν+z+
(F Dα2 L I0ν−z−
L i f +Dα2t−−Dα2 −kt−− st−+k + s
)cosh
(√s+k
D (x −L)
)Fp
D(s +k)z+z−ν+ν−(Dα2 −k − s)sinh
(√s+k
D L
)
+L i f ν−z−
(−F Dα2 L I0ν+z+
L i f e−αL −Dα2t++Dα2 +kt++ st+−k − s
)cosh
(√s+k
D x
)Fp
D(s +k)z+z−ν+ν−(Dα2 −k − s)sinh
(√s+k
D L
)− L I0α2
Dα2 −k − se−αx
L c =−L i f cosh
(√s+k
D (x −L)
)Fp
D(s +k)sinh
(√s+k
D L
)−F Dα2 L I0ν−z−
L i f −Dα2t−+Dα2 +kt−+ st−−k − s
(Dα2 −k − s)z−ν−
−L i f cosh
(√s+k
D x
)Fp
D(s +k)sinh
(√s+k
D L
) F Dα2 L I0ν+z+e−αL
L i f +Dα2t+−Dα2 −kt+− st++k + s
(Dα2 −k − s)z+ν+
− L I0α2
Dα2 −k − se−αx
L c =− L i f
Fp
D(s +k)sinh
(√s+k
D L
) (1− t−z−ν−
− F Dα2 L I0
L i f (Dα2 −k − s)
)cosh
√s +k
D(x −L)
+ L i f
Fp
D(s +k)sinh
(√s+k
D L
) (1− t+z+ν+
− F D L I0
L i f (Dα2 −k − s)e−αL
)cosh
√s +k
Dx
− L I0α2
Dα2 −k − se−αx
Appendix D 96
L c =− Li f
F
√D( jω+k)sinh
(√jω+k
D L
)
·K1(ω)cosh
√jω+k
D(x −L)
−K2(ω)cosh
√jω+k
Dx
− L
I0
α
Dα2 −k − jωexp(−αx)
(D.43)
where
K1(ω) = 1− t−z−ν−
− F D L
I0α2
Li f
(Dα2 −k − jω)
(D.44)
K2(ω) = 1− t+z+ν+
− F D L
I0α2
Li f
(Dα2 −k − jω)
exp(−αL) (D.45)
D.2 Electrochemical impedance
For the electrochemical impedance we assume that the Laplace transform of the light intensity is
zero. That is
L I0( jω) = 0, I0(x,0) = 0 (D.46)
We assume the faradaic current, i f , to be a function of the proton concentration in the oxide film and
the potential differenceΦ1 −Φ2 as
i f =(∂i f
∂c
)Φ1−Φ2
c(0)+[
∂i f
∂(Φ1 −Φ2)
]c,x=0
[Φ1(0)−Φ2(0)
](D.47)
= A0c(0)+B0[Φ1(0)−Φ2(0)
](D.48)
where
A0 =(∂i f
∂c
)Φ1−Φ2
, B0 =[
∂i f
∂(Φ1 −Φ2)
]c,x=0
(D.49)
The Laplace transform of i f is thus
L i f = A0 L c(0)+B0 L Φ1(0)−Φ2(0) (D.50)
The local admittance at the solution interface, Y0, at x = 0 is given by the expression
Y0 =L i f
L Φ1(0)−Φ2(0)(D.51)
Appendix D 97
The Laplace transform of the current can be found by using the result in Eq. (2.58)
L c(0) =− L i f
Fp
D(s +k)sinh(
s+kD L
) ·K1(ω)cosh
√s +k
DL
−K2(ω)cosh(0)
where we have used that cosh(−x) = cosh(x). Inserting K1(ω) and K2(ω) gives
L c(0) =− L i f
Fp
D(s +k)sinh
(√s+k
D L
)1− t−
z−ν−cosh
√s +k
DL
− 1− t+z+ν+
(D.52)
which combined with (D.48) gives
L c(0)
Z ′D
= A0 L c(0)+B0 L Φ1(0)−Φ2(0)
L c(0) = B0 L Φ1(0)−Φ2(0)1
1Z ′
D− A0
(D.53)
where
Z ′D =− 1
Fp
D(s +k)sinh
(√s+k
D L
)1− t−
z−ν−cosh
√s +k
DL
− 1− t+z+ν+
(D.54)
Combination of (D.50) and (D.53) gives
L i f = A0
B0 L Φ1(0)−Φ2(0)1
1Z ′
D− A0
+B0 L Φ1(0)−Φ2(0)
= B0 L Φ1(0)−Φ2(0)
1(1
Z ′D− A0
)1
A0
+1
= B0 L Φ1(0)−Φ2(0)
11−Z ′
D A0
Z ′D A0
+1
= B0 L Φ1(0)−Φ2(0)
(Z ′
D A0
1−Z ′D A0
+ 1−Z ′D A0
1−Z ′D A0
)= B0 L Φ1(0)−Φ2(0)
(1
1− A0Z ′D
)(D.55)
Thus, (D.51) becomes
Y0 =B0 L Φ1(0)−Φ2(0) 1
1−A0 Z ′D
L Φ1(0)−Φ2(0)= B0
1− A0Z ′D
(D.56)
Appendix D 98
The local admittance at the metal-oxide interface, YL at x = L is given by the expression
YL = L i f
L ΦM (L)−Φ1(L)(D.57)
where ΦM (L) is the potential at the metal, and i f at x = L is given by
i f =(∂i f
∂c
)ΦM−Φ1
c(L)+[
∂i f
∂(ΦM −Φ1)
]c,x=L
[ΦM (L)−Φ1(L)
](D.58)
= AL c(L)+BL[ΦM (L)−Φ1(L)
](D.59)
where
AL =(∂i f
∂c
)ΦM−Φ1
, BL =[
∂i f
∂(ΦM −Φ1)
]c,x=L
(D.60)
The Laplace transform of i f is thus
L i f = AL L c(L)+BL L ΦM (L)−Φ1(L) (D.61)
We use the result in Eq. (2.58) at x = L to find an expression for L i f
L c(L) =− L i f
Fp
D(s +k)sinh
(√s+k
D L
) ·K1(ω)cosh(0)−K2(ω)cosh
√s +k
DL
(D.62)
Inserting K1(ω) and K2(ω) gives
L c(L) =− L i f
Fp
D(s +k)sinh
(√s+k
D L
)1− t−
z−ν−− 1− t+
z+ν+cosh
√s +k
DL
(D.63)
which combined with (D.59) gives
L c(L)
Z ′′D
= AL L c(L)+BL L ΦM (L)−Φ1(L)
L c(L) = BL L ΦM (L)−Φ1(L)1
1Z ′′
D− AL
(D.64)
where
Z ′′D =− 1
Fp
D(s +k)sinh
(√s+k
D L
)1− t−
z−ν−− 1− t+
z+ν+cosh
√s +k
DL
(D.65)
Appendix D 99
Combination of (D.61) and (D.64) gives
L i f = AL
BL L ΦM (L)−Φ1(L)1
1Z ′′
D− AL
+BL L ΦM (L)−Φ1(L)
= BL L ΦM (L)−Φ1(L)
1(1
Z ′′D− AL
)1
AL
+1
= BL L ΦM (L)−Φ1(L)
11−Z ′′
D AL
Z ′′D AL
+1
= BL L ΦM (L)−Φ1(L)
(Z ′′
D AL
1−Z ′′D AL
+ 1−Z ′D AL
1−Z ′D AL
)= BL L ΦM (L)−Φ1(L)
(1
1− AL Z ′′D
)(D.66)
Thus, (D.57) becomes
YL =BL L ΦM (L)−Φ1(L) 1
1−AL Z D ′′
L ΦM (L)−Φ1(L)= BL
1− AL Z ′′D
(D.67)
To relate the potential difference L Φ1(0)−Φ2(0) to the potential as measured with respect to a
reference electrode, we assume the iridium oxide system and the speciation given above. We write
the proton-transfer reaction as
HxH*)V′
H +H+(aq) (D.68)
The interface to the electrode support is written as
0*) h·+e ′ (D.69)
We assume (D.69) to be in equilibrium. The electrochemical potentials of electrodes in the connecting
leads, µe is thus
−µe =µh =µ0h +RT ln[c(L)]+FΦ1(L) (D.70)
The measured potential, V , is related to the electrochemical potential in a reference electrode, µrefe as
µe −µrefe =−FV (D.71)
We assume that µrefe can e measured by a reference electrode so that its value is representative of Φ2
at x = 0 plus a constant. The amplitude and phase of the measured electrode potential is thus derived
by taking the time dependent part of the linearized Eq. (D.70)
FV =µrefe +µ0
h +RT ln[c(L)]+FΦ1(L)
Appendix D 100
=−FΦ2(0)+C +µ0h +RT ln[c(L)]+FΦ1(L)
FV = RT
cec(L)+F Φ1(L)−F Φ2(0) (D.72)
where C is a constant as explained above.
The potential Φ1 can be related to the concentration c and the faradaic current i f through
i f
z+ν+F= (z+u+− z−u−)F
(ce∂Φ1
∂x+ c
∂Φ1e
∂x
)+ (D+−D−)
∂c
∂x(D.73)
We expect ce to be independent of the position in the electrode. For zero current we get
0 = (z+u+− z−u−)F
(ce∂Φ1
∂x+ c
∂Φ1e
∂x
)+ (D+−D−)
∂c
∂x
F∂Φ1e
∂x=− D+−D−
z+u+− z−u−∂ lnce
∂x︸ ︷︷ ︸=0
(D.74)
which implies∂Φ1e
∂x= 0 (D.75)
Putting this result into (D.72) and integrating for x = 0 to x = L gives
i f
z+ν+F= (z+u+− z−u−)F
(ce∂Φ1
∂x
)+ (D+−D−)
∂c
∂x
∂Φ1
∂x=− D+−D−
F ce (z+u+− z−u−)
∂c
∂x+ i f
F 2ce (z+u+− z−u−)(z+ν+)
∂Φ1
∂x=− D+−D−
F ce (z+u+− z−u−)
∂c
∂x+ i f
F 2 ∑ci z2
i ui∫ x=L
x=0∂Φ1 =− D+−D−
F ce (z+u+− z−u−)
∫ x=L
x=0∂c − i f
F 2 ∑ci z2
i ui
∫ x=L
x=0∂x
Φ1(L)− Φ1(0) =− D+−D−F ce (z+u+− z−u−)
[c(L)− c(0)]+ i f L
κ
Φ1(L) = Φ1(0)− D+−D−F ce (z+u+− z−u−)
[c(L)− c(0)]+ i f L
κ(D.76)
where κ= F 2 ∑ci z2
i ui .
With the result from (D.76), Eq. E F tilde V 1 becomes
FV = RT
cec(L)+F Φ1(0)− D+−D−
ce (z+u+− z−u−)[c(L)− c(0)]+ F i f L
κ−F Φ2(0)
FV = F Φ1(0)−F Φ2(0)
+[
RT
ce− D+−D−
ce(z+u+− z−u−)
]c(L)+ D+−D−
ce (z+u+− z−u−)c(0)+ F i f L
κ
Appendix D 101
We introduce the Nernst-Einstein relation ,Di = RTui , and get
FV = F Φ1(0)−F Φ2(0)
+[
RT
ce− RTu+−RTu−
ce (z+u+− z−u−)
]c(L)+ D+−D−
ce (z+u+− z−u−)c(0)+ F i f L
κ
FV = F Φ1(0)−F Φ2(0)
+
z+D+︷ ︸︸ ︷
RT (z+u+)−z−D−︷ ︸︸ ︷
RT (z−u−)−D+︷ ︸︸ ︷
RTu++D−︷ ︸︸ ︷
RTu−ce (z+u+− z−u−)
c(L)+ D+−D−ce (z+u+− z−u−)
c(0)+ F i f L
κ
FV = F Φ1(0)−F Φ2(0)
+[
(z+−1)D+− (z−−1)D−ce (z+u+− z−u−)
]c(L)+ D+−D−
ce (z+u+− z−u−)c(0)+ F i f L
κ(D.77)
The faradaic impedance is given by
Z f = L V
L i f (D.78)
We use the expression in (D.77) to find L V
L V =L Φ1(0)−Φ2(0) (D.79)
+ (z+−1)D+− (z−−1)D−F ce (z+u+− z−u−)
L c(L) (D.80)
+ D+−D−F ce (z+u+− z−u−)
L c(0) (D.81)
+ L i f L
κ(D.82)
Eq. (D.79) is found by the local impedance at x = 0, Y0, from Eqs. (D.51) and (D.56)
Y0 =L i f
L Φ1(0)−Φ2(0)
L Φ1(0)−Φ2(0) = L i f
Y0= L i f
B01−A0 Z ′
D
(D.83)
From (D.63) and (D.65) it can be shown that
L c(L) =L i f Z ′′D (D.84)
From (D.52) and (D.54) it can be shown that
L c(0))L i f Z ′D (D.85)
Appendix D 102
Thus, L V becomes
L V = L i f
Y0+ (z+−1)D+− (z−−1)D−
F ce (z+u+− z−u−)L i f Z ′′
D
+ D+−D−F ce (z+u+− z−u−)
L i f Z ′D + L
κL i f
(D.86)
The faradaic impedance becomes
Z f = L V
L i f = 1
Y0+ D+−D−
F ce (z+u+− z−u−)Z ′
D
+ (z+−1)D+− (z−−1)D−F ce (z+u+− z−u−)
Z ′′D + L
κ
= Z0 +ZΦ+ZΩ
(D.87)
where
Z0 = Y −10 (D.88)
ZΩ = L
κ(D.89)
ZΦ = D+−D−F ce (z+u+− z−u−)
Z ′D + (z+−1)D+− (z−−1)D−
F ce (z+u+− z−u−)Z ′′
D (D.90)
Appendix E 103
E Finite difference method
E.1 Coupled, linear, difference equations
The numerical approach described below is given in [9].
In general, a set of n coupled linear, second-order differential equations can be represented as [9]
n∑k=1
ai ,k (x)d 2ck
d x2 +bi ,k (x)dck
d x+di ,k (x)ck = gi (x) (E.1)
where ck (x) are the n unknown functions, i denotes the equation number, and each of the equations
can involve all of the unknowns ck through the sum.
For central difference approximations of the derivatives with a mesh distance h,
d 2ck
d x2 = ck (x j +h)+ ck (x j −h)−2ck (x j )
h2 +O(h2) (E.2)
dck
d x= ck (x j +h)− ck (x j −h)
2h+O(h2) (E.3)
we obtain the difference equations
n∑k=1
Ai ,k ( j )Ck ( j −1)+Bi ,k ( j )Ck ( j )+Di ,kCk ( j +1) =Gi ( j ) (E.4)
where j denotes a certain point number in the mesh, and the coefficients are given by [9]
Ck ( j ) = ck (x) (E.5)
Ai ,k ( j ) = ai ,k (x j )− h
2bi ,k (x j ) (E.6)
Bi ,k ( j ) =−2ai ,k (x j )+h2di ,k (x j ) (E.7)
Di ,k ( j ) = ai ,k (x j )+ h
2bi ,k (x j ) (E.8)
Gi ( j ) = h2gi (x j ) (E.9)
At j = 1, the equations are
n∑k=1
Bi ,k (1)Ck (1)+Di ,k (1)Ck (2)+Xi ,kCk (3) =Gi (1) (E.10)
where the third term at j = 3 has been added to allow the treatment of complex boundary conditions
[9]. General boundary conditions for equation number 1 at x = 0 would read
n∑k=1
pi ,kdck
d x+ei ,k ck = fi (E.11)
Appendix E 104
A forward difference accurate to order h2 has the coefficients in Eq. (E.10)
Xi ,k =−0.5pi ,k , Di ,k (1) = 2pi ,k ,
Bi ,k (1) = hei ,k −1.5pi ,k , Gi (1) = h fi
(E.12)
A central-difference form given in Eq. (E.3) introduces an image point at x = −h, outside the
domain of interest has the coefficients in Eq. (E.10)
Xi ,k =−Bi ,k (1) = pi ,k
2, Di ,k (1) = hei ,k , Gi (1) = h fi (E.13)
For similar reasons, the difference equations at j = jmax are written
n∑k=1
Yi ,kCk ( j −2)+ Ai ,k ( j )Ck ( j −1)+Bi ,k ( j )Ck ( j ) =Gi ( j ) (E.14)
where the coefficients Yi ,k again allow the introduction of complex boundary conditions at x = L.
E.2 Solution of coupled, linear difference equations
For j = 1, we let Ck ( j ) take the form
Ck (1) = ξk (1)+n∑
l=1Ek,l (1)Cl (2)+xk,l Cl (3) (E.15)
Substitution into Eq. (E.10) shows that ξk , Ek,l , and xk,l satisfy the equations [9]
n∑k=1
Bi ,k (1)ξk (1) =Gi (1),
n∑k=1
Bi ,k (1)Ek,l (1) =−Di ,l (1),
n∑k=1
Bi ,k (1)xk,l =−Xi ,l
(E.16)
where Bi ,k can be readily solved.
For the remaining points, except j = jmax, we assume the form of the unknowns Ck
Ck ( j ) = ξk ( j )+n∑
l=1Ek,l ( j )Cl ( j +1) (E.17)
Substitution of Eq. (E.17) into Eq. (E.4) to eliminate first Ck ( j −1) and then Ck ( j ) and setting the
remaining coefficients of each Ck ( j +1) equal to zero yield a set of equations for the determination of
Appendix E 105
ξk and Ek,l
n∑k=1
bi ,k ( j )ξk ( j ) =Gi ( j )−n∑
l=1Ai ,l ( j )ξl ( j −1) (E.18)
n∑k=1
bi ,k ( j )Ek,m( j ) =−Di ,m( j ) (E.19)
where
bi ,k ( j ) = Bi ,k ( j )+n∑
l=1Ai ,l ( j )El ,k ( j −1) (E.20)
and is not to be confused with bi ,k (x) in Eq. (E.1).
Here the Thomas method for tridiagonal matrices has been generalized to the banded matrix
defined by Eqs. (E.4), (E.10) and (E.14). Finally, one has Eq. (E.14) for j = jmax. If Ck ( j − 2) and
Ck ( j −1) are eliminated by means of Eq. (E.17), then the values of Ck ( jmax) can be determined from
the resulting equations
n∑k=1
bi ,k ( j )Ck ( j ) =Gi ( j )−n∑
l=1Yi ,lξl ( j −2)−
n∑l=1
ai ,l ( j )ξl ( j −1), (E.21)
where
ai ,l ( j ) = Ai ,l ( j )+n∑
m=1Yi ,mEm,l ( j −2) (E.22)
bi ,k ( j ) = Bi ,k ( j )+n∑
l=1ai ,l ( j )El ,k ( j −1) (E.23)
Having the values for Ck ( jmax), one is now in a position to determine Ck ( j ) in reverse order in j
from Eq. (E.17) and finally determine Ck (1) from Eq. (E.15).
Since the boundary-value problem involves boundary conditions at both x = 0 and x = L, it is
not possible to start at either end and obtain final values of the unknowns. Instead, one makes two
passes through the domain in opposite directions. Ek,l in Eq. (E.17) allows the effect of the boundary
conditions at x = L to be reflected back through this domain, the effect not being realized until the
back subsitution is carried out [9].
All the steps to solve the coupled, linear difference equations are given above. However, to pro-
gram these steps can be a bit tricky. Newman has given a program code to solve these steps in New-
man’s BAND routine, where you supply the coefficients A, B , D and G and run the program for each
value of j [9].
Appendix F 106
F Program codes for the Matlab environment
The scripts used to calculate the impedance spectra in this study are given below. The frequency
distribution subroutine, the BAND subroutine, and the MATINV subroutine are required in order to
execute the scripts as written. Including the subroutines in the same folder as the impedance spectra
scripts is recommended, for simplicity.
F.1 Finite difference method
F.1.1 Newman’s BAND subroutine
Newman’s BAND subroutine that is supplied matrix coefficients defined by the differential equations
and boundary conditions to be solved. Calls the subroutine MATINV to invert the matrices. Estab-
lished for the Matlab environment by J. W. Evans and P. Kar at Dept. of Materials Science and Engi-
neering at University of California, Berkley.
% ************************************************************************
% MATLAB version of BAND
% The following is a version of BAND (and MATINV) obtained by
% "translating" the FORTRAN version in Newman.
% ************************************************************************
function [y]=band(J) % i.e. ’J’ = ’I’ from test_band
global A B C D E G X Y NPOINTS N NJ NP1 DETERMINANT
% ************************************************************************
% Case where J negative
% ************************************************************************
if (J-2) < 0 % i.e. image point (I=1) outside boundary (left hand side)
NP1=N+1; % I runs from 1:NPOINTS and NPOINTS = NJ, while N = # eqn
for I=1:N
D(I,2*N+1)=G(I);
for L=1:N
LPN=L+N;
D(I,LPN)=X(I,L);
end
end
% *******************************************************************
Appendix F 107
% Calling Matinv
% *******************************************************************
matinv(N,2*N+1);
if (DETERMINANT == 0)
sprintf(’DETERMINANT = 0 at J= %6.3f’,J)
return % stod tidligere break
else
for K=1:N
E(K,NP1,1)=D(K,2*N+1);
for L=1:N
E(K,L,1) = -D(K,L);
LPN = L+N;
X(K,L) = -D(K,LPN);
end
end
return
end
end
% ************************************************************************
% Case where J-2 equal to zero (I=2, i.e. first real point)
% ************************************************************************
if((J-2)==0)
for I=1:N
for K=1:N
for L=1:N
D(I,K)=D(I,K)+A(I,L)*X(L,K);
end
end
end
end
% ************************************************************************
% Case where J-2 is greater than zero
% ************************************************************************
if (J-NJ) < 0
Appendix F 108
for I=1:N
D(I,NP1) = -G(I);
for L=1:N
D(I,NP1)=D(I,NP1)+A(I,L)*E(L,NP1,J-1);
for K=1:N
B(I,K)=B(I,K)+A(I,L)*E(L,K,J-1);
end
end
end
else
for I=1:N
for L=1:N
G(I)=G(I)-Y(I,L)*E(L,NP1,J-2);
for M=1:N
A(I,L)=A(I,L)+Y(I,M)*E(M,L,J-2);
end
end
end
for I=1:N
D(I,NP1) = -G(I);
for L=1:N
D(I,NP1)=D(I,NP1)+A(I,L)*E(L,NP1,J-1);
for K=1:N
B(I,K)=B(I,K)+A(I,L)*E(L,K,J-1);
end
end
end
end
% ***********************************************************************
% Calling Matinv
% ***********************************************************************
matinv(N,NP1);
if DETERMINANT ~= 0
Appendix F 109
for K=1:N
for M=1:NP1
E(K,M,J) = -D(K,M);
end
end
else
sprintf(’DETERMINANT = 0 at J= %6.3f’,J)
return % stod tidligere break
end
if (J-NJ)<0
return
else
for K = 1:N
C(K,J) = E(K,NP1,J);
end
for JJ = 2:NJ
M = NJ - JJ + 1;
for K=1:N
C(K,M) = E(K,NP1,M);
for L=1:N
C(K,M)=C(K,M)+E(K,L,M)*C(L,M+1);
end
end
end
for L=1:N
for K=1:N
C(K,1) = C(K,1) + X(K,L)*C(L,3);
end
end
end
return
Appendix F 110
%*********************************************************************
Appendix F 111
F.1.2 Newman’s matrix inversion subroutine MATINV
Newman’s program code for inverting matrices, used in the BAND subroutine. Established for the
Matlab environment by J. W. Evans and P. Kar at Dept. of Materials Science and Engineering at Uni-
versity of California, Berkley.
function [DETERMINANT] = matinv(N,M)
global A B C D E G X Y NPOINTS NJ DETERMINANT
DETERMINANT=1.0;
for I=1:N
ID(I)=0;
end
for NN=1:N
BMAX = 1.1;
for I=1:N
if(ID(I)~=0)
continue
else
BNEXT = 0;
BTRY = 0;
for J=1:N
if(ID(J)==0)
if(abs(B(I,J)) > BNEXT)
BNEXT=abs(B(I,J));
if(BNEXT > BTRY)
BNEXT = BTRY;
BTRY = abs(B(I,J));
JC = J;
end
end
end
end
end
if(BNEXT < (BMAX*BTRY))
BMAX=BNEXT/BTRY;
IROW=I;
Appendix F 112
JCOL=JC;
end
end
if(ID(JC)==0)
ID(JCOL)=1;
else
DETERMINANT=0;
return
end
if(JCOL==IROW)
F=1/(B(JCOL,JCOL));
for J=1:N
B(JCOL,J)=B(JCOL,J)*F;
end
for K=1:M
D(JCOL,K)=D(JCOL,K)*F;
end
else
for J=1:N
SAVE=B(IROW,J);
B(IROW,J)=B(JCOL,J);
B(JCOL,J)=SAVE;
end
for K=1:M
SAVE=D(IROW,K);
D(IROW,K)=D(JCOL,K);
D(JCOL,K)=SAVE;
end
end
for I=1:N
if(I ~= JCOL)
F=B(I,JCOL);
for J=1:N
B(I,J)=B(I,J)-F*B(JCOL,J);
Appendix F 113
end
for K=1:M
D(I,K)=D(I,K)-F*D(JCOL,K);
end
end
end
end
return
Appendix F 114
F.1.3 Frequency array establisher
Subroutine that establishes a logarithmically distributed frequency array. Established by Professor
Svein Sunde at Department of Materials Science, Norwegian University of Science and Technology.
%**************************************************************************
% *
% Establishes a frequency array *
% *
% Svein Sunde *
% NTNU, Sem Saelands veg 12, NO-7491 Trondheim *
% Ver. 0.9 Jan 24, 2013 *
% *
% *
% *
% References: *
% *
% Uses: *
% *
%**************************************************************************
function [w] = freqs(fmin,fmax,pperd)
wmax = 2*pi*fmax; %maximum value of angular frequency (s-1)
wmin = 2*pi*fmin; %minimum value of angular frequency (s-1)
logwmax = log10(wmax);
logwmin = log10(wmin);
points = (logwmax-logwmin)*pperd;
steplogw = (logwmax-logwmin)/points;
w = zeros(points+1,1);
i = 0;
for logw = logwmax:-steplogw:logwmin;
i = i+1;
w(i) = 10^logw;
Appendix F 115
end
end
Appendix F 116
F.2 Warburg impedance
F.2.1 Infinite Warburg diffusion impedance, forward difference
Matlab code to calculate numerically the Warburg infinite diffusion impedance spectrum in Eq. (2.18)
with a forward difference with an accuracy to H 2 for the boundary conditions.
%**************************************************************************
% *
% TMT4500 - Prosjekt *
% *
% Warburg infinite impedance - Forward difference *
% *
% Didrik Smabraten *
% *
% Fall 2013 *
% *
%**************************************************************************
clear all % Clear all variables
close all % Close all plots etc
clc % Clear command window
%**************************************************************************
% Defining simulation parameters
%**************************************************************************
% Table 1 (Sunde et al.)
c0 = 0.025; % mol cm-3
ce = 0.004; % mol cm-3
zp = 1; % -
zn = -1; % -
nup = 1; % -
nun = 1; % -
up = 0.1; % cm2 (Vs)-1
un = 1.7e-8; % cm2 (Vs)-1
Appendix F 117
i0 = 0.69e-3; % A cm-2
dE = -20.27; % V cm3 mol-1
kappa = 55e-6; % S cm-1
L = 100e-4; % cm
Cd = 10e-6; % F cm-2
T = 352.15; % K
RT = 8.314*T; % J mol-1
F = 96495; % A s mol-1
kb = 8.6173324e-5; % eV K-1
kbT = kb*T; % eV
n =1; % [-]
% Table 2 (Sunde et al.)
B0 = 44.06^-1; % Ohm cm2
A0 = -0.46; % A cm mol-1
sigma = 1.3; % S cm-1
tp = (zp*up)/(zp*up-zn*un); % -
tn = 1-tp; % -
% Calculated
Dp = kbT*up; % cm2 s-1
Dn = kbT*un; % cm2 s-1
diff =(zp*up*Dn-zn*un*Dp)/(zp*up-zn*un); % cm2 s-1
omega = freqs(0.1e-3,100e3,10); % Frequency distributed over log
s = j*omega; % Laplace variable
i = 1; % Lif=1 for simplicity
%**************************************************************************
% Setting up points for BAND subroutine
%**************************************************************************
%*********************
% Points for BAND(J) *
Appendix F 118
%*********************
global A B C D G X Y N NJ
NPOINTS=input(’Number of steps: ’)+3; % Number of steps
N=1; % Number of equations
NJ=NPOINTS; % Number of points
H=1/(NPOINTS-3); % Step size
%**************************************************************************
% Setting up equations for BAND to solve tilde(c)
%**************************************************************************
for J=1:length(s)
for I=1:NPOINTS
A(1,1) = diff;
B(1,1) = -2*diff-H^2*s(J);
D(1,1) = A(1,1);
G(1) = 0;
if I==1;
X(1,1) = -0.5*diff;
B(1,1) = -1.5*diff;
D(1,1) = 2*diff;
G(1) = H*i/n/F;
end
if I==NPOINTS
Y(1,1) = 0;
B(1,1) = 1;
D(1,1) = 0;
G(1) = 0;
end
band(I)
end
%**************************************************************************
Appendix F 119
% Numerical solution at x=0
%**************************************************************************
ZWnum(J)=dE*C(1)/i;
%**************************************************************************
% Analytical solution at x=0
%**************************************************************************
ZWan(J)=-dE/n/F/sqrt(diff*s(J));
end
%**************************************************************************
% Plotting solutions
%**************************************************************************
% Nyquist plot
figure(’units’,’normalized’,’outerposition’,[0 0 1 1])
plot(real(ZWan),-imag(ZWan),’-b’,’LineWidth’,1.5)
hold on
plot(real(ZWnum),-imag(ZWnum),’.r’,’MarkerSize’,20)
xlabel(’Re(Z) [\Omega cm^2]’,’FontSize’,12)
ylabel(’-Im(Z) [\Omega cm^2]’,’FontSize’,12)
legend(’Z_f, Analytical’,’Z_f, Numerical’)
title(’Nyquist plot of infinite Warburg impedance’)
axis equal
set(gca,’FontSize’,12)
Appendix F 120
F.2.2 Infinite Warburg diffusion impedance, central difference
Matlab code to calculate numerically the Warburg infinite diffusion impedance spectrum in Eq. (2.18)
with a central difference with an accuracy to H 2 for the boundary conditions.
%**************************************************************************
% *
% TMT4500 - Prosjekt *
% *
% Warburg infinite impedance - Central difference *
% *
% Didrik Smabraten *
% *
% Fall 2013 *
% *
%**************************************************************************
clear all % Clear all variables
close all % Close all plots etc
clc % Clear command window
%**************************************************************************
% Defining simulation parameters
%**************************************************************************
% Table 1 (Sunde et al.)
c0 = 0.025; % mol cm-3
ce = 0.004; % mol cm-3
zp = 1; % -
zn = -1; % -
nup = 1; % -
nun = 1; % -
up = 0.1; % cm2 (Vs)-1
un = 1.7e-8; % cm2 (Vs)-1
i0 = 0.69e-3; % A cm-2
dE = -20.27; % V cm3 mol-1
Appendix F 121
kappa = 55e-6; % S cm-1
L = 100e-4; % cm
Cd = 10e-6; % F cm-2
T = 352.15; % K
RT = 8.314*T; % J mol-1
F = 96495; % A s mol-1
kb = 8.6173324e-5; % eV K-1
kbT = kb*T; % eV
n =1; % [-]
% Table 2 (Sunde et al.)
B0 = 44.06^-1; % Ohm cm2
A0 = -0.46; % A cm mol-1
sigma = 1.3; % S cm-1
tp = (zp*up)/(zp*up-zn*un); % -
tn = 1-tp; % -
% Calculated
Dp = kbT*up; % cm2 s-1
Dn = kbT*un; % cm2 s-1
diff =(zp*up*Dn-zn*un*Dp)/(zp*up-zn*un); % cm2 s-1
omega = freqs(0.1e-3,100e3,10); % Frequency distributed over log
s = j*omega; % Laplace variable
i = 1; % Lif=1 for simplicity
%**************************************************************************
% Setting up points for BAND subroutine
%**************************************************************************
%*********************
% Points for BAND(J) *
%*********************
global A B C D G X Y N NJ
Appendix F 122
NPOINTS=input(’Number of steps: ’)+3; % Number of steps
N=1; % Number of equations
NJ=NPOINTS; % Number of points
H=1/(NPOINTS-3); % Step size
%**************************************************************************
% Setting up equations for BAND to solve tilde(c)
%**************************************************************************
for J=1:length(s)
for I=1:NPOINTS
A(1,1) = diff;
B(1,1) = -2*diff-H^2*s(J);
D(1,1) = A(1,1);
G(1) = 0;
if I==1;
X(1,1) = 1/2*diff;
B(1,1) = -X(1,1);
D(1,1) = 0;
G(1) = H*i/n/F;
end
if I==NPOINTS
Y(1,1) = 0;
B(1,1) = 1;
D(1,1) = 0;
G(1) = 0;
end
band(I)
end
%**************************************************************************
% Numerical solution at x=0
Appendix F 123
%**************************************************************************
ZWnum(J)=dE*C(2)/i;
%**************************************************************************
% Analytical solution at x=0
%**************************************************************************
ZWan(J)=-dE/n/F/sqrt(diff*s(J));
end
%**************************************************************************
% Plotting solutions
%**************************************************************************
% Nyquist plot
figure(’units’,’normalized’,’outerposition’,[0 0 1 1])
plot(real(ZWan),-imag(ZWan),’-b’,’LineWidth’,1.5)
hold on
plot(real(ZWnum),-imag(ZWnum),’.r’,’MarkerSize’,20)
xlabel(’Re(Z) [\Omega cm^2]’,’FontSize’,12)
ylabel(’-Im(Z) [\Omega cm^2]’,’FontSize’,12)
legend(’Z_f, Analytical’,’Z_f, Numerical’)
title(’Nyquist plot of infinite Warburg diffusion impedance’)
axis equal
set(gca,’FontSize’,12)
Appendix F 124
F.2.3 Transmissive Warburg diffusion impedance
Matlab code to calculate numerically the transmissive Warburg diffusion impedance spectrum in Eq.
(2.19).
%**************************************************************************
% *
% TMT4500 - Prosjekt *
% *
% Warburg finite impedance - Transmissive boundary *
% - Central difference *
% *
% Didrik Smabraten *
% *
% Fall 2013 *
% *
%**************************************************************************
clear all % Clear all variables
close all % Close all plots etc
clc % Clear command window
%**************************************************************************
% Defining simulation parameters
%**************************************************************************
% Table 1 (Sunde et al.)
c0 = 0.025; % mol cm-3
ce = 0.004; % mol cm-3
zp = 1; % -
zn = -1; % -
nup = 1; % -
nun = 1; % -
up = 0.1; % cm2 (Vs)-1
un = 1.7e-5; % cm2 (Vs)-1
i0 = 0.69e-3; % A cm-2
Appendix F 125
dE = -20.27; % V cm3 mol-1
kappa = 55e-6; % S cm-1
L = 100e-4; % cm
Cd = 10e-6; % F cm-2
T = 352.15; % K
RT = 8.314*T; % J mol-1
F = 96495; % A s mol-1
kb = 8.6173324e-5; % eV K-1
kbT = kb*T; % eV
n =1; % [-]
% Table 2 (Sunde et al.)
B0 = 44.06^-1; % Ohm cm2
A0 = -0.46; % A cm mol-1
sigma = 1.3; % S cm-1
tp = (zp*up)/(zp*up-zn*un); % -
tn = 1-tp; % -
% Calculated
Dp = kbT*up; % cm2 s-1
Dn = kbT*un; % cm2 s-1
diff =(zp*up*Dn-zn*un*Dp)/(zp*up-zn*un); % cm2 s-1
omega = freqs(0.1e-3,100e3,10); % Frequency distributed over log
s = j*omega; % Laplace variable
i = 1; % Lif=1 for simplicity
%**************************************************************************
% Setting up points for BAND subroutine
%**************************************************************************
%*********************
% Points for BAND(J) *
%*********************
Appendix F 126
global A B C D G X Y N NJ
NPOINTS=input(’Number of steps: ’)+3; % Number of steps
N=1; % Number of equations
NJ=NPOINTS; % Number of points
H=1/(NPOINTS--3); % Step size
%**************************************************************************
% Setting up equations for BAND to solve tilde(c)
%**************************************************************************
for J=1:length(s)
for I=1:NPOINTS
A(1,1) = diff/L^2;
B(1,1) = -2*diff/L^2-H^2*s(J);
D(1,1) = A(1,1);
G(1) = 0;
if I==1;
X(1,1) = 0.5*diff;
B(1,1) = -X(1,1);
D(1,1) = 0;
G(1) = H*L*i/n/F;
end
if I==NPOINTS
Y(1,1) = 0;
B(1,1) = 1;
D(1,1) = 0;
G(1) = 0;
end
band(I)
end
%**************************************************************************
% Numerical solution at x=0
Appendix F 127
%**************************************************************************
ZWnum(J)=dE*C(2)/i;
%**************************************************************************
% Analytical solution at x=0
%**************************************************************************
ZWan(J)=-dE/n/F*tanh(L*sqrt(s(J)/diff))/sqrt(s(J)*diff);
end
%**************************************************************************
% Plotting solutions
%**************************************************************************
% Nyquist plot
figure(’units’,’normalized’,’outerposition’,[0 0 1 1])
plot(real(ZWan),-imag(ZWan),’-k’,’LineWidth’,1.5)
hold on
plot(real(ZWnum),-imag(ZWnum),’ob’,’MarkerSize’,10,’LineWidth’,1.5)
xlabel(’Re(Z) [\Omega cm^2]’,’FontSize’,15)
ylabel(’-Im(Z) [\Omega cm^2]’,’FontSize’,15)
legend(’Z_W, Analytical’,...
[’Z_W, Numerical. Steps = ’,num2str(NPOINTS-3)])
%title(’Nyquist plot of finite Gerischer impedance’)
axis equal
for J=[length(s)-10 length(s)-5 length(s)-2]
text(real(ZWan(J)),-imag(ZWan(J)),...
[’\leftarrow \omega = ’,num2str(omega(J))],...
’HorizontalAlignment’,’left’,’FontSize’,15)
end
set(gca,’FontSize’,12)
Appendix F 128
F.2.4 Reflective Warburg diffusion impedance
Matlab code to calculate numerically the reflective Warburg diffusion impedance spectrum in Eq.
(2.22).
%**************************************************************************
% *
% TMT4500 - Prosjekt *
% *
% Warburg finite impedance - Reflective boundary *
% - Central differenciation *
% *
% Didrik Smabraten *
% *
% Fall 2013 *
% *
%**************************************************************************
clear all % Clear all variables
close all % Close all plots etc
clc % Clear command window
%**************************************************************************
% Defining simulation parameters
%**************************************************************************
% Table 1 (Sunde et al.)
c0 = 0.025; % mol cm-3
ce = 0.004; % mol cm-3
zp = 1; % -
zn = -1; % -
nup = 1; % -
nun = 1; % -
up = 0.1; % cm2 (Vs)-1
un = 1.7e-8; % cm2 (Vs)-1
i0 = 0.69e-3; % A cm-2
Appendix F 129
dE = -20.27; % V cm3 mol-1
kappa = 55e-6; % S cm-1
L = 100e-4; % cm
Cd = 10e-6; % F cm-2
T = 352.15; % K
RT = 8.314*T; % J mol-1
F = 96495; % A s mol-1
kb = 8.6173324e-5; % eV K-1
kbT = kb*T; % eV
n =1; % [-]
% Table 2 (Sunde et al.)
B0 = 44.06^-1; % Ohm cm2
A0 = -0.46; % A cm mol-1
sigma = 1.3; % S cm-1
tp = (zp*up)/(zp*up-zn*un); % -
tn = 1-tp; % -
% Calculated
Dp = kbT*up; % cm2 s-1
Dn = kbT*un; % cm2 s-1
diff =(zp*up*Dn-zn*un*Dp)/(zp*up-zn*un); % cm2 s-1
omega = freqs(0.1e-3,100e3,10); % Frequency distributed over log
s = j*omega; % Laplace variable
i = 1; % Lif=1 for simplicity
%**************************************************************************
% Setting up points for BAND subroutine
%**************************************************************************
%*********************
% Points for BAND(J) *
%*********************
Appendix F 130
global A B C D G X Y N NJ
NPOINTS=input(’Number of steps: ’)+3; % Number of steps
N=1; % Number of equations
NJ=NPOINTS; % Number of points
H=1/(NPOINTS-3); % Step size
%**************************************************************************
% Setting up equations for BAND to solve tilde(c)
%**************************************************************************
for J=1:length(s)
for I=1:NPOINTS
A(1,1) = diff;
B(1,1) = -2*diff-H^2*(s(J))*L^2;
D(1,1) = A(1,1);
G(1) = 0;
if I==1;
B(1,1) = -0.5*diff/L;
X(1,1) = -B(1,1);
D(1,1) = 0;
G(1) = H*i/F/n;
end
if I==NPOINTS
B(1,1) = -0.5/L;
Y(1,1) = -B(1,1);
D(1,1) = 0;
G(1) = 0;
end
band(I)
end
%**************************************************************************
Appendix F 131
% Numerical solution at x=0
%**************************************************************************
ZWnum(J)=dE*C(2)/i;
%**************************************************************************
% Analytical solution at x=0
%**************************************************************************
ZWan(J)=-dE/n/F/sqrt(diff*s(J))*coth(sqrt((s(J))/diff)*L);
end
%**************************************************************************
% Plotting solutions
%**************************************************************************
% Nyquist plot
figure(’units’,’normalized’,’outerposition’,[0 0 1 1])
plot(real(ZWan),-imag(ZWan),’-k’,’LineWidth’,1.5)
hold on
plot(real(ZWnum),-imag(ZWnum),’ob’,’MarkerSize’,10,’LineWidth’,1.5)
xlabel(’Re(Z) [\Omega cm^2]’,’FontSize’,15)
ylabel(’-Im(Z) [\Omega cm^2]’,’FontSize’,15)
legend(’Z_W, Analytical’,...
[’Z_W, Numerical. Steps = ’,num2str(NPOINTS-3)])
%title(’Nyquist plot of finite Gerischer impedance’)
axis equal
for J=[length(s)-10 length(s)-5 length(s)-2]
text(real(ZWan(J)),-imag(ZWan(J)),...
[’\leftarrow \omega = ’,num2str(omega(J))],...
’HorizontalAlignment’,’left’,’FontSize’,15)
end
set(gca,’FontSize’,15)
Appendix F 132
F.3 Gerischer impedance
F.3.1 Transmissive Gerischer impedance
Matlab code to calculate numerically the transmissive Gerischer impedance spectrum in Eq. (2.29).
%**************************************************************************
% *
% TMT4500 - Prosjekt *
% *
% Finite Gerischer impedance - Transmissive boundary *
% - Central difference *
% *
% Didrik Smabraten *
% *
% Fall 2013 *
% *
%**************************************************************************
clear all % Clear all variables
close all % Close all plots etc
clc % Clear command window
%**************************************************************************
% Defining simulation parameters
%**************************************************************************
% Table 1 (Sunde et al.)
c0 = 0.025; % mol cm-3
ce = 0.004; % mol cm-3
zp = 1; % -
zn = -1; % -
nup = 1; % -
nun = 1; % -
up = 0.1; % cm2 (Vs)-1
un = 1.7e-8; % cm2 (Vs)-1
Appendix F 133
i0 = 0.69e-3; % A cm-2
dE = -20.27; % V cm3 mol-1
k = input(’k=’); % s-1
kappa = 55e-6; % S cm-1
L = 100e-4; % cm
Cd = 10e-6; % F cm-2
T = 352.15; % K
RT = 8.314*T; % J mol-1
F = 96495; % A s mol-1
kb = 8.6173324e-5; % eV K-1
kbT = kb*T; % eV
n =1; % [-]
% Table 2 (Sunde et al.)
B0 = 44.06^-1; % Ohm cm2
A0 = -0.46; % A cm mol-1
sigma = 1.3; % S cm-1
tp = (zp*up)/(zp*up-zn*un); % -
tn = 1-tp; % -
% Calculated
Dp = kbT*up; % cm2 s-1
Dn = kbT*un; % cm2 s-1
diff =(zp*up*Dn-zn*un*Dp)/(zp*up-zn*un); % cm2 s-1
omega = freqs(0.1e-3,100e3,10); % Frequency distributed over log
s = j*omega; % Laplace variable
i = 1; % Lif=1 for simplicity
%**************************************************************************
% Setting up points for BAND subroutine
%**************************************************************************
%*********************
Appendix F 134
% Points for BAND(J) *
%*********************
global A B C D G X Y N NJ
NPOINTS=input(’Number of steps: ’)+3; % Number of steps
N=1; % Number of equations
NJ=NPOINTS; % Number of points
H=1/(NPOINTS-3); % Step size
%**************************************************************************
% Setting up equations for BAND to solve tilde(c)
%**************************************************************************
for J=1:length(s)
for I=1:NPOINTS
A(1,1) = diff/L^2;
B(1,1) = -2*diff/L^2-H^2*(s(J)+k);
D(1,1) = A(1,1);
G(1) = 0;
if I==1;
B(1,1) = -0.5*diff/L;
X(1,1) = -B(1,1);
D(1,1) = 0;
G(1) = H*i/n/F;
end
if I==NPOINTS
Y(1,1) = 0;
B(1,1) = 1;
D(1,1) = 0;
G(1) = 0;
end
band(I)
end
Appendix F 135
%**************************************************************************
% Numerical solution at x=0
%**************************************************************************
ZGnum(J)=dE*C(2)/i;
%**************************************************************************
% Analytical solution at x=0
%**************************************************************************
ZGan(J)=-dE*1/n/F/sqrt(diff*(s(J)+k))*tanh(sqrt((s(J)+k)/diff)*L);
end
%**************************************************************************
% Plotting solutions
%**************************************************************************
% Nyquist plot
figure(’units’,’normalized’,’outerposition’,[0 0 1 1])
plot(real(ZGan),-imag(ZGan),’-k’,’LineWidth’,1.5)
hold on
plot(real(ZGnum),-imag(ZGnum),’ob’,’MarkerSize’,10,’LineWidth’,1.5)
xlabel(’Re(Z) [\Omega cm^2]’,’FontSize’,15)
ylabel(’-Im(Z) [\Omega cm^2]’,’FontSize’,15)
legend(’Z_G, Analytical’,...
[’Z_G, Numerical. Steps = ’,num2str(NPOINTS-3)])
title(’Nyquist plot of finite Gerischer impedance’)
axis equal
for J=[length(s)-30 length(s)-25 length(s)-20]
text(real(ZGan(J)),-imag(ZGan(J)),...
[’\leftarrow \omega = ’,num2str(omega(J))],...
’HorizontalAlignment’,’left’,’FontSize’,15)
end
Appendix F 136
set(gca,’FontSize’,15)
Appendix F 137
F.3.2 Reflective Gerischer impedance
Matlab code to calculate numerically the reflective Gerischer impedance spectrum in Eq. (2.32).
%**************************************************************************
% *
% TMT4500 - Prosjekt *
% *
% Finite Gerischer impedance - Reflective boundary *
% - Central difference *
% *
% Didrik Småbråten *
% *
% Fall 2013 *
% *
%**************************************************************************
clear all % Clear all variables
close all % Close all plots etc
clc % Clear command window
%**************************************************************************
% Defining simulation parameters
%**************************************************************************
% Table 1 (Sunde et al.)
c0 = 0.025; % mol cm-3
ce = 0.004; % mol cm-3
zp = 1; % -
zn = -1; % -
nup = 1; % -
nun = 1; % -
up = 0.1; % cm2 (Vs)-1
un = 1.7e-8; % cm2 (Vs)-1
i0 = 0.69e-3; % A cm-2
dE = -20.27; % V cm3 mol-1
Appendix F 138
k = input(’k=’); % s-1
kappa = 55e-6; % S cm-1
L = 100e-4; % cm
Cd = 10e-6; % F cm-2
T = 352.15; % K
RT = 8.314*T; % J mol-1
F = 96495; % A s mol-1
kb = 8.6173324e-5; % eV K-1
kbT = kb*T; % eV
n =1; % [-]
% Table 2 (Sunde et al.)
B0 = 44.06^-1; % Ohm cm2
A0 = -0.46; % A cm mol-1
sigma = 1.3; % S cm-1
tp = (zp*up)/(zp*up-zn*un); % -
tn = 1-tp; % -
% Calculated
Dp = kbT*up; % cm2 s-1
Dn = kbT*un; % cm2 s-1
diff =(zp*up*Dn-zn*un*Dp)/(zp*up-zn*un); % cm2 s-1
omega = freqs(0.1e-3,100e3,10); % Frequency distributed over log
s = j*omega; % Laplace variable
i = 1; % Lif=1 for simplicity
%**************************************************************************
% Setting up points for BAND subroutine
%**************************************************************************
%*********************
% Points for BAND(J) *
%*********************
Appendix F 139
global A B C D G X Y N NJ
NPOINTS=input(’Number of steps: ’)+3; % Number of steps
N=1; % Number of equations
NJ=NPOINTS; % Number of points
H=1/(NPOINTS-3); % Step size
%**************************************************************************
% Setting up equations for BAND to solve tilde(c)
%**************************************************************************
for J=1:length(s)
for I=1:NPOINTS
A(1,1) = diff;
B(1,1) = -2*diff-H^2*(s(J)+k)*L^2;
D(1,1) = A(1,1);
G(1) = 0;
if I==1;
B(1,1) = -0.5*diff/L;
X(1,1) = -B(1,1);
D(1,1) = 0;
G(1) = H*i/n/F;
end
if I==NPOINTS
B(1,1) = -0.5;
Y(1,1) = -B(1,1);
D(1,1) = 0;
G(1) = 0;
end
band(I)
end
%**************************************************************************
Appendix F 140
% Numerical solution at x=0
%**************************************************************************
ZGnum(J)=dE*C(2)/i;
%**************************************************************************
% Analytical solution at x=0
%**************************************************************************
ZGan(J)=-dE/n/F/sqrt(diff*(s(J)+k))*coth(sqrt((s(J)+k)/diff)*L);
end
%**************************************************************************
% Plotting solutions
%**************************************************************************
% Nyquist plot
figure(’units’,’normalized’,’outerposition’,[0 0 1 1])
plot(real(ZGan),-imag(ZGan),’-k’,’LineWidth’,1.5)
hold on
plot(real(ZGnum),-imag(ZGnum),’ob’,’MarkerSize’,10,’LineWidth’,1.5)
xlabel(’Re(Z) [\Omega cm^2]’,’FontSize’,15)
ylabel(’-Im(Z) [\Omega cm^2]’,’FontSize’,15)
legend(’Z_G, Analytical’,...
[’Z_G, Numerical. Steps = ’,num2str(NPOINTS-3)])
title(’Nyquist plot of reflective Gerischer impedance’)
axis equal
for J=[length(s)-30 length(s)-25 length(s)-20]
text(real(ZGan(J)),-imag(ZGan(J)),...
[’\leftarrow \omega = ’,num2str(omega(J))],...
’HorizontalAlignment’,’left’,’FontSize’,15)
end
set(gca,’FontSize’,15)
Appendix F 141
F.4 Electrochemical impedance
Matlab code to calculate numerically the electrochemical impedance spectrum for a semiconducting
thin film electrode in Eq. (2.75).
%**************************************************************************
% *
% TMT4500 - Prosjekt *
% *
% Electrochemical impedance - Central difference *
% *
% Didrik Smabraten *
% *
% Fall 2013 *
% *
%**************************************************************************
clear all % Clear all variables
close all % Close all open windows
clc % Clear command window
%**************************************************************************
% Defining simulation parameters
%**************************************************************************
% Table 1 (Sunde et al.)
c0 = 0.025; % mol cm-3
ce = 0.004; % mol cm-3
zp = 1; % -
zn = -1; % -
nup = 1; % -
nun = 1; % -
up = 0.1; % cm2 (Vs)-1
un = 1.7e-8; % cm2 (Vs)-1
i0 = 0.69e-3; % A cm-2
dE = -20.27; % V cm3 mol-1
Appendix F 142
k = input(’k = ’);
kappa = 55e-6; % S cm-1
L = 100e-4; % cm
Cd = 10e-6; % F cm-2
T = 352.15; % K
RT = 8.314*T; % J mol-1
F = 96495; % A s mol-1
kb = 8.6173324e-5; % eV K-1
kbT = kb*T; % eV
n =1; % [-]
% Table 2 (Sunde et al.)
B0 = 44.06^-1; % Ohm cm2
A0 = -0.46; % A cm mol-1
sigma = 1.3; % S cm-1
tp = (zp*up)/(zp*up-zn*un); % -
tn = 1-tp; % -
% Calculated
Dp = kbT*up; % cm2 s-1
Dn = kbT*un; % cm2 s-1
diff =(zp*up*Dn-zn*un*Dp)/(zp*up-zn*un); % cm2 s-1
omega = freqs(0.1e-3,100e3,10); % Frequency distributed over log
s = j*omega; % Laplace variable
i = 1; % Lif=1 for simplicity
kappa =F^2*ce*(zp*up-zn*un)*(zp*nup);
%**************************************************************************
% Setting up points for BAND subroutine
%**************************************************************************
%*********************
% Points for BAND(J) *
Appendix F 143
%*********************
global A B C D G X Y N NJ
NPOINTS=input(’Number of steps: ’)+3; % Number of steps
N=1; % Number of equations
NJ=NPOINTS; % Number of points
H=1/(NPOINTS-3); % Step size
%**************************************************************************
% Setting up equations for BAND to solve tilde(c)
%**************************************************************************
for J=1:length(s)
for I=1:NPOINTS
A(1,1) = diff;
B(1,1) = -2*diff-H^2*(s(J)+k)*L^2;
D(1,1) = A(1,1);
G(1) = 0;
if I==1;
B(1,1) = -0.5*diff/L/(1-tn);
X(1,1) = -B(1,1);
D(1,1) = 0;
G(1) = H*i/(zn*nun*F);
end
if I==NPOINTS
B(1,1) = -0.5*diff/L/(1-tp);
Y(1,1) = -B(1,1);
D(1,1) = 0;
G(1) = H*i/(zp*nup*F);
end
band(I)
end
%**************************************************************************
Appendix F 144
% Numerical solution at x=0
%**************************************************************************
Zfnum(J) = (i-A0*C(2))/B0/i+...
((zp-1)*Dp-(zn-1)*Dn)/(F*i*ce*(zp*up-zn*un))*...
C(NPOINTS-1)+(Dp-Dn)/(F*i*ce*(zp*up-zn*un))*C(2)+L/kappa;
%**************************************************************************
% Analytical solution at x=0
%**************************************************************************
x = 0;
prefactor(J) = -1/(F*sqrt(diff*(s(J)+k))*sinh(sqrt((s(J)+k)/diff)*L));
K1(J) = (1-tn)/(zn*nun);
K2(J) = (1-tp)/(zp*nup);
Can(J) = i*prefactor(J)*(K1(J)*cosh(sqrt((s(J)+k)/diff)*(x-L))-...
K2(J)*cosh(sqrt((s(J)+k)/diff)*x));
ZD(J) = prefactor(J)*((1-tn)/(zn*nun)*cosh(sqrt((s(J)+k)/diff)*L)-...
(1-tp)/(zp*nup));
ZDD(J) = prefactor(J)*((1-tn)/(zn*nun)-...
(1-tp)/(zp*nup)*cosh(sqrt((s(J)+k)/diff)*L));
Z0 (J) = (1-A0*ZD(J))/B0;
Zomega = L/kappa;
Zphi(J) = (Dp-Dn)/(F*ce*(zp*up-zn*un))*ZD(J)-...
((zp-1)*Dp-(zn-1)*Dn)/(F*ce*(zp*up-zn*un))*ZDD(J);
Zfan(J) = Z0(J)+Zomega+Zphi(J);
end
%**************************************************************************
% Plotting solutions
Appendix F 145
%**************************************************************************
% Nyquist plot
figure(’units’,’normalized’,’outerposition’,[0 0 1 1])
plot(real(Zfan),-imag(Zfan),’-k’,’LineWidth’,1.5)
for J=[length(s)-30 length(s)-25 length(s)-20]
text(real(Zfan(J)),-imag(Zfan(J)),...
[’\leftarrow \omega = ’,num2str(omega(J))],...
’HorizontalAlignment’,’left’,’FontSize’,15)
end
hold on
plot(real(Zfnum),-imag(Zfnum),’ob’,’LineWidth’,1.5,...
’MarkerSize’,10,’MarkerFaceColor’,’auto’)
xlabel(’Re(Z) [\Omega cm^2]’,’FontSize’,15)
ylabel(’-Im(Z) [\Omega cm^2]’,’FontSize’,15)
legend(’Z_f, Analytical’,’Z_f, Numerical’)
title(’Nyquist plot of electrochemical impedance’)
axis equal
set(gca,’FontSize’,15)
Appendix F 146
F.5 Photoelectrochemical impedance
F.5.1 Steady state concentration profile
Matlab code to calculate numerically the steady-state concentration profile for a semiconducting thin
film electrode in Eq. (2.78).
%**************************************************************************
% *
% TMT4500 - Prosjekt *
% *
% Steady state concentration- Central difference *
% *
% Didrik Smabraten *
% *
% Fall 2013 *
% *
%**************************************************************************
clear all % Clear all variables
close all % Close all open windows
clc % Clear command window
%**************************************************************************
% Defining simulation parameters
%**************************************************************************
% Table 1 (Sunde et al.)
c0 = 0.025; % mol cm-3
ce = 0.004; % mol cm-3
zp = 1; % -
zn = -1; % -
nup = 1; % -
nun = 1; % -
up = 0.1; % cm2 (Vs)-1
un = 1.7e-8; % cm2 (Vs)-1
Appendix F 147
i0 = 0.69e-3; % A cm-2
dE = -20.27; % V cm3 mol-1
k = input(’k = ’);
kappa = 55e-6; % S cm-1
L = 100e-4; % cm
Cd = 10e-6; % F cm-2
T = 352.15; % K
RT = 8.314*T; % J mol-1
F = 96495; % A s mol-1
kb = 8.6173324e-5; % eV K-1
kbT = kb*T; % eV
n =1; % [-]
% Table 2 (Sunde et al.)
B0 = 44.06^-1; % Ohm cm2
A0 = -0.46; % A cm mol-1
sigma = 1.3; % S cm-1
tp = (zp*up)/(zp*up-zn*un); % -
tn = 1-tp; % -
% Calculated
Dp = kbT*up; % cm2 s-1
Dn = kbT*un; % cm2 s-1
diff =(zp*up*Dn-zn*un*Dp)/(zp*up-zn*un); % cm2 s-1
omega =freqs(0.01e-3,100e3,10); % Frequency distributed over log
s =j*omega; % Laplace variable
i =1; % Lif=1 for simplicity
I0 =[1e3 1 1e-3 1e-5 0]; % mol cm-2 s-1
alpha =0.25e-5; % cm-1
steps=input(’Number of steps: ’);
markerstyles = ’o’,’x’,’*’,’s’,’d’,’v’,’^’,’<’,’>’,’p’,’h’;
Appendix F 148
linestyles = ’-o’,’-x’,’-*’,’-s’,’-d’,’-v’,’-^’,’-<’,...
’->’,’-p’,’-h’,’:’,’-.’,’--’;
figure(’units’,’normalized’,’outerposition’,[0 0 1 1])
%**************************************************************************
% Setting up points for BAND subroutine
%**************************************************************************
%*********************
% Points for BAND(J) *
%*********************
global A B C D G X Y N NJ
NPOINTS=steps+3; % Number of steps
N=1; % Number of equations
NJ=NPOINTS; % Number of points
H=1/(NPOINTS-3); % Step size
y=-H:H:1+H;
x=y*L;
for K=1:length(I0)
%**************************************************************************
% Setting up equations for BAND to solve tilde(c)
%**************************************************************************
for I=1:NPOINTS
A(1,1) = diff;
B(1,1) = -2*diff-H^2*k*L^2;
D(1,1) = A(1,1);
G(1) = -H^2*alpha*I0(K)*L^2*(exp(-alpha*x(I)));
if I==1;
B(1,1) = -0.5*diff/(1-tn);
Appendix F 149
X(1,1) = -B(1,1);
D(1,1) = 0;
G(1) = H*i/(zn*nun*F)*L;
end
if I==NPOINTS
B(1,1) = -0.5*diff/(1-tp);
Y(1,1) = -B(1,1);
D(1,1) = 0;
G(1) = H*i/(zp*nup*F)*L;
end
band(I)
end
Cnum(K,:) = C;
%**************************************************************************
% Analytical solution at x=0
%**************************************************************************
prefactor = -1/(F*sqrt(diff*k)*sinh(sqrt(k/diff)*L));
K1 = (1-tn)/(zn*nun)-F*diff*I0(K)*alpha^2/(i*(diff*alpha^2-k));
K2 = (1-tp)/(zp*nup)-F*diff*I0(K)*alpha^2/(i*(diff*alpha^2-k))*...
exp(-alpha*L);
for J=2:length(x)-1
Can(J) = i*prefactor*(K1*cosh(sqrt(k/diff)*(x(J)-L))-...
K2*cosh(sqrt(k/diff)*x(J)))-...
I0(K)*alpha/(diff*alpha^2-k)*exp(-alpha*x(J));
end
%**************************************************************************
% Plotting solutions
Appendix F 150
%**************************************************************************
figure(1)
plot(x(2:20:NPOINTS-1),Cnum(K,2:20:NPOINTS-1),linestylesK,...
’LineWidth’,1.5,...
’MarkerSize’,10,’MarkerFaceColor’,’auto’)
hold all
legendnameK = [’C_e, numerical. I_0 = ’,num2str(I0(K))];
legend(legendname)
xlabel(’Length [cm]’,’FontSize’,20)
ylabel(’Concentration [mol cm^-3]’,’FontSize’,20)
set(gca,’FontSize’,20)
end
Appendix F 151
F.5.2 Photoelectrochemical impedance
Matlab code to calculate numerically the photoelectrochemical impedance spectrum for a semicon-
ducting thin film electrode in Eqs. (2.78), (2.83) and (2.70).
%**************************************************************************
% *
% TMT4500 - Prosjekt *
% *
% Photoelectrochemical impedance - Central differenciation *
% *
% Didrik Smabraten *
% *
% Fall 2013 *
% *
%**************************************************************************
clear all % Clear all variables
close all % Close all open windows
clc % Clear command window
%**************************************************************************
% Defining simulation parameters
%**************************************************************************
% Table 1 (Sunde et al.)
c0 = 0.025; % mol cm-3
ce = 0.004; % mol cm-3
zp = 1; % -
zn = -1; % -
nup = 1; % -
nun = 1; % -
up = 0.1; % cm2 (Vs)-1
un = 1.7e-8; % cm2 (Vs)-1
i0 = 0.69e-3; % A cm-2
dE = -20.27; % V cm3 mol-1
Appendix F 152
k = input(’k = ’);
kappa = 55e-6; % S cm-1
L = 100e-4; % cm
Cd = 10e-6; % F cm-2
T = 352.15; % K
RT = 8.314*T; % J mol-1
F = 96495; % A s mol-1
kb = 8.6173324e-5; % eV K-1
kbT = kb*T; % eV
n =1; % [-]
% Table 2 (Sunde et al.)
B0 = 44.06^-1; % Ohm cm2
A0 = -0.46; % A cm mol-1
sigma = 1.3; % S cm-1
tp = (zp*up)/(zp*up-zn*un); % -
tn = 1-tp; % -
% Calculated
Dp = kbT*up; % cm2 s-1
Dn = kbT*un; % cm2 s-1
diff =(zp*up*Dn-zn*un*Dp)/(zp*up-zn*un); % cm2 s-1
omega =freqs(0.1e-3,100e3,10); % Frequency distributed over log
s =j*omega; % Laplace variable
i =1; % Lif=1, for simplicity
I0 =[1e5 1e3 1 1e-3 1e-5 0]; % mol cm-2 s-1
alpha =0.25e-5; % cm-1
markerstyles = ’o’,’x’,’*’,’s’,’d’,’v’,’^’,’<’,’>’,’p’,’h’;
linestyles = ’-o’,’-x’,’-*’,’-s’,’-d’,’-v’,’-^’,’-<’,...
’->’,’-p’,’-h’,’:’,’-.’,’--’;
steps = input(’Number of steps: ’);
Appendix F 153
%**************************************************************************
% Creating full screen figures to plot
%**************************************************************************
figure(’units’,’normalized’,’outerposition’,[0 0 1 1])
xlabel(’Re(Z) [\Omega cm^2]’,’FontSize’,15)
ylabel(’-Im(Z) [\Omega cm^2]’,’FontSize’,15)
%**************************************************************************
% Setting up points for BAND subroutine
%**************************************************************************
%*********************
% Points for BAND(J) *
%*********************
global A B C D G X Y N NJ
NPOINTS=steps+3; % Number of steps
N=1; % Number of equations
NJ=NPOINTS; % Number of points
H=1/(NPOINTS-3); % Step size
y=-H:H:1+H;
x=y*L;
%**************************************************************************
% Setting up equations for BAND to solve tilde(c)
%**************************************************************************
for K=1:length(I0)
for J=1:length(s)
Appendix F 154
%**************************************************************************
% Steady state concentration
%**************************************************************************
for I=1:NPOINTS
A(1,1) = diff;
B(1,1) = -2*diff-H^2*k*L^2;
D(1,1) = A(1,1);
G(1) = -H^2*L^2*alpha*I0(K)*(exp(-alpha*x(I)));
if I==1;
B(1,1) = -0.5*diff/(1-tn);
X(1,1) = -B(1,1);
D(1,1) = 0;
G(1) = H*i/(zn*nun*F)*L;
end
if I==NPOINTS
B(1,1) = -0.5*diff/(1-tp);
Y(1,1) = -B(1,1);
D(1,1) = 0;
G(1) = H*i/(zp*nup*F)*L;
end
band(I)
end
Ce = C;
%**************************************************************************
% Diffusion concentration
%**************************************************************************
for I=1:NPOINTS
A(1,1) = diff;
B(1,1) = -2*diff-H^2*(s(J)+k)*L^2;
D(1,1) = A(1,1);
Appendix F 155
G(1) = 0;
if I==1;
B(1,1) = -0.5*diff/(1-tn);
X(1,1) = -B(1,1);
D(1,1) = 0;
G(1) = H*i/(zn*nun*F)*L;
end
if I==NPOINTS
B(1,1) = -0.5*diff/(1-tp);
Y(1,1) = -B(1,1);
D(1,1) = 0;
G(1) = H*i/(zp*nup*F)*L;
end
band(I)
end
dCe=gradient(Ce,H);
dC=gradient(C,H);
integrate = 0;
for I = 2:NPOINTS-1
integrate = integrate+...
(-i/(F*Ce(I)*zp*nup*(zp*up-zn*un))*(1-C(I)/Ce(I))+...
(Dp-Dn)/(Ce(I)*(zp*up-zn*un))*...
(dC(I)-C(I)/Ce(I)*dCe(I)))*H/F;
end
Zfnum(K,J) = RT/F/Ce(NPOINTS-1)/i*C(NPOINTS-1)+(i-A0*C(2))/B0/i-...
integrate/i/F;
end
%**************************************************************************
Appendix F 156
% Plotting numerical solution
%**************************************************************************
% Nyquist plot
figure(1)
plot(real(Zfnum(K,:)),-imag(Zfnum(K,:)),linestylesK,’LineWidth’,1.5,...
’MarkerSize’,10,’MarkerFaceColor’,’auto’)
xlabel(’Re(Z) [\Omega cm^2]’,’FontSize’,15)
ylabel(’-Im(Z) [\Omega cm^2]’,’FontSize’,15)
legendinfoK = [’Z_f, numerical. I_0 = ’,num2str(I0(K))];
legend(legendinfo)
title([’Nyquist plot of photoelectrochemical impedance. k = ’,...
num2str(k),’ , steps = ’,num2str(NPOINTS-1)])
axis equal
set(gca,’FontSize’,15)
hold all
end