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Research Collection
Doctoral Thesis
The deployment of digital simulation tools to verify cyclicvoltammetry experiments
Author(s): Bolinger, Roman Wilhelm
Publication Date: 2000
Permanent Link: https://doi.org/10.3929/ethz-a-004041922
Rights / License: In Copyright - Non-Commercial Use Permitted
This page was generated automatically upon download from the ETH Zurich Research Collection. For moreinformation please consult the Terms of use.
ETH Library
Diss. ETH No. 13637
The Deployment of Digital Simulation Tools to
Verify Cyclic Voltammetry Experiments
Investigation Using Examples of
«-Hydroxyketones Systems and
the Electrochemical Oregonator
A dissertation submitted to the
SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZÜRICH
for the degree of
DOCTOR OF TECHNICAL SCIENCES
presented by
Roman Wilhelm Bolinger
Dipl. Chem.-Ing. ETH
born January 4, 1971
from Kaiseraugst, Aargau
Accepted on the recommendation of
Prof. Dr. P. Rys, referee
Prof. Dr. B. Jaun, co-referee
Prof. Dr. B. Speiser, co-referee
Zürich 2000
The optimist says, the glass is half full.
The pessimist says, the glass is half
empty.
The engineer says, the glass is twice as
big as it needs to be.
To my parents
Meinen Eltern
Acknowledgements
There are several individuals whom without their help and expertise the completion of
my thesis would not have been possible:
Prof. P. Rys: Many thanks for your professional insight, your valuable guidanceand the countless fruitful discussions.
Prof. B. Jaun: Your generous support with the experimental stages of my thesis as
well as your organic chemistry expertise were crucial to my success.
Prof. B. Speiser: Our pivotal work together on the simulation of cyclic voltammetry
experiments as well as my time spent in Tübingen was extremely important in takingthe first step.
Dr. A. Klaus: Your love for languages, which was reflected in the modifications you
provided to my writing, is greatly appreciated.
Dr. E. Dedeoglu: Your administrative skilfulness in providing me with the neces¬
sary computer tools and scrutinisation of my mathematical derivations was essential
for my research.
Mr. A. Dutly: In addition to my thanks for your proficient GC-MS analyses I en¬
joyed our conversations over "Omi's Hackbraten".
I also owe special recognition to Prof. Rys' group members, and my friends, who
provided assistance in the lab and were professionally and personally supportive.
Meinen Eltern danke ich besonders für Ihre wertvolle Unterstützung während meiner
ganzen Ausbildung.
Contents
Zusammenfassung 1
Abstract 3
1 Introduction and Objectives 5
1.1 Introduction 5
1.2 Objectives 7
2 Electrochemistry 9
2.1 Introduction 9
2.2 Principles of Electrochemistry 9
2.2.1 Electrode Potentials 9
2.2.2 Measuring Electrode Potentials 11
2.2.3 Double Layer 12
2.2.4 Kinetics 13
2.2.5 The Influence of Mass Transport 15
2.2.6 Reversible, Irreversible and Quasi-Reversible Reactions 17
2.2.7 E & C Nomenclature 18
2.3 Introduction to Cyclic Voltammetry 19
2.3.1 Linear Sweep and Cyclic Voltammetry 19
2.3.2 Solvents and Electrolytes 19
2.3.3 Electrodes 20
2.3.4 Working Electrodes 21
2.3.5 Reference Electrodes for Use in Aqueous Solvents 21
2.3.6 Reference Electrodes for Use in Organic Solvents 21
2.3.7 Counter Electrodes 22
2.3.8 The Electrochemical Cell 22
3 Digital Simulation in Cyclic Voltammetry 25
3.1 Introduction 25
3.2 The EqrCEqr/COMP Mechanism 26
3.3 Derivation of the Orthogonal Collocation Model 27
3.3.1 Basic Equations for the EqrCEqr/COMP Mechanism 27
3.3.2 Initial and Boundary Conditions 28
3.3.3 Discretisation with Orthogonal Collocation 29
I
II Contents
3.3.4 Spline Collocation 33
3.3.5 Transformations of the Space Coordinate X 38
3.4 Results and Discussion 43
3.4.1 Measurements in Aqueous Media 43
3.4.2 Simulations with the EqrCEqr/COMP Mechanism 43
3.4.3 Influence of Starting Potential 44
3.4.4 Special Simulation Aspects 45
3.4.5 Discussion of the Simulation Results 49
3.4.6 Experiments in Acetonitrile 51
3.4.7 Cyclic Voltammetry Experiments 51
3.5 Hydratisation Effects 53
4 Oscillating Reactions 55
4.1 Introduction 55
4.2 A Brief Historical Overview 55
4.3 The Belousov-Zhabotinsky Reaction 56
4.3.1 General Aspects 56
4.3.2 The Mechanism of the BZ Reaction 56
4.4 The Oregonator 57
4.5 Electrochemical Oscillators 58
4.6 The Mercury/Chloropentammine Co(III) Oscillator 58
4.7 Oscillations in Quinone Systems 59
5 An Electrochemical Oregonator 61
5.1 Design of an Electrochemical Oregonator 61
5.2 Mathematical Formulation 62
5.3 Oscillating Behaviour of the EqrCCCCC Mechanism 64
5.4 Simulating the Electrochemical Oregonator 65
5.4.1 General Aspects 65
5.4.2 Testing the CVSIM Model 67
5.4.3 Simulations with the Full Electrochemical Oregonator 68
5.4.4 Results 70
5.5 Numerical Instabilities 76
6 Outlook 81
7 Experimental Section 83
7.1 Instrumental Setup 83
7.2 Reference and Working Electrodes 84
7.2.1 General Remarks 84
7.2.2 Electrodes in Aqueous Media 84
7.2.3 Reference Electrode in Acetonitrile 84
7.2.4 Working Electrode in Acetonitrile 85
7.3 Solvent and Reagent Preparation 85
7.3.1 Procedure for the Aqueous Electrolyte 85
Contents III
7.3.2 Purification of the Organic Electrolyte 85
7.3.3 Preparation of the Acyloins 87
Bibliography 92
A Symbols 93
B Abbreviations 99
Curriculum Vitae 103
Zusammenfassung
Die vorliegende Arbeit beschäftigt sich mit der Untersuchung des Reduktionsmechanis¬
mus von Küpenfarbstoffen mit a-Hydroxyketonen mittels experimenteller zyklischerVoltammetrie und dem Einsatz von digitalen Simulationstechniken für die zyklischeVoltammetrie.
Im ersten Teil dieser Arbeit führte man zyklisch-voltammetrische Experimente mit
den zwei unterschiedlichen a-Hydroxyketonen Acetoin und Adipoin in einer alkali¬
schen wässrigen Lösung durch. Die so erhaltenen Voltammogramme und die bereits
existierenden experimentellen Daten wurden mit den Simulationsergebnissen des ur¬
sprünglich vorgeschlagenen EqrCEqr/COMP-Mechanismus verglichen. CVSIM, das in
dieser Arbeit verwendete Simulationsprogramm für zyklische Voltammetrie, setzt die
orthogonale Collocation als mathematische Methode für die numerische Integrationder partiellen Differentialgleichungen ein. Der Vergleich zwischen den experimentellenund den simulierten Daten zeigte einige Ähnlichkeiten, aber auch gewichtige Unter¬
schiede. Das experimentell untersuchte System folgt entweder einem anderen als dem
EqrCEqr/COMP-Reaktionsmechanismus oder die experimentellen Daten können nur
in einem sehr engen, bisher noch nicht gefundenen Bereich von Parameterwerten rech¬
nerisch nachgebildet werden.
Die symmetrische Form des ersten anodischen Peaks in den experimentellen zy¬
klischen Voltammogrammen deutet auf Adsorptionseffekte hin. Das CVSIM-Simula-
tionsmodell berücksichtigt diese Adsorptionseffekte nicht, obwohl diese grundsätzlichin das Modell implementiert werden könnten. Da man sich aber dadurch keine we¬
sentliche Verbesserung in der Übereinstimmung zwischen experimentellen und berech¬
neten Daten erhoffen konnte, wurde auf die entsprechende Erweiterung im CVSIM-
Simulationsmodell verzichtet.
Um die Bedeutung möglicher vorgelagerter Adsorptions- und Hydratationsgleich¬
gewichte auszuloten, eliminierte man diese vorerst dadurch, dass man die zyklisch-voltammetrischen Experimente in einem nicht-wässrigen Lösungsmittel, dem Aceto-
nitril, durchführte, in Anwesenheit von Adipoin und seinem Oxidationsprodukt 1,2-
Cyclohexandion. Im Potentialbereich zwischen +1.6 V und —2.3 V vs. Ag/Ag+ und
für Vorschubgeschwindigkeiten zwischen 50 mV/s und 200 V/s konnten keine Peaks
beobachtet werden. Die Reaktanden schienen unter diesen Bedingungen elektroche¬
misch vollständig inaktiv zu sein. Um die Reduktionskraft von Adipoin zu steigern,erhöhte man den pH-Wert der Elektrolytlösung. Zu diesem Zweck wurde BEMP (2-tert-Butylimino-2-diethylamino-l,3-dimethyl-perhydro-l,3,2-diazaphosphorin), eine
so genannte Schwesinger Base, eingesetzt. Selbst nach Zugabe von BEMP zur Elek-
1
2 Zusammenfassung
trolytlösung konnten aber keine Oxidationspeaks beobachtet werden. Es muss ange¬
nommen werden, dass Adipoin entweder durch die Bildung des entsprechenden Car-
binolammonium-Additionsproduktes mit den Amino-Gruppen von BEMP oder durch
die Bildung seines zyklischen Dimers als reduktives Reagens desaktiviert wird.
Die Experimente in Acetonitril zeigten, dass vorgelagerte Hydratationsgleichge¬wichte von eminenter Wichtigkeit für die Geschwindigkeit der Redox- und der Kom-
proportionierungsschritte sein müssen. Sie beeinflussen offensichtlich die effektiven
Konzentrationen der aktiven reduzierenden Spezies entscheidend. Deshalb müssen
zukünftige Untersuchungen diese zusätzlichen Reaktionen berücksichtigen und können
nur in wässrigen Lösungen durchgeführt werden.
Da es nicht gelang, die experimentellen zyklischen Voltammogramme dieses Redox-
Systems mit a-Hydroxyketonen unter Zugrundelegung des ursprünglich vorgeschla¬
genen Reaktionsmechanismus (EqrCEqr/COMP) mittels digitaler Simulation nachzu¬
bilden, war es naheliegend zu vermuten, dass dieser hypothetische Mechanismus die
Reaktionsschritte im experimentell untersuchten Redox-System nicht vollständig und
korrekt wiedergibt. Dies wurde unterdessen in anderen experimentellen Studien inso¬
fern bestätigt, als dass das experimentelle Redox-System einen oszillierenden Reakti¬
onsverlauf zeigt, welcher durch den EqrCEqr/COMP-Mechanismus nicht beschrieben
wird.
Im zweiten Teil der Arbeit wurde daher evaluiert, inwieweit oszillierende Reaktions¬
systeme mittels zyklischer Voltammetrie analysiert und charakterisiert werden können.
Als Modellreaktion wurde die Belousov-Zhabotmsky-R.eaktion ausgewählt, deren Re¬
aktionsschritte sich vereinfacht mit Hilfe des Oregonators beschreiben lassen. Um die
zyklischen Voltammogramme dieses oszillierenden Reaktionssystems rechnerisch simu¬
lieren zu können, führte man eine Mediatorreaktion ein, welche die heterogene Elektro¬
nenübertragung an der Elektrode mit der Sequenz der homogenen Reaktionsschritte
verbindet. Die CVSIM-Simulationen konzentrierten sich darauf, jenes Zeitfenster zu
finden, in welchem sich das oszillierende Verhalten der Reaktion manifestiert. Durch
Variieren der Vorschubgeschwindigkeit gelang es schliesslich einen Messbereich zu fin¬
den, in welchem die über eine Anzahl von Messzyklen auftretende Reihe von anodischen
Peaks periodisch steigt und fällt.
Wie sich umgekehrt aus entsprechenden experimentellen zyklischen Voltammogram-men eines oszillierenden Reaktionssystems die reaktionsmechanistischen Informationen
ableiten Hessen, soll Gegenstand zukünftiger Studien über die Verküpungsreaktion mit
CK-Hydroxyketonen werden.
Abstract
In this work the mechanism of the vat dye reduction by a-hydroxyketones was inves¬
tigated by means of cyclovoltammetric experiments and digital simulation techniquesfor cyclic voltammetry.
In the first part cyclovoltammetric experiments with the two a-hydroxyketonesacetoin and adipoin were carried out in an alkaline aqueous solution. Together with
already existing experimental data these voltammograms were compared with the sim¬
ulated output of the postulated EqrCEqr/COMP mechanism. CVSIM, the simulation
program for cyclic voltammetry employed in this thesis, uses orthogonal collocation
as the mathematical method for the numerical integration of the partial differential
equations. The comparison between the experimental and the simulated data showed
some similarities but also some discrepancies. The system investigated experimentallyeither seems to match with a reaction mechanism different from the EqrCEqr/COMPmechanism or the experimental data can only be simulated with a very narrow range
of parameters which has not been found yet.
The symmetrical shape of the first anodic peak in the experiments suggests the
presence of adsorption effects. The simulation model, however, does not take into
account any adsorption effects, although the implementation of such effects would be
feasible. But because the implementation of adsorption phenomena into the model
was not considered to enhance substantially the conformity between experimental and
simulated data, the effect of extending the CVSIM simulation model accordingly was
not studied further.
For a more detailed investigation of the potential influence of pre-equilibria from
adsorption or hydratation phenomena on the mechanism it was necessary to replace the
aqueous solvent for the cyclovoltammetric experiments by a non-aqueous solvent like
acetonitrile. Also, in addition to the reducing agent adipoin its oxidation product 1,2-
cyclohexanedione was present in the solutions investigated. Neither for adipoin nor for
1,2-cyclohexanedione redox peaks were observed between +1.6 V and —2.3 V vs. the
Ag/Ag+ reference electrode and for sweep rates between 50 mV/s and 200 V/s. The
reactants seemed to be completely electrochemically inactive under these conditions.
To enhance the reductive power of adipoin, the pH of the electrolyte solution had
to be increased. BEMP (2-£er£-butylimino-2-diethylamino-f ,3-dimethyl-perhydro-f,3,2-diazaphosphorine), the so-called Schwesmger base, was chosen for this purpose.
However, even after the addition of BEMP to the electrolyte solution no oxidation
peaks could be observed. Obviously the adipoin is deactivated as reducing agent either
by forming the corresponding carbinolammonium addition product with the amino
3
4 Abstract
function of BEMP or by forming a cyclic dimer.
The experiments in acetonitrile showed that hydratation pre-equilibria of the a-
hydroxyketone in aqueous solvents must be of crucial importance for the rates of the
redox as well as the comproportionation steps. They obviously influence decisively the
effective concentrations of the active reducing species. Therefore, future studies have
to take into account these additional reactions and can be run solely in aqueous media.
Because the experimental cyclovoltammograms of this redox system with a-hydro-
xyketones could not be imitated by means of digital simulation on the base of the
postulated EqrCEqr/COMP mechanism, this hypothetical mechanism was not assumed
to reflect completely and correctly the reaction steps of the redox system studied ex¬
perimentally. Meanwhile these findings are corroborated by other experimental studies
revealing that the experimental redox system exhibits an oscillating reaction behaviour
which cannot be described by the EqrCEqr/COMP mechanism.
The second part of this work treats the evaluation as to the question how far
oscillating reaction systems can be analysed and characterised with the aid of cyclo-
voltammetry. As a model reaction the Belousov-Zhabotmsky reaction was chosen, the
reaction steps of which can be described in a simplified way by the Oregonator. For
the numerical simulation of the cyclovoltammograms of this oscillating reaction systemfirst a mediator reaction had to be introduced connecting the heterogeneous electron
transfer at the electrode with the sequence of homogeneous reaction steps. The CVSIM
simulations were focused on finding the time window within which the oscillating be¬
haviour of the reaction occurs. By varying the sweep rates it was possible at last to
find that measuring range displaying a series of anodic peaks periodically increasingand decreasing over a number of measuring cycles.
Future studies on the vatting reaction with a-hydroxyketones will have to show how
reaction mechanistic information can be retrieved from the corresponding experimental
cyclovoltammograms.
Chapter 1
Introduction and Objectives
1.1 Introduction
Cotton is the most important textile fibre used worldwide. Its current market share
is more than fifty percent. Cotton fibres are almost ubiquitous in our daily life. The
colouring is effected by means of several types of dyes. The use of vat dyes makes
up about twenty percent of the total consumption [1]. Typical examples for vat dyesare indigo (1) or pyranthrone (2). In general, they are structurally characterised by a
system of conjugated carbonyl groups (3).
R1 R2 R3 R4I r I I
-,I
o=cfc=cfc=oL Jn
Vat dyes are applied in a specific way (equations 1.1 - 1.3): Due to their insolubilityin water vat dyes have to be transformed first into the water-soluble form, the so-
called leuco form. The corresponding reduction is usually run with sodium dithionite
at pH 13.
5
6 Introduction and Objectives
R1 R2 R3 R4 R1 R2 R3 R4
S2042-+ 4 OH" + 0=cfc=cfC =0—-2 S032-+ 4 H20 +"0-cfC-C^C-O" (1.1)
l JnL Jn
leuco form
2 Na2S204 + 2 NaOH —*- Na2S203 + 2 Na2S03 + H20 (1.2)
Na2S204 + 02 + 2 NaOH —*~ Na2S03 + Na2S04 + H20 (1.3)
The reduced dye (leuco form) subsequently diffuses into the fibre and is finallyreoxidised. One of the major drawbacks of this industrial dyeing process is the highamount (worldwide approx. f80,000 t) of sulfate, sulfite and thiosulfate in the waste
water and of the toxic sulfide formed subsequently by biogenetic corrosion of the waste
water pipelines. Increasingly severe environmental laws force textile industry to developmore biocompatible vatting systems. Thus, our research efforts are focused on the
replacement of sodium dithionite in the vat dyeing process by a biodegradable organic
compound [2],[3],[4]. The class of CK-hydroxyketones seems to meet the requirementsin terms of reductive efficiency and biodegradability. One industrial process is already
employing hydroxyacetone instead of sodium dithionite [5],[6].The ability of CK-hydroxyketones to reduce vat dyes is demonstrated in equations
1-4 and 1.5.
SHN sNHs6 O HO OH O^O
_
R5 R611 1 l l + B i\fc) 1 + B I
_
I._ t,
R5-C—C-R6 «* R5-C= C-R6^« *- R5-C—C-R6^=^"0-C-C-0_ (1.4)I - HB - HBH
R1 R2 R3 R4 R5 R6 R1 R2 R3 R4 R5 R6I r I I -, I II I r I I -, I II
._, _.
o=c-Hc=c-Hc=o + ~o-c=c-o~ "0-CfC-CfC-0"+ o=c-c=o (1.5)L Jn L Jn
The oxidation products of the CK-hydroxyketones were investigated by Fédérer [2].The biodegradability of these compounds was the subject of Haaser's work [3], and
Jermini [4] focused on the reactivity and the reaction mechanism.
To complement the investigations on the reaction mechanism, Matic [7] studied
the CK-hydroxyketone systems by means of cyclic voltammetry: The electrode replacesthe vat dye in this redox process, i.e. the electron transfer takes place between the
CK-hydroxyketone and the electrode instead.
However, the apparent complexity of this system and the fact that the output
from cyclic voltammetry experiments does not contain structure related information
and that the shape of experimental cyclic voltammograms cannot be easily predicted,made it impossible until now to determine the underlying mechanism.
The complexity of the redox system of the vatting process with CK-hydroxyketonesbecomes evident if one reflects on various surprising results: Some cyclic voltammo-
f. 2 Objectives 7
grams show an irregular dependence of the anodic current peaks on the concentra¬
tions of the CK-hydroxyketones, on the starting potentials and on the sweep cycles
[7]. From this point of view a chaotic or an oscillating behaviour of the redox systemcannot be excluded. This idea is supported by the observation that the oxidation of
CK-hydroxyketones is an autocatalytic process. Two different explanations have been
given for this surprising kinetic behaviour: At first, it was shown that the correspond¬
ing diketones formed during the primary reduction step are highly unstable and thus
readily converted into a secondary reducing species by aldol condensation (see equa¬
tion 1.6). This new CK-hydroxyketone reduces the dye concurrently with the primary
reducing agent.
H,0
+ H20
oh o
(1.6)
The autocatalytic reduction behaviour of CK-hydroxyketones could also be explained
by a mechanism shown in figure 1.1.
Such a reaction mechanism is fulfilling all requirements for the occurrence of an os¬
cillatory behaviour under specific reaction conditions. This would also explain the ob¬
servation of oscillatory reactions during the oxidation of CK-hydroxyketones by iron(III)salts [81.
1.2 Objectives
The main goal of this thesis is to verify or to dismiss the reaction mechanism for
the reduction of vat dyes by CK-hydroxyketones proposed by Matic [7] and Jermini [4].The mechanism was suggested on the basis of many experimental cyclovoltammograms
characterising a reduction system (with CK-hydroxyketones) in which the vat dye is
substituted by the oxidising surface of an electrode. To meet the goal mentioned
above, digital simulation techniques for cyclic voltammetry are employed. In a first
step the voltammograms for the reaction sequence shown in figure 1.1 will be simulated
and compared with already existing experimental data.
The complex reaction system proposed might exhibit an oscillatory behaviour under
certain, yet unknown, conditions. Therefore, a well-known chemical oscillator, e.g. the
Oregonator [9], will also be simulated. A comparison of the characteristic propertiesof the obtained simulated cyclovoltammograms with experimental data might givesome answer to the question whether the very complicated pattern observed in the
experimental cyclovoltammograms is caused by the oscillatory behaviour of the redox
system.
8 Introduction and Objectives
Dye (Dye-) Dye- (Dye2")
Dye (Dye-)
Figure 1.1: Reaction scheme for the reduction mechanism of vat dyes by
a-hydroxyketones. Example: HKH = H3CC(0)C(OH)HCH3;HR" = H3CC(0-)C(OH)CH3; HK* = H3CC(0*)C(OH)CH3; K- =
H3CC(0*)C(0-)CH3; K = H3CC(0)C(0)CH3; B" = base; aid = aldol
condensation (see equation 1.6).
Chapter 2
Electrochemistry
2.1 Introduction
This chapter summarises briefly the most relevant topics and equations of electro¬
chemistry. For a more profound understanding of the electrochemical theory, one has
to resort to one of the numerous textbooks (e.g. [10],[11],[12],[13]).The principles of electrochemistry also apply to cyclic voltammetry (CV). The
symbols and abbreviations used in this chapter are enumerated at the end of the thesis
in appendices A and B.
2.2 Principles of Electrochemistry
2.2.1 Electrode Potentials
The electrode potential is the main thermodynamic variable of electrochemistry. Before
the other variables and aspects are considered, the origin of electrode potentials has to
be defined and explained.The chemical potential of a species I in solution is given by
fj,i = f^ + RT\nai . (2.1)
The activity ai is defined as the product of the dimensionless activity coefficient ji
and the standardised molality mi/m0 (m0 = 1 mole/kg):
ai = li— • (2.2)m0
In ideal solutions 7« equals 1 and ai only depends on the molality of component
I. The chemical potential can also be expressed as the partial derivative of the free
energy G with respect to the mole fraction xf.
-(It •
9
10 Electrochemistry
Therefore, the total free energy of a solution is:
G = ^Xlin . (2.4)i
Hence, the free energy change during a reaction at equilibrium is described by the
fundamental expression:
AIG = Y,vlfil = 0. (2.5)
i
In electrochemistry frequently the situation is encountered where a species I exists
within two different solutions, with the two phases I and II being in contact with each
other. If the chemical equilibrium is established at the phase boundary, the following
expression is valid:
MI) = MU) . (2.6)
A simple example for this situation is a metal electrode being placed in a solution
containing the corresponding metal ion. The reaction leading to the equilibrium is
generally described as
Mra+(solv) + n-e-— M , (2.7)
where M is the metal and n is the number of electrons transferred. If the chemical
potentials of the two phases are different, either metal ions from the solution will be
deposited on the metal surface or metal ions from the electrode will be dissolved into the
solution. At equilibrium, dissolution is as fast as deposition, whereas if the equilibriumstate is left into one or the other direction, one of the two reactions is taking placemore rapidly. Thus, equation (2.6) is extended to:
MI) + THFiptO) = MU) + n^MH) , (2.8)
where MI) and MH) are the electrical potentials within the phases (I) and (II),and are referred to as the Galvani potentials. The term riiFtpi stands for the work
needed to transfer one mole of n-valent ions from a remote position to the interior of
the solution with a potential </?. The electrochemical potential fri is therefore defined
as
fii = Hi+ ntFLpi = [i° + RT In at +riiFtpi . (2.9)
According to equation (2.5) the condition for the electrochemical equilibrium is:
J>^ = 0. (2.10)
i
The stoichiometric coefficient vi is a positive number for the products and a negativenumber for the reagents.
2.2 Principles of Electrochemistry ff
If equation (2.9) is taken into account and the equilibrium Galvani potential differ¬ence is written for equation (2.8), equation (2.11) is obtained:
M[ + RT In aM = ßM«+ + RT m aM«+ + nFipi
+M + nRT In ae- - nFcpn , (2.11)
where </?i is the Galvani potential of the metal electrode, and </?n is the Galvani
potential of the solution. The activity terms of the metal atoms and the electrons in
the electrode can be neglected, because they are constant. Thus, equation (2.11) is
rewritten as
—jlnaMn+ . (2.12)
A</?° is referred to as the standard Galvani potential difference in the case where
aMn+ = 1. Now two problems arise. First of all, the Galvani potential inside the metal
depends on the potential in the solution. Second, the Galvani potential of the solution
itself is inaccessible by experiments, and hence it cannot act as reference potentialfor the electrode itself. It is a well-known fact that potentials can only be measured
relative to some standard. However, if a suitable reference electrode system is set up,
directly the electrode potential E can be used instead of A</?. The exchange of these
two variables in equation (2.12) leads to the Nernst equation:
(RF\— jlnaMn+ • (2.13)
The potential of a redox couple l/m (I + e~ ^ m) which is reduced or oxidised at
an inert metal electrode is a fairly representative situation for most electroanalyticalmethods. The adapted version of the Nernst equation yields the equilibrium potential
El/m'.
RT\ ai
^^{wrt (2-i4)
If the activities are replaced by the product of the activity coefficient (7) and the
solution concentration (cqo), the equilibrium potential E°, is calculated as follows:
E?,m = E?,m+ -77I* —+ -T7
In-^-. (2.15)
2.2.2 Measuring Electrode Potentials
As mentioned in the last section, the Galvani potential difference between the electrode
and the solution is only accessible by the experiment with an additional reference elec¬
trode. The original version of such an electrode is the Normal Hydrogen Electrode,NHE. It is also called Standard Hydrogen Electrode (SHE). Generally, these measure¬
ments are conducted in specially designed electrochemical cells. They consist of two
12 Electrochemistry
so-called half-cells, which are often separated by an ion-permeable membrane or a salt
bridge with a frit. One of the half-cells contains the reference electrode and the other
one the metal/metal ion system of interest. The two electrodes are interconnected byan external control and measurement circuit. A possible construction is depicted in
figure 3-5 in [12] p. 83. To measure the real standard potential of the metal/metalion, all ions have to be present in unit activity, and the hydrogen pressure has to be
1 atmosphere. The most frequently used reference electrodes are the normal hydro¬
gen electrode, the Saturated Calomel Electrode (SCE, Hg/Hg2Cl2) and the Ag/AgClelectrode. Reaction equations and standard electrode potentials for these reference
electrodes are given by:
H+(aq) + e~ = \ H2(g, 1 atm) E° = 0.000 V
Hg2Cl2(s) + 2e" = 2Hg(l) + 2C1" E° = 0.241 V
AgCl(s) + 2e- = Ag(s)+Cl- E° = 0.197 V
Numerous tables for all different kinds of electrode reactions and potentials can
be found e.g. in [14],[15],[16]. Reference electrodes and their experimental details are
described in the book of Ives and Janz [17].
2.2.3 Double Layer
In the previous sections an equilibrium state was described, where a metal electrode is
immersed in a solution with the corresponding metal ions. Each time the equilibrium is
disturbed, either the forward or the reverse reaction (see equation 2.7) will be favoured
until the system is back to the original state. If the deposition of metal ions onto the
metal electrode is faster than the reverse reaction, the electrode surface will become
positively charged. The electron deficiency on the metal surface attracts anions from
the solution. This process builds up the so-called double layer (see figure 2.1). To
disturb the system a potential that is either lower or higher than the equilibrium
potential can be imposed on the electrodes. If the potential is driven slowly away
from the original state, first only the charge of the double layer will be altered. If the
potential difference is large enough to drive either one of the reactions, a net current
flowing between the two electrodes can be observed and measured. This so-called
overpotential highly depends on the electroactive species itself and on the electrode
material.
The most simple model for the double layer is the parallel-plate capacitor, as shown
in figure 2.1. In analogy to the capacitor, the first plate is the electrode and the
other one the solution adjacent to the layer. Helmholtz termed the region between the
electrode and the center of the solvated counter ions (OHP = Outer Helmholtz Plane)the compact double layer. Today this layer is known as Helmholtz layer. If the distance
between the plates Ax equals I nm and the voltage drop AE equals f V, the resultingfield strength is Ax/AE = f09 V/m or f07 V/cm. This field strength is of six orders
of magnitude higher than the one observed with conventional capillary electrophoresis,where a 60 cm capillary and a voltage drop of 30 kV is employed.
2.2 Principles of Electrochemistry 13
Figure 2.1: The parallel plate capacitor model for the double layer. The distance
between the two plates is a/2, where a is the diameter of the solvated
counter ion.
This model does not take into account the possibility that some forces and influences
weaken the attraction of the solvated ions by the electrode surface. Goüy and Chapman
(see e.g. [f f] p. 80) tried to consider the effect of thermal motion of the ions. Theyintroduced a so-called diffuse double layer adjacent to the Helmholtz layer. This new
layer contains ions of either charge and is more extended.
Nonetheless, the idea of specifically adsorbed ions on the electrode surface still was
not incorporated. All kind of species (ions, neutral molecules, solvent dipoles, etc.)
may be adsorbed on a metal electrode surface either by van der Waals or coulombic
interactions. By stripping off their solvent sheath, particularly anions undergo spe¬
cific adsorption by van der Waals interaction. The possibility of having anions closer
adsorbed to the electrode surface than cations leads to the proposition of an Inner
Helmholtz Plane (IHP). This type of adsorption even takes place if the electrode sur¬
face is below the Potential of Zero Charge (PZC). The PZC is the potential where no
free excess charge exists on the electrode surface. Figure 2.2 represents this final model
with an inner and an outer Helmholtz plane.
2.2.4 Kinetics
The former sections focused mainly on the thermodynamics of the electron transfer. By
extending the view to kinetics, the rate limiting factors for the overall process certainlyhave to be considered. A short analysis of a complete electrochemical reaction reveals
that the following steps are involved:
• Mass transport from the solution to the electrode
• Adsorption on the electrode
14 Electrochemistry
solvated anion
specifically adsorbed anion
solvated cation
double layer diffusion layer
Figure 2.2: The extended 3-layer model consisting of the double layer (between metal
surface and outer Helmholtz layer), diffusion layer (adjacent to the double
layer) and electrolyte solution (adjacent to the diffusion layer, not shown
here).
• Electron transfer at the electrode
Desorption from the electrode
• Mass transport from the electrode to the solution
The redox system in equilibrium supplies thermodynamic information. In order to
elucidate the kinetics, the overpotential r/ = E— E°, has to be applied to the electrode,which in turn induces a current. The potential drop in the electrolyte between the two
electrodes (iR) and the overpotentials at the cathode (rjc) and the anode (tja) yield
together the total shift of the cell voltage:
Ecell — £cell,rev — ÏR + TJc + TJa
The current density j is a function of 77, because the relations between these variables
are defined as:
2.2 Principles of Electrochemistry 15
J = Jc + Ja = -nF (kcci>00 - kACm,oo) (2.16)
f ~aUmnFrl 1kc = kc,o exp <^
— V
(l-a°/m)nFV-kA = kA,o exp <( —
The transfer coefficient of the oxidation m — I (cx°,m) describes the symmetry of
the activation barrier. There is a corresponding coefficient for the reduction reaction,but to reduce the amount of variables it had to be substituted by f — ot°,. Actually,this simplification only applies for one-electron transfers, but it is generally acceptedfor most of the multiple electron transfers as well.
There is no net current at equilibrium. This means that jc and ja are equal and
the corresponding current density is called exchange current density j0:
jo = nFci>00kc,o = nFcmtQOkA,o
Substitution of this equation into equation (2.16) leads to the well-known Butler-
Volmer equation:
J =Jo
(! - aî/JnFV \ f -a°/mnFri ]exp
' v — '
RT |--FfF}(2.17)
2.2.5 The Influence of Mass Transport
For sufficiently high positive or negative overpotentials the limiting form of equation
(2.17) is reached. This dependence was already observed empirically. In the logarith¬mic form it is known as Tafel equation:
V = ~^-ß On bo| + In \j\) = a + bin \j\ . (2.18)ai/mnP
The variables a and b stand for a = bin \j0\ and b = RT/(a°,nF), respectively. The
rate of reaction can be determined simply by controlling the potential of the electrode.
The limiting factor for this control is the mass transport, which can be expressed in
terms of the current density for each species. The cathodic current density ji is negative
by definition:
ji = -nF (q)00 - Q,o) 3m = nF (cm>00 - cm>0)
On the assumption that the limiting current densities at both electrodes are equal
(ji,L = Jm,L = Jl), the combination of the electron transfer reaction with the mass
16 Electrochemistry
transport results in equation (2.19). The corresponding curves for a = 0.5 are shown
in figure 2.3:
exp(1-of/JniV
RTexp
-a?/mnFr) \RT J
JL .
h expJo
(1 " aî/JnFr,RT
+ exp-tf/mnFnRT
(2.19)
1.0
0.5
0.0
-0.5
-1.0_
-1000
' 3l/Jo = 1
•3l/3o = 100
Jl/Jo = 100000
-500 0
n [mV]
500 1000
Figure 2.3: j/Jl as a function of n for different values o/Jl/Jo-
It is evident that mass transport plays an important role as far as the total reaction
rate is concerned. There are three possible contributions for mass transport in solu¬
tion, namely diffusion, convection and migration. If the set-up consists of a stationaryelectrode without a stirrer, then convection will occur only due to small temperature
or concentration changes within the solution. Migration can mostly be neglected con¬
sidering the high concentration of the inert electrolyte. Therefore, diffusion represents
the main mechanism for mass transport under these conditions.
If the system is pushed to the mass transport limit by the application of a sufficiently
high overpotential, Fick's first law can be applied:
j = nFD 1^-\dx)x=
= nFD-Co
x=0 in(2.20)
2.2 Principles of Electrochemistry 17
In this case, cq
calculated from
0 and the so-called diffusion-limited current density can be
j = nFD^-On
(2.21)
To visualise the influence of diffusion, the concentration profiles for both a diffusion-
controlled and a kinetically controlled reaction are sketched in figure 2.4-
0 >N distance from the electrode surface
Figure 2.4: Concentration profiles for kinetically controlled (la and lb) and diffusion-controlled (2a and 2b) electrochemical reactions at an electrode surface,la and 2a, respectively, show the situation right after the start of the
reaction, whereas lb and 2b depict the profiles after a time interval.
The curves in figure 2.4 are simplified by means of the Nernst linearisation. This
simplification is shown in figure 2.5.
2.2.6 Reversible, Irreversible and Quasi—Reversible Reactions
The kinetics of the electron transfer themselves largely determine the overall charac¬
teristics of the electrochemical reaction. A deeper understanding of the interactions
between electron transfer and mass transport is important for the interpretation of the
results from electroanalytical experiments.
Very fast electron transfer reactions (ETRs) have only one transition range, where
mass transport control changes rapidly from the anodic to the cathodic region and vice
versa. Thus, an equilibrium plateau with no measurable net current is not observed.
Very little overpotential is needed to push the whole process to the mass transport limit.
Because of the thermodynamic equilibrium at the phase boundaries these reactions are
reversible. The rate constant of the ETR exceeds f0_1 cm/s and Nernst conditions are
fulfilled.
18 Electrochemistry
electrode diffusion layer
true concentration profile
Nernst approximation
x = 0 x = 5n
Figure 2.5: Nernst linearisation of the concentration profile within the diffusion layer.
For extremely slow ETRs (i.e., heterogeneous rate constant < f0~5 cm/s) high
overpotentials have to be applied for the reaction to leave the reaction rate determined
range. Then the electrochemical conversion will be limited to either the anodic or
the cathodic reaction. The corresponding back reaction can be neglected and the
process becomes irreversible. This state is far from the thermodynamic equilibriumand equation (2.14) does not apply any more.
Between these two extreme situations (very fast vs. very slow ETRs) the reaction
will become quasi-reversible, the heterogeneous rate constant lying between f0_1 cm/sand f0~5 cm/s. In contrast to completely reversible reactions these quasi-reversible re¬
actions require a considerable amount of the activation overpotential to drive the chargetransfer reactions. This situation cannot be described suitably if for its mathematical
treatment the above simplifications are presumed.
Cyclic voltammetry provides a variation of sweep rates over several orders of mag¬
nitudes (see section 2.3.1). Therefore, the time scale of the experiment has to be con¬
sidered carefully, because the reaction system may become reversible, quasi-reversibleor irreversible at different sweep rates. Thus, the values for the heterogeneous rate
constant mentioned in this section are only valid for sweep rates around O.f V/s.
2.2.7 E & C Nomenclature
The electrochemists use a well-known notation to describe their reaction mechanisms:
the E & C nomenclature. E stands for a heterogeneous electron transfer at the elec¬
trode, whereas C refers to a homogeneous chemical reaction in solution. The indices r,
i and qr mean reversible, irreversible and quasi-reversible reactions (see section 2.2.6).The mechanism can virtually be any conceivable combination of E's and C's. Each
2.3 Introduction to Cyclic Voltammetry 19
E and C has to be mapped to a clearly defined electron transfer or chemical reaction to
be really meaningful. The example below shows an E^ mechanism, where I is reduced
to m which reacts further to a product p:
l + e~ ^ m E°, ki, a°/mm > p fehom
2.3 Introduction to Cyclic Voltammetry
The general electrochemical basis was already treated in section 2.2. The peculiaritiesof cyclic voltammetry are the issue of the following subsection.
Hemze gives a brief overview in [f 8]. The most important aspects of the theory can
be found in Gosser's book [19], where some experimental examples are explained and
even the simulation of cyclic voltammetry experiments by means of an explicit finite
difference method is shown.
2.3.1 Linear Sweep and Cyclic Voltammetry
The potential can be varied in different ways. It depends on the analytical method,whether potential steps or linear sweeps are applied. Bard and Faulkner [20] describe
the most common methods. In this work the linear sweep technique will be preferred.
Usually, a cyclic voltammetry experiment consists of two or several linear potential
sweeps. Thus, start, switching and end potential(s) have to be defined. Typically two
linear units, so-called half-cycles, are combined, and the start and the end potentialare often identical. The number of possibilities to carry out such an experiment is
almost unlimited. Figure 2. # presents a small collection of conceivable variations.
The sweep rates can be varied in the range from 10~2 V/s to f 06 V/s. At the upper
end of this scale the requirements for the device and the solutions grow significantly.
Sweep rates up to fOOO V/s suffice for most experiments except for investigations of
radicals with very short lifetimes.
2.3.2 Solvents and Electrolytes
The crucial point of combining the right solvent with the right electrolyte for a certain
reactant is discussed in countless papers. General descriptions and recommendations
are collected in [21], [22] and [23]. Two compendia [24], [25] provide more detailed
information to this topic.
The purity of the solvents and the electrolytes strongly influences the potential range
which can be used for the experiments. Careful distillation of the solvents and drying of
the electrolytes considerably improves the quality of the voltammograms. Otherwise,
electrochemically active impurities could possibly undergo a reaction, which inevitablywould result in higher currents.
To remove the cathodically active oxygen the solution in the electrochemical cell
is purged with nitrogen or argon. The purity of the solvent/electrolyte system can
20 Electrochemistry
E„
E„
E„
E„
E„
E„
E„
E„
t t
Figure 2.6: A sample collection of potential variations in cyclic voltammetry.
be verified by simply scanning the whole experimental potential range. The baseline
should not show any aberrant peaks nor clearly visible fluctuations. This so-called
background voltammogram should be run before the addition of the reactants. Purifi¬
cation of the solvents and the background scan have to be repeated until the system
meets the requirements.
Frequently used organic solvents are acetonitrile (MeCN), propylene carbonate
(PC), dimethylformamide (DMF) and dimethylsulfoxide (DMSO). Tetraalkylammo-nium salts are the best choice of electrolyte for these solvents because of their rather
high solubility and their relatively low solution resistance. Tetraethylammonium and
tetrabutylammonium cations in connection with Perchlorate, tetrafluoroborate or hexa-
fluorophosphate anions are most commonly used.
2.3.3 Electrodes
The advent of modern electrochemistry created the need for new electrodes and elec¬
trode set-ups. The most common arrangement today is the electrochemical cell with
three different electrodes:
• working electrode (WE)
• reference electrode (RE)
• counter electrode (CE)
2.3 Introduction to Cyclic Voltammetry 21
The purpose and popular versions of these electrodes are described below. However,there are some general requirements for all kinds of electrodes. Electrochemicallyand chemically inert behaviour in a large potential region is required for high qualityelectrochemical measurements. Strong adsorption, i.e. poisoning and reformation of
the electrode surface, is a rather undesirable effect. Furthermore, low resistance is
essential for proper potential measurements. Finally, the handling of the material
during the manufacturing process often imposes restrictions on the later use. The best
choice of material from the electrochemical point of view does not always lead to the
best electrode in practice.
2.3.4 Working Electrodes
Normally the heterogeneous electron transfers take place at the working electrode.
Hence, the catalytic influence of the electrode material with respect to this transfer
reaction has to be adequate in order not to decrease the overall reaction rate. The
electrode should not interact with the solvent or the supporting electrolyte in a wide
potential range. To obtain a reproducible electrode surface composition it is an im¬
portant prerequisite that a standard procedure for the pretreatment of the WE is
established.
2.3.5 Reference Electrodes for Use in Aqueous Solvents
The vast amount of reference electrodes clearly underlines the importance of electro¬
chemical measurements in aqueous solutions. Many electrochemically useful support¬
ing electrolytes show very high conductivities in water. Therefore, for this solvent it is
relatively easy to build a reference electrode.
The Normal Hydrogen Electrode (NHE) is the most famous reference electrode
because it is used as the primary reference electrode to define standard potentials in
aqueous solutions. It is advisable to consult one of the numerous books, e.g. [17], for
further detailed information about the properties and the design of the NHE.
The Saturated Calomel Electrode (SCE, Hg/Hg2Cl2) used to belong to the most
popular reference electrodes for pH measurements and Polarographie investigations.The SCE was abandoned in favor of the even more reproducible and reliable silver-
silver chloride (Ag/AgCl) electrode.
2.3.6 Reference Electrodes for Use in Organic Solvents
Measurements in nonaqueous solvents require reference electrodes which differ from
their aqueous counterparts. The exclusion of water from the whole system is crucial
for the stability and the reproducibility of the electrode. Some reagents are very sen¬
sitive even to traces of water. Reference electrodes containing an aqueous electrolyte,
e.g. Ag/AgCl in aqueous KCl solution, have to be carefully separated from the rest
of the electrochemical cell. To regain electrical conductivity between the reference and
22 Electrochemistry
the working electrode a salt bridge is needed. However, this set-up causes a signifi¬cant increase of the junction potentials at the phase boundaries between the organicsolvent in the cell and the salt bridge and the aqueous electrolyte around the reference
electrode and the salt bridge.Calomel and other mercurous halides disproportionate in a variety of organic sol¬
vents. Therefore, it is not advisable to replace aqueous electrolytes in calomel electrodes
with electrolytes dissolved in aprotic solvents. Ag/AgCl electrodes can be only used in
a chloride free electrolyte/solvent system, because in many aprotic solvents the highlysoluble anionic AgCln_n+1 complexes formed with chloride ions represent a remarkable
addition to the overall junction potential.Water-free reference electrodes, e.g. Ag/Ag(solv)+ or Li/Li(solv)+ where 'solv'
stands for an organic solvent, require a more sophisticated glassware and a supporting
electrolyte with very high purity. The Ag+ cation is mostly added in the form of either
AgN03 orAgC104.Various other reference electrodes for use in nonaqueous solvents were proposed. A
wide range of information about this topic can be found in [17],[21],[22].
2.3.7 Counter Electrodes
The measurements with the original two-electrode set-up suffers from the fact that
the whole cell current flows through the reference electrode. In modern voltammetrythere is an additional electrode called the counter or auxiliary electrode (CE). In the
so-called three-electrode set-up the cell current flows between the working electrode
and the counter electrode, protecting the reference electrode.
2.3.8 The Electrochemical Cell
The typical electrochemical cell consists of a glass container with a removable cap
having inlets for the electrodes and the gas purge. A stirring bar ensures the homoge¬neous distribution of the reactants within the cell. This forced mixing is carried out
only between the experiments. This procedure ensures that any concentration profiles
throughout the whole cell disappear before the new experiment is started. Figure 2.7
shows a possible arrangement for a three-electrode cell. It is similar to the device that
was used for the experiments performed in this work.
2.3 Introduction to Cyclic Voltammetry 23
from thermostat
to thermostat
working electrode
reference electrode
solution level
counter electrode
stirring bar
Figure 2.7: A three-electrode cell for electrochemical experiments.
24 Electrochemistry
Chapter 3
Digital Simulation in Cyclic
Voltammetry
3.1 Introduction
Cyclic Voltammetry (CV) provides a sophisticated electroanalytical tool to study re¬
action systems which include electron transfers. The simultaneous investigation of
the time dependence by means of sweep rate variation as well as the potential depen¬dence allows to recognise the presence of different species. This recognition can be
tedious based on % vs. t curves gained from potential step experiments alone. However,its use as a technique to investigate reaction mechanisms is hindered by two majordrawbacks. First of all, voltammetrie curves of complex systems cannot be easily pre¬
dicted, i.e. the visualisation of models containing electron transfers and homogeneouschemical reactions demands a high level of expertise and experience. Therefore, the
analytical response from such experiments does not necessarily prove or falsify the pro¬
posed mechanism. Furthermore, structural information cannot be gained by means of
electroanalytical methods. The isolation and subsequent identification of intermedi¬
ates and products is required to clearly verify a proposed model based on top of these
results.
Digital simulation techniques can be used to create powerful tools to visualise and
verify electrochemical mechanisms. The transport phenomena, electron transfers and
chemical reactions in an electrochemical experiment can be described with Partial Dif¬
ferential Equations (PDEs). The first algorithm specially designed for electrochemical
purposes was developed by Feldberg [26] on the basis of Explicit Finite Differences
(EFD). Since its initial use in f969 the original algorithm has been extended and im¬
proved by many scientists. The application of Implicit Finite Differences (IFD) to
the simulation of electroanalytical problems led to a considerable gain in accuracy and
stability of the results. This type of algorithm made significant progress with the de¬
ployment of new techniques (e.g. Hemze and Störzbach [27] and Rudolph [28]). Britz
[29] explains the use of Finite Differences (FD) methods and their derivatives. Tasia
[30] compares FD algorithms for electroanalytical problems.Another method for electroanalytical simulators was developed by Whiting and
25
26 Digital Simulation in Cyclic Voltammetry
Carr [31]. Orthogonal Collocation (OC) [32] was already commonly used in the en¬
gineering and technical chemistry sciences at this time. This method is based on
the approximation of the diffusion equation by polynomials. The partial differential
equations are transformed into Ordinary Differential Equations (ODEs). They can be
solved by numerical integration with less effort than the PDEs. Speiser et al. (see
e.g. [33],[34],[35],[36]) made significant advances in this field. The main advantagesturned out to be low values of CPU time. The application of Spline Collocation (SC,see section 3.3.4) and the transformation of the X space coordinate (see section 3.3.5)will enable the simulation of homogeneous second order reactions with very high di-
mensionless rate constants up to fO10.
This chapter demonstrates the OC equation derivation for the EqrCEqr/COMPmechanism, which involves two quasi-reversible electron transfers (Eqr) together with
a pseudo-first order reaction (C) and a second order comproportionation reaction
(COMP). Thus, special terms are used for the handling of the quasi-reversibiliiy and
SC is employed for the comproportionation reaction. This model has been recently
developed for Speiser's program system EASI [36], because Matic [7] had proposed this
mechanism for the oxidation of acetoin and other acyloins.
3.2 The EqrCEqr/COMP Mechanism
The EqrCEqr/COMP mechanism can be represented schematically as follows:
C: B
COMP: B + C
The standard equilibrium potentials E°,, the heterogeneous rate constants ks>i/m,the transfer coefficients ot°, with the corresponding potential dependent gradients
dai/m/dE and the homogeneous rate constants for the forward reactions kt and the
reverse reactions k_% are independent simulation parameters. The value of the backward
reaction rate constant A;_2 linearly depends on the equilibrium constant K2 of the
comproportionation reaction. The sweep rate v is the controlling variable for the
time dependency of the system. There are twelve parameters to be varied in the
simulation. The task of finding reasonable values for all of these parameters can onlybe accomplished if either suitable literature values can be found or the parameters can
be fitted to the experimental data by means of the simulation.
±e"
±e"
B
C
D
A + D
EA/B, kStA/B, a°A/B> daA/B/0E
Kx = fci/fc_i
E.
c/Di Kc/d, a°c/D, dac/D/dE
K2 = k2/k_2 = expl±JL (Ec/D - EA/B) |
3.3 Derivation of the Orthogonal Collocation Model 27
3.3 Derivation of the Orthogonal Collocation Model
3.3.1 Basic Equations for the EqrCEqr/COMP Mechanism
The diffusion equations with the reaction terms for all the species considered are first
specified in the dimensional form:
dcA d2cAj
k2
W=
Da1)^ + hCBCc -
k~2CaCd (3-1}
dcB d2cBj.j j
.famm
-KT= Db~ô~Y
~
klCB + ^-icc ~ k2cBcc + -ttCaCd (3.2)
decn
d2cc. , , ,
.fa,Q „A
-7TT=
Dc^rir + facB-
k_xcc-
k2cBcc + —cAcD (3.3)at ox1 K2
dcD d2cD fa
W= DD— + k2cBcc - -CAC» . (3.4)
To ease the handling of the equations and to facilitate the comparison of the results,the variables are made dimensionless. The following conversions apply:
c* = djé] ^ q = é\c\ (3.5)
FT' = rt -<=>- t = T'/t t = a = -pr=v for cyclic voltammetry (3.6)
RI
X = x/L ^^ x = XL (3.7)
ß = -^-^D = ßrL2 (3.8)tL2
faJTfacVr
k,t for first order rate constants
^H/;0/r^h=< (3.9)
3 ' k%t/(P for second order rate constants
4>i/m = kS}i/m/Va~D ^^ kS)ijm = ip'l/my/äD . (3.10)
L is the distance from the electrode where no diffusion occurs during the simula¬
tion. The sweep rate v in equation (3.6) is used to integrate the time dependency of
the system into the model. To simplify the derivation and the programming of the
simulation model only one diffusion coefficient D, i.e. Da = DB = Dc = Do = D, is
defined. This will eliminate further complications by reducing the amount of unknown
parameters. It could be taken into consideration to implement different diffusion coef¬
ficients for every species in less complex mechanisms. Since the diffusion coefficients of
28 Digital Simulation in Cyclic Voltammetry
similar species in solution are approximately of the same order of magnitude and since
in most cases the diffusion coefficient of a species cannot be exactly specified, only one
single coefficient will be applied for all species.The diffusion equations (3.1) - (3.4) can be rewritten in the dimensionless form:
dc\ _ß&C*A K2
aT ßIx^ + K2CßCc~
K?A°D
dTßd2cBdx2
KiCB + K_iC,C~
K2CBCG + ~~j~TcAcD
~ = 'J^V2 + KlCB~
K-l°C~
K2CBCC + -^-CACDÔT' dx2
dc*D d2c*D K2
ÖT7-
ßIx^ + K2CßCc~
K~2CaCd
(3.11)
(3.12)
(3.13)
(3.14)
Finally, the dimensionless current function %, which was originally proposed byNicholson and Sham [37], is included:
V^x = ±
c°3FA^hD= ±Vß f
derdX
x=o.
(3.15)
In the case of the EqrCEqr/COMP mechanism equation (3.15) expands to the fol¬
lowing expression:
V^x = vpd^AdX
1dX L_r (3.16)
x=o v^^1- / X=0J
From now on the dimensionless form will be used for all equations, if not stated
otherwise.
3.3.2 Initial and Boundary Conditions
The solution of the diffusion equations depends on the initial and boundary conditions
of the system. The initial conditions are normally defined first. When the experiment
is started, there is solely species A present in the cell. Therefore, all concentration
values are standardised with ca(T = 0):
0<X<f;T' = 0: c*A(X,0) = l, c*B(X, 0) = cc(X, 0) = c*D(X, 0) = 0
The boundary conditions have to be evaluated in the bulk and at the electrode
surface. It is assumed that the bulk concentrations of all species remain constant
throughout the experiment or simulation:
X = 1;T' > 0 : cA(l,T') = I, cB(l,T') = cc(l,T') = c*D(l,T') = 0
3.3 Derivation of the Orthogonal Collocation Model 29
The situation at the electrode surface is more complicated. The temporal variation
of the concentration gradients at the electrode surface is described by the followingPDEs:
X = 0; T' > 0 :
dc*A\_
^A/Bq (T^-(i-aAIK)ft-{l-»A/B)
dXx=o ^sx(T')-{1-aA'B)eA)B-
[cA(0,T') - cB(0,T')Sx(T')9A/B] (3.17)
dc*A\ fdcBdX J x=o V dx J x=o
dcC\ _^ZADo rTM_(i_« ),-(l-«C/D)
dx)x=0- ^ßS^T) e^
[c*c(0, T') - c*D(0, T')Sx(T')9c/d] (3.19)
dc%\ (delC \ I UL-D
dX > x=o \ dX / x=o
where
=
. exp{-T'} V < Tx' '
expjT' - 2T{} T'X<V <2T'X
(3.20)
f nF 1Oi/m = exp I
— (^/ro - £start) | . (3.21)
T'x is the time when the sweep is reversed, and -Estart stands for the starting potentialof the sweep. The derivation of the right-hand side terms of equations (3.17) and (3.19)can be found in [34].
3.3.3 Discretisation with Orthogonal Collocation
The model equations have to be converted into a form that is appropriate for numerical
integration. The discretisation is based on the approximation by orthogonal polyno¬mials where PDEs will be transformed into ODEs. The first and the second partialderivatives of the concentration with respect to the space coordinate are written as
sums:
30 Digital Simulation in Cyclic Voltammetry
<j4_dX
d2c\dX<
N+2
= Y,a,3c:(xj,t')x, }=1
N+2
x, 3=1
(3.22)
(3.23)
In equation (3.24) and equation (3.25) the two boundary terms have been extracted
from the sum. The boundary concentrations play an important role in the simulation,and this extraction facilitates their mathematical treatment.
deldX
N+l
= A,lCf(0,T') + At>N+2cl(l,T') + J2 At,3cl(X3,T')xt 3=2
(3.24)
d2c\dX2
N+l
= BhlC;(o, r) + BhN+2djßi, r) + J2 Bt>3c;(x3,r)xt 3=2
The same discretisation applies to the diffusion equation
<j4_dV
N+2
ßY,BhJc;(XJ,T')-P*[c;(Xl,T')}Xt
J = i
= ß
N+l
BMcf (0, T') + BhN+2C;(l, T') + J2 Bt>3c;(X3,T')3=2
P*[cî(X,T>)] ,
(3.25)
(3.26)
where p* is the dimensionless chemical reaction term. Discretisation of equations
(3.11) - (3.14) leads to the following four expressions:
dc*AdV
= ßxt
N+l
Bhlc*A(0, T') + Bl>N+2c\(l, T') + J2 Bu<?A{X3,T')3=2
K2+ k2cb(X%, T')c*c(Xt, T') - -^cA(Xt, T')c*D(Xl, T')
lV2(3.27)
dc*BdV
N+l
= ß Bhlc*B(0, T') + Bl>N+2cB(l, T') + Y, b^jCb(Xj,T')x> L 0=2
- KlCB(Xt, T') + K.lcUXl, T') - K2cB(Xt, T')c*c(Xt, T')
+ ^cA(Xt,T>)CD,(Xi,T>)lV2
(3.28)
3.3 Derivation of the Orthogonal Collocation Model 31
dc*cdV
N+l
= ß Bhlcc(0, T') + BhN+2c*c(l, T') + J2 B,^cc(XJ,T')X> I 3=2
+ KlCB(Xt, T') - K.lCc(Xt, T') - K2cB(Xt, T')cc(Xt, T')
+ ^cA(xt,ryD(^Tf)lV2
dc*DdV
ßxt
N+l
Bttlc*D(0, T') + BhN+2c*D(l, T') + J2 B%,3c*D(X3,T')3=2
H2*
+ k2cb(X%, T')cc(X%, T') - -=±cA(Xt, T')c*D(X, T')lV2
(3.29)
(3.30)
The current function, equation (3.16), has to be discretised as well:
V*x = VßN+l
AhlcA(Q,T') + AhN+2cA(l,T') + J2 Ai,3cA(X3,T')0=2
N+l
+ AlAcc(0,T') + AhN+2cc(l,T') + J2 Ai,0cUx^Tr)3=2
(3.31)
In general, the application of OC to the boundary conditions has to be carried out
in the same way as for the diffusion and current function equations. For the solution of
this equation system the coefficients of the (unknown) electrode surface concentrations
have to be extracted from the previously defined bulk concentration terms:
Aii _ t^Sx(Tr{l-aA^eA%aA/B) cA(0, T>) + ^Sx(TTA/BÔAA/BBcB(0, T>)
N+l
AhN+2cA(l,T') + Y,Ai,3cA(X3,T')3=2
mSx(T')-(l-aA'B)eA%aA/B)c\(Q, T>) +'AjB
Va
Ahl-^-sx(TTA^eAA/BBN+l
AhN+2cB(l,T') + J2A^cb(X^T')3=2
(3.32)
4(0, T')
(3.33)
32 Digital Simulation in Cyclic Voltammetry
All _
^Sxiryv-^e-f-*0'^Vß
c*c(0, T') + -^Sx(T')ac/DOcc/DDc*D(0, T')Vß
N+l
Al,N+2CG(l,T') + Y,Ai,3cc(X3,T>)3=2
^Sx(T')-{l-ac/D)ÔcmaC/D)c*c(,0, T') +VP
Altl - ^sx(TTc/Dec%DN+l
AhN+2c*D(l,T') + J2Ai,3c*D(X3,T')3=2
(3.34)
4(0, TO
(3.35)
Such an equation system can best be represented as a matrix. For a proper presen¬
tation and for the sake of readability, two abbreviations are introduced:
Qa/b = SX(T')9A/B
Qc/d = SX(T')9C/D
(3.36)
(3.37)
This leads to this boundary condition matrix:
i>'
A/B maA/I( ^1,1
^A/Bn-^-»A/B) -/JT^A/BVß ^A/B
^A/B(n-^-»A/B) AlflVß ^A/B
0
0
/ c*A(0,T) \
4(0, T')
4(0, T)
V 4(0, T') /
<A/B inpA/lVß ^A/B
0
0
Vc/Dfnr(l-<xc/D) J^ ^C/DVß UC/D
Vß ^C/DVß vc/ß
Al>N+2cB(l,V) + E =2 Al>3cB(X3,T)
AltN+2cA(l, V) + ES Alt3cA(X3,T')
.N+l
'3=2
.N+l
'3=2
AltN+2<fD(l, V) + ES1 Alt3c*D(X3,T')
\
Ai,N+2C*c(l,T>) + E =2 Ai,3c*c(X3,T)
. (3.38)
3.3 Derivation of the Orthogonal Collocation Model 33
3.3.4 Spline Collocation
Fast homogeneous reactions coupled to electron transfers require a special mathemat¬
ical treatment within the simulation. Spline collocation has been used to tackle this
problem for the first time by Hertl and Speiser [35]. There has been an earlier attempt
with a similar method by Pons [38]. The full derivation can be found in [39]. Fig¬ure 3.1 schematically shows the principle of spline collocation, i.e. the partition of the
simulation space into two layers and their corresponding approximation functions.
c(xs)i = c(xs)2dc I dc_ I
dx \xs,l dx \xs,2
outer function
Lx
1X
X1=0 X, =1 X2 = l
Figure 3.1: The inner and outer approximation functions used for spline collocation.
In this case there are two layers: the inner "reaction layer" with the index '1' and
the outer "diffusion layer" with index '2'. The number of diffusion equations doubles:
dc*AldT>
ß'x1%
N+l
BhlcAl(0,T>) + BhN+2c*Al(l,T>) + J2 Bh3cX(Xh3,T>)3=2
H2*
+ ^(X^T'Yc^X^T') - -^cAl(Xltt,T')c*Dl(x^T') (3-39)Ä2
34 Digital Simulation in Cyclic Voltammetry
dc\2dV
= ßx2.
N+l
BhlcA2(0, T') + Bl>N+2c%(\, T1) + J2 Bh3cA2(X2,3,T')3=2
K2+ ^42 (X2„ T')c*C2 (X2>t, T') - -±cA2 (X2>t, T')c*D2 (X2>t, T') (3.40)
Ä2
dcBldT>
= ß'xu
N+l
BhlcBl(0, T') + 5^+24,(1, T') + J2 BtjCBl(Xlt3,r)3=2
- KlC*Bl(Xltt,T') + K-ic*Cl(Xltt,r) - K2cBl(Xht,T')cCi(Xht,T')
+ ^(x^ry^x^T')Ä2
(3.41)
dcB2dT>
x2.
ß
N+l
BhlcB2(0,T>) + BhN+2cB2(l,T>) + J2 Bl>3c%(X2>3,T')3=2
mcB2(X2tl,Tr) + k.iCc2(X2>1,T') - K2cB2(X2>l,T')cC2(X2>l,T')K2
+ ^cA2(X2,t,T')c*D2(X2,t,T') (3.42)
dcCldV
= ß'xu
N+l
Bt>1c*Cl(0, TO + BhN+2cCl(l, T') + J2 B^Cl(Xi,3, TO3=2
+ Klc*Bl (Xltt, T') - k-iC*Ci (Xltt, T') - k2c*Bi (Xltt, T')cCi (Xltt, T')
+ ^(x^ry^x^T')Ä2
(3.43)
dr*acc2
dV
r iv+i
= ß TV42(0, TO + BhN+2c*C2(l, T') + Y, Bh3Cc2(X2t3,T>)*2,. L 3=2
+ kiCb2(X%1,T') - k^Cc^X^T1) - K2cB2(X2tt,T')cC2(X2tt,T')
+ ^cX(X2„T'yD2(X2„T')Ä2
(3.44)
dc*DldT>
= ß'Xi,
N+l
Bt>1cDl(0,T') + Bt>N+2cDl(l,T') + J2 Bt>3cDl(X1>3,T')3=2
K2+ k2cBi (Xltt, T')cCl (Xltt, T') - -^c*Al (Xltt, T')c*Dl (Xltt, T')
J^2(3.45)
dc*p2dV
= ßx2.
N+l
BhlcC2(0, T') + BhN+2cC2(l, T1) + J2 Bh3cC2(X2„T')3=2
H2*
+ ^42 (X2,t, T')cC2 (X2>t, T') - ^c% (X2>t, T')c*D2 (X2>t, T') . (3.46)Ä2
These equations introduce the new parameter ß' and change the definition for ß:
3.3 Derivation of the Orthogonal Collocation Model 35
0 =4 0=
D
TX2 t(L — Xs)2
The diffusion coefficient for the inner and outer function is now ß' and ß, respec¬
tively.The initial and boundary conditions from normal orthogonal collocation still apply
with some minor changes:
0<X1<l;T' = 0: cAl(Xu0) = I, c*Bl (Xx, 0) = cCl (Xx, 0) = c*Dl (Xx, 0) = 0.
Again the boundary conditions have to be calculated for the bulk and at the elec¬
trode surface. The assumption that the initial bulk concentrations of all species remain
constant throughout the experiment or simulation is still valid:
X2 = 1;T'>0: 42(f,T0 = l, cB2(l,T>) = cC2(l,T>) = Cd,2(1,T>) = 0.
The boundary conditions at the electrode surface will actually not change except
for ß being replaced by ß':
Xi = 0; T' > 0 :
öd \"Ax j^-sx(Tr)-^-a^eAfBaA/B)
dxJXi=0 ^AV > A'B
• [4^0, TO - cBl(0, T')Sx(T')eA/B] (3.47)
(3.48)^V\ (9cBl\dX
,
'
x1=o \dXjXi=0
9cc:
dX,'
x1=o^sx(T')-{l-ac/D)e~c%ac,D)[cCi(0,T')-c*Di(0,T')Sx(T')9c/d}
9cc:) -
(9c*Dl\dX,'
x1=o \dXjXi=0
(3.49)
(3.50)
Another condition defines the concentrations at the boundary between the inner
and outer spline layer:
)x,=rm)^ (3-51)
The concentrations at the boundary between the inner layer (c*(l,T0) and the
outer layer (c* (0, TO) have to be equal, and this condition also has to be fulfilled in
the simulation.
36 Digital Simulation in Cyclic Voltammetry
After discretisation and extraction of the terms with unknown concentrations equa¬
tion (3.51) for species A finally turns to:
AN+2AcAl(0,T') + ( AN+2>N+2 - xl^,Altl ] cAl(l,T')
N+l
Y,AN+2jcAl(X1J,T') + xl%3=2
Iß'
N+l
AhN+2cA2(l,T') + J2Ai,3cA2(X2,3,T') . (3.52)3=2
Discretising equations (3.47) - (3.50) works analogously in the orthogonal colloca¬
tion case. The discretisation for component A yields the following expression:
/de* \N+l
(^J = AltlcAl(0,r) + AhN+2cAl(l,T>) + YtA1jCAl(X1j,r)
= ^s^T'y^-^o-^^[cAl(0,T')-cBl(0,T')eA/BSx(T')}
The extraction of the unknown concentration terms leads to:
(3.53)
/L,_
^A/B Q(r'\-{l-aAIK)9~{l~aA/B)^i.i rgj
^U ) Va/b
N+l
cAi(o,t>) + ^sx(TTA'BeaAA/BBcBl(^r)+ AhN+2cAl(l,T') = -J2Ah3cAl(Xh3,T') . (3.54)
3=2
All these terms together build up a matrix. In order to express this 8x8 matrix
as a whole it is subdivided into four 4x4 matrices:
/
4,1 —
<A/1
<A/I
IW
Al,l-(l-aA/B)
'^A/B
-(l-aA/B)^A/B
0
0
A.N+2
0
0
<A/I
fWrf*A/B"-A/B
Ai,i>'
A/I
rw
o
0
f.aA/BZA/B
\
AlN+2
0
0
(3.55)
4,2 —
0 0
0 0
_^£ADfn,-(l-«C/D) Aï}N+2VÏÏ ^c/d
VF ^c/d u
^'c/p (TfC/DVW UC/D
/l,l^'c/D AttC/JVW ^C/D
0 \0
0
A, N+2
(3.56)
3.3 Derivation of the Orthogonal Collocation Model 37
^2,1 —
AN+2,1
U2,2 —A
0
0
0
0
N+2,1
AN+2,N+2
A
~\jiAi>i
AN+2,1
0 0
0 0
0 0
0 0
+2,W+2
AN+2,1
AN+2^N+2Ja«7^1,1
ß
0
0
0
0
AN+^N+2Ja«7^1,1
(3.57)
(3.58)
As a consequence, the concentration vector splits into two subvectors:
<Ca/b
( 4,(o, to \
4,(1,^0
4,(0, to
V^(i,to y
-CID
I 4,(o, to \
4,(i,n
4,(0, to
V4,(i,ny
The constant right-hand side terms are divided in the same manner:
PA/B
/ N+l \
3=2
N+l
-EAi,3cBl(xh3,r)3=2
N+l
-Y.Ai,3cCl(Xl>3,T)3=2
N+l
. -J2Ai,3cDl(Xlt3,T')\ 3=2 /
38 Digital Simulation in Cyclic Voltammetry
WC/D =
( N+l
3=2
N+l
-E3=2
N+l
-E3=2
N+l
EAn+2,jcAi(X1,3,T>) + JJ
EAN+2,3cBl(Xh3,T) + J]7
V+l ;
EAN+2,Jc*Cl(xlt3,r) + ^
EAN+2,JcDl(Xlt3,T>) + j£\ 3=2
V '
N+l
AhN+2c*A2(l,T)+ EAi,3cA2(X2,3,T)3=2
N+l
Al>N+2cB2(l,T) + E Al>3cB2(X2>3,T)3=2
N+l
AltN+2cC2(l,T>)+ EAi,3cC2(X2t3,T>)3=2
N+l
Ai,n+2CD2(1,T')+ EAi,3cD2(X2t3,T>)3=2
Finally, the "abbreviated" form of the matrix equation is obtained:
Hl,l 1*111,2
Œ2,i M2,2
pA/B(3.59)
^A/B
Cc/D J \ Pc/D
The current function equation according to Nicholson and Sham [37] for splinecollocation is given by:
V^x = Vß*
Its discretisation yields
04,dX
+
x1=o
04,dX
x1=o
(3.60)
N+l
JX =V^ A1AcAl(0,T') + AltN+2cAl(l,T') + Y,A1,3cAl(X3,T')3=2
N+l
+ AMc^(0,TO + A1)W+2c^(f,T0 + Y,Ai,Jc*Cl(X3,T') . (3.61)3=2
3.3.5 Transformations of the Space Coordinate X
Sometimes extreme parameter combinations with fast second order reactions are en¬
countered which cannot be simulated with SC alone. In this case an additional math¬
ematical transformation is needed, most preferably the one of the space coordinate X.
The goal of this transformation is to increase the accuracy of the calculation in the
regions with the highest demand, mainly the reaction layer. Speiser incorporated two
different spatial transformations into his program system (see [39] pp. 71-84). Both
transformations will be discussed in the sections below.
Urban Transformation
This transformation was proposed by Urban (see [39] p. 71), formerly at the "Institut
für Chemische Pflanzenphysiologie, Universität Tübingen". The space coordinate X
3.3 Derivation of the Orthogonal Collocation Model 39
is transformed into the new coordinate V (V for the inner layer and V for the outer
layer). This conversion process is depicted in figure 3.2. The mathematical execution
adheres to these equations:
UX
V
~
i + x
V =
u
u --us
I-Us
Xu
l + U
u = usv
U = (l-Us)V + Us
0 < U < [/,
U3 < U < I
U = 0 Us £7 = 0.5
V' = l
(V = 0)
x -^ oo
X^oo
U = 1
V = IV = 0
Figure 3.2: Urban transformation of the dimensionless space coordinate X.
The general boundary equation for component A (see equation (3.53)) then turns
to:
40 Digital Simulation in Cyclic Voltammetry
fdc* \N+1
{-g^) f
= AhlcAl(0,T') + AhN+2cAl(l,T') + Y/Ai,3cAl(V;,T')
_
1 + Vß/ß' ^A/Bc ,Tn_(i_a )fl-(l-«A/B)
~
i + 2yß!p VFx{ > A/B
[cAl(o,r)-cBl(o,r)9A/Bsx(r)} (3.62)
The SC boundary condition at the reaction/diffusion-only layer interface yields for
the Urban transformation:
'k\=
vw fdci\ßV)yl
= l 1 + VPV^j^O'
Discretisation of this condition leads to the following expression:
(3.63)
-4w+2,l4,(0,TO + AN+2>N+2VWF
i + vWß'-.Altl cAl(l,T>)
N+l
E3=2
ç^^^^^îtSN+l
Al>N+2c%(l, T') + J2 Ai,0c%(V3, T')3=2
(3.64)
The diffusion equations for the inner and the outer layer also have to be transformed.
The discretised general versions are shown here:
dc* vrf i + (2-7')VP
dT>
l+vW) (l + 2^ß/ß)'l + {2-V')yßJß' del 2dct
54dT'
VWß
ß(l - Vf
dV'2 dV
(l-V)92ct 2dctdV2 dV
+ p*K)
+ p*K)l + 2Vß/ßJJ
Transformation of the current function equation (3.16) yields:
(3.65)
(3.66)
V/7TX = ±"
ß'(l + 2yß/ß>
i + Vß/ß1
N+l
AhlcAl(0, T') + AhN+2cAl(l, T') + J2 Ai,3cAl(V;, T')3=2
N+l
+AhlcCl(0, TO + Al>N+2cCl(l, TO + J2 Ai^v;, T')3=2
(3.67)
3.3 Derivation of the Orthogonal Collocation Model 41
Exponential Transformation
The rules for the exponential transformation are subsequently shown. X is converted
to W and W respectively, with concomitant inversion (figure 3.3). The guidelines for
the exponential transformation are listed below:
U' = exp{-X} X InU'
U' = 1 - (1 - U'8)W
[/' = (!- W)U'a
u's < U' < 1
o < u' < m
x = 0 L oo
x = o Xs = xs/L X = l X oo
U' = 0 U's
W = 0 W = 1
(W = 0)
U' = l
W' = 1
Figure 3.3: Exponential transformation of the dimensionless space coordinate X.
Therefore, the discretised boundary equation for species A is altered correspond¬
ingly:
42 Digital Simulation in Cyclic Voltammetry
(de* \N+l
[-Q^j f
=AhlcAl(0,T') + AhN+2cAl(l,T') + Y/Ai,3cAl(W;,T')
IiJÄV HÎAIRq, (T')-(l-»A/B)ß-^-a^B)
\ß'J y ß' Vß7{ } A/B
• [4,(0, TO - c^(0,TO^/ß5A(TO] . (3.68)
Exponential transformation leads to an expression similar to equation (3.63):
(3.69)w=o
The discretised form with the unknown concentration terms on the left-hand side
of the equation yields:
^v+2,i4,(0, TO + [ AN+2>N+2 + [ 1 - exp {i V4^ I ] ^1,1 ] c^,(l, T')1 +
N+l
Y,AN+2,3cAl(W;,T')3=2
1 — expVWw V N+l
AhN+2cA2(l,T>) + Y,Ai,3cA2(W3,T>)3=2
(3.70)i + VWf,
According to the same procedure the discretisation of the diffusion equations results
in:
dV
ß
\+vw;
W'+(l- W) expJ vß/W I
li + vWï7/
f — exp+ vW^J
W'+(l- W) exp| vPIF Ili + vWÏ7/ a24 del
VßH_\dW'2 dW>
dT>
ß
1 +
! _ expy/Pir I
\i + vWJ
"(f-lU)^-^; JdW2 dW
+ P*K) (3.71)
vW^7F(1-1U) + 4«) • (3.72)
3.4 Results and Discussion 43
Finally, the Nicholson and Sham current function equation is transformed expo¬
nentially:
7TY =ß
expIßlß'
i+Jß/ß'
i + Vß/ß1 x expißlß'
i+Jß/ß'
AhlcAl(0,Tr) + AltN+2cAl(l,Tr)
N+l
+ J2a^caAKt')3=2
+AhlcCl(0, TO + AhN+2cCl(l, TO + J2 Ai^cAK T')
N+l
3=2
(3.73)
3.4 Results and Discussion
3.4.1 Measurements in Aqueous Media
Matic [7] proposed a mechanism for the electrochemical oxidation of acetoin in an aque¬
ous alkaline solution. This reaction system consists of two electron transfers connected
by a homogeneous chemical reaction and a comproportionation reaction. Its corre¬
sponding mathematical model for the simulation in CVSIM was developed in section
3.3. Matic's experiments build the main part of the base for the following simulations
and the comparisons between experimental and simulated data. The instrumental set¬
up, the electrodes and the preparation of the reactants and the solvents are described
in chapter 7.
3.4.2 Simulations with the EqrCEqr/COMP Mechanism
Different parameter combinations were chosen for the simulations. The following sim¬
ulation parameters remained constant for all of the calculations:
• Tswitch = 0.6 V, switching potential
• AE = O.OOf V, potential step size
• T = 298 K, temperature
• c°A = 0.024 M, initial concentration of compound A
• A = 0.406 cm2, electrode area
• D = f0~5 cm2/s, diffusion coefficient (all species)
• ola/b = cyc/d = 0.5, transfer coefficients for both ETs
44 Digital Simulation in Cyclic Voltammetry
• Acya/b/AE = Acyc/d/AE = 0, potential dependence of both transfer coefficients
• N = 9, number of collocation points
The integrations were performed using the ddebdf integrator. The expandingsimulation space was turned on to obtain a higher accuracy.
The standard equilibrium potentials and the heterogeneous rate constants of the
ETs, the forward and reverse rate constants of the first chemical reaction and the
forward rate constant of the comproportionation reaction (COMP) were altered from
run to run. The starting potential was varied, according to the experiments, between
—0.6 V, —0.7 V and —0.8 V. From time to time the sweep rate v was changed from
O.f V/s to higher values. Table 3.1 lists the lower and upper limits of the varied
parameters and the combination that corresponded most closely to the experimentaldata (column Best Fit).
3.4.3 Influence of Starting Potential
The influence of the starting potential on the whole voltammogram was the first pa¬
rameter to be studied. The peak potentials and the shape of the curves of three
voltammograms, one from every experimental series, were compared. Figure 3.4 shows
three voltammograms that were all taken at the beginning of their corresponding exper¬
imental series. The sweep rate was O.f V/s for all of them. In addition, the background
current, measured before the reactant was added, was subtracted from every voltam¬
mogram.
A different starting potential mainly influences the height of the first anodic peak.The differences are due to additional electrode surface reactions, i.e. the reduction of
the platinum oxide. It is very likely that the composition and the active area of the
electrode surface itself was different for these three voltammograms, because local pH
changes of the alkaline aqueous electrolyte affect the activity of the surface.
Simulations with the 'Best Fit' parameter combination mentioned in table 3.1 and
varying starting potentials should yield similar voltammograms. Neither the peak po¬
tentials and currents nor the shape of the curves altered during the first full cycle.This behaviour changed in the second, consecutive cycle, where inversion of the exper¬
imental situation was observed, i.e. the peak currents of the first peak increased with
decreasing starting potential. These simulated voltammograms are displayed togetherin figure 3.5.
The following figures are added to clarify the difference between the first and the
second cycle of two consecutive voltammograms. The experimental and the simulated
comparisons are depicted in figures 3.6 and 3.7 respectively.The comparison of voltammograms with different sweep rates turned out to be more
difficult, because the experimental data at higher sweep rates showed a low signal-to-noise ratio. Thus, scrutiny is limited to the experiments with sweep rates of O.f V/s,0.2 V/s and 0.5 V/s. Figures 3.8 and 3.9 clearly exhibit the influence of the sweep
rate.
3.4 Results and Discussion 45
Table 3.1: Tower/upper limit and best fit parameter
Eqr CEqr/COMP mechanism.
values from the simulation of the
Parameter Description Lower Limit Upper Limit Best Fit
El Standard equilibrium
potential of the first
ET
-0.6 V -0.4 V -0.45 V
E°2 Standard equilibrium
potential of the second
ET
-0.55 V -0.2 V -0.55 V
ks,i Rate constant of the
first ET
f0~2 cm/s f0 cm/s f0_1 cm/s
ks,2 Rate constant of the
second ET
f0~6 cm/s fO-3 cm/s fO-6 cm/s
h Forward reaction rate
constant of the first ho¬
mogeneous reaction
Is"1 fO3 s"1 fO2 s"1
fc_i Reverse reaction rate
constant of the first ho¬
mogeneous reaction
f0"6 s"1 fO3 s"1 fO"6 s"1
k2 Forward reaction rate
constant of the sec¬
ond homogeneous reac¬
tion (COMP)
fO"1 M-^s"1 fO10 M-^s"1 fO"1 M-^s"1
V Sweep rate O.f V/s 20 V/s —
-'-'start Starting potential -0.8 V -0.6 V —
Peak currents increase and peak potentials are shifted to higher potentials with
increasing sweep rate. Thus, there is no difference between the experiment and the
simulation as far as this parameter is concerned.
3.4.4 Special Simulation Aspects
It is apparent in figures 3.4 to 3.7 that the peak currents in the experimental and
simulated voltammograms differed from each other by one order of magnitude. With
regard to the experimentally deduced concentration and electrode area values this
deviation seems to be extremely high. There are two possibilities to reduce the peakcurrents and of course the current in general. The first changeable parameter would
be the electrode area. The diffusion coefficient D would be another parameter that
46 Digital Simulation in Cyclic Voltammetry
<
0.20
0.15
0.10
0.05
0.00
-0.05
-1000 -500
£start = -0.6 V.
Tstart = —0.7 V
Tstart = —0.8 V
E / mV500 1000
Figure 3.4: Three experimental background corrected voltammograms with different
starting potentials Estori (v = 0.1 V/s).
<
1.5
1.0
0.5
0.0
-0.5,
-1000 -500
£start = -0.6 V
-Estart = —0.7 V
-Estart = —0.8 V
E / mX500 1000
Figure 3.5: Three simulated voltammograms with different starting potentials Estori
(v= 0.1 V/s).
3.4 Results and Discussion 47
<
0.15
0.10
0.05
0.00
-0.05
-1000
First Cycle
Second Cycle
-500
E / mV500 1000
Figure 3.6: Two consecutive experimental voltammograms with D = 10 5 cm2/s(Estart = -0.8 V,V=0.1 V/S).
<
-21 ,
-1000
First Cycle
Second Cycle
-500
E / mX500 1000
Figure 3.7: Two consecutive simulated voltammograms (Estart = —0.8 V, v =
0.1 V/s).
48 Digital Simulation in Cyclic Voltammetry
<
0.15
0.10
0.05
0.00
-0.05
-0.10
-1000 -500
E / mV500 1000
Figure 3.8: Three experimental voltammograms with different sweep rates v (Estart-0.8 V).
<
-21 ,
-1000 -500
E / mX500 1000
Figure 3.9: Three simulated voltammograms with different sweep rates v (Estart-0.8 V).
3.4 Results and Discussion 49
1.0
0.8
0.6
<S
0.4
0.2
0.0
-0.2
-1000 -500 0 500 1000
E / mV
Figure 3.10: Two consecutive simulated voltammograms with D = 10~6 cm2/s(Estart = -0.8 V,V=0.1 V/S).
could be used for this purpose. If the diffusion coefficient decreases by one order of
magnitude, i.e. from fO-5 cm2/s to f0~6 cm2/s, it will result in four times lower peakcurrents. The two simulated voltammograms from figure 3.7 have been recalculated
with the new value for the diffusion coefficient. The result from this simulation is
shown in figure 3.10.
The 20 mV lower anodic peak potentials are a side effect of the smaller diffusion
coefficient. The definition of Vi/m m equation (3.10) holds the explanation of this
potential shift. Choosing a value for D that is lower by one order of magnitude results
in vTÖ times higher value for Vum- Therefore, the peak current is reached at a lower
potential.
3.4.5 Discussion of the Simulation Results
The discrepancies between the experiment and the simulation are obvious. Neverthe¬
less, the simulated output contains some similarities with the experimental one. Some
of the main similarities and differences are discussed in the following paragraphs.The combination of two single electron transfers with at least one homogeneous
chemical reaction exists in both the experiment and the simulation model. The fact
that there is no direct evidence of the assumed comproportionation reaction neither
from the experimental nor from the simulated data shows that the possible existence
of such a comproportionation cannot be deduced from the obtained cyclovoltammo¬
grams. Nevertheless, the compound A which is regenerated by this comproportionation
First Cycle
Second Cycle
50 Digital Simulation in Cyclic Voltammetry
process may still have such a strong influence on the obtained results that the model
calculation can only fit the experimental data in a very narrow range of parameter
values. Obviously, this range has not yet been found.
The symmetrical shape of the first experimental anodic peak is an indication for
the presence of adsorption effects, whereas the simulation shows a diffusion-controlled
signal. The introduction of adsorption effects into the computer model would be feasible
though not advisable, because the mathematical derivation of complex models is error-
prone and more mostly unknown parameters are introduced.
The experimental and the simulated peak current ratios of the two anodic peaksdiffer clearly as well. The first experimental peak is always higher than the second
one. The simulations show either two peaks of equal height or the situation where the
second peak is higher than the first one.
Another possible way to reduce these deviations is the employment of different
diffusion coefficients for every species. The magnitude of this parameter plays an
important role for the amplitude of the peaks (see section 3.4-4)- In order to better
adapt the simulated to the experimental data the ratio of DA (diffusion coefficient of
species A) to Dc (diffusion coefficient of species C) should be greater than f. The
same rule accordingly applies to the ratio of DB to Do- However, it is not very likelythat the diffusion coefficients of the compounds in this reaction system differ by one
order of magnitude or more. Therefore, it can be assumed that the approximationmade by using a single diffusion coefficient does not account for the differences in peak
amplitudes alone. In addition, the growing number of mostly unknown simulation
parameters increases the complexity of the model. Its high configurability augmentsthe probability of finding a parameter combination which fits a wrong model to the
experimental data. This situation should be avoided, unless there is a strong indication
for the existence of such a mechanism.
The simulation work included the examination of some other mechanistic models in
order not to miss any additional, probable solutions. The EqrEqr/COMP (36), ErErCir
(14) and ErErCirCcat (23) models have been scrutinised. The indices ir, qr and r
stand for irreversible, quasi-reversible and reversible respectively, and the numbers in
parentheses indicate the model numbers in CVSIM. The voltammograms from these cal¬
culations mostly deviate from the experiments in the cathodic sweep, i.e. the cathodic
part of the first ET (B — A) is reversible or quasi-reversible instead of irreversible.
The assumed simulation model does not match the experimental data. The sys¬
tem either follows a different reaction mechanism or additional effects like adsorptionand oxidation processes on the metal surface of the electrode and hydratisation pre-
equilibria present in the experiments have to be integrated into the computer model.
In order to eliminate these possible adsorption and hydratisation effects cyclovoltam¬metric experiments were also performed in nonaqueous solvents such as acetonitrile
(see section 3.4-6). It was expected that a comparison of the experimental data with
the model calculation could give some information about e.g. the decisive role of the
hydratisation equilibria on the redox process of a-hydroxyketones in aqueous media.
3.4 Results and Discussion 51
3.4.6 Experiments in Acetonitrile
The voltammograms in acetonitrile first revealed that the resistance of the reference
electrode was too high. The iR drop measurements on the BAS 100B/W workstation
yielded a resistance of about 3 kO. Normally this value should be at least below
f kO, or even better below 500 O. Thus, a miniaturised version of the same reference
electrode was built. The following iR compensation tests on the BAS 100B/W showed
resistances around 300 O for the new electrode during the first experiments of a new
series. The full resistance could be electronically compensated. However, after some
voltammograms were made, the tests resulted in error messages indicating that the
resistance was too high to be measured. Sometimes these errors only occurred for
specific parameter values in the filter options. The instability of the RC circuit used
for calculations of the solution resistance (see [40] pp. 5-3ff.) was probably the cause
for this behaviour. Nonetheless, the peak potential difference for ferrocene was around
65 mV up to a scan rate of 2 V/s. The theoretical value for this difference is 58 mV.
3.4.7 Cyclic Voltammetry Experiments
The CV measurements with adipoin and its oxidation product f ,2-cyclohexanedionewere mainly made with the BAS 100B/W electrochemical workstation. Scan rates
varied from 50 mV/s to 200 V/s and the potential range was between +1.6 V and
—2.3 V vs. the Ag/Ag+ reference electrode. No redox peaks were observed in this range
and for any scan rate used. Both reactants seemed to be completely electrochemicallyinactive under these conditions. The employment of a glassy carbon working electrode
did not change anything either.
Thus, adipoin had to be activated by increasing the pH of the electrolyte solu¬
tion. It is not convenient to use sodium hydroxide or any of the usual aqueous bases.
Therefore, BEMP (2-tert.-butylimino-2-diethylamino-f,3-dimethyl-perhydro-f,3,2-diazaphosphorine, fluka, purum), a so-called Schwesmger base, was chosen. Schwe-
singer [41],[42],[43] proposed this kind of bases for many syntheses, where deproto-nation of the reactant was critical. The pKa value of BEMP in acetonitrile is 27.58
[43].The addition of BEMP to the electrolyte solution (without any acyloins) yielded
voltammograms with two oxidation peaks in the positive (vs. Ag/Ag+) potential
area. No corresponding reduction peaks could be observed. The main reason for this
behaviour could be either electrochemically irreversible electron transfers or chemicallyirreversible homogeneous reactions in between and/or after the two electron transfers.
In order to elucidate the mechanism, further experiments were performed. No visible
reduction occurred even during fast scans (> 100 V/s). The variation of the scan
rate revealed that the two electrochemical steps were connected by a homogeneouschemical reaction. The height of the second peak decreased when the scan rate was
increased (see figure 3.11). There was not a corresponding reduction peak for the
second electron transfer over the whole scan rate range. However, a small reduction
peak, corresponding to the first oxidation peak, appears for scan rates > 50 V/s as it
52 Digital Simulation in Cyclic Voltammetry
<
2.0
1.5
1.0
0.5
-1.0
-2000
v = 1.0 V/sv = 0.5 V/s
v = 0.2 V/s
-1000 0
E j mV
1000 2000
Figure 3.11: Three experimental voltammograms of BEMP m acetonitrile.
<
20
15
10
v = 50 V/s
v = 20 V/s
u = 10 V/s
2000
Figure 3.12: Three experimental voltammograms of BEMP m acetonitrile.
3.5 Hydratisation Effects 53
can be seen in figure 3.12. Hence, the first electron transfer must be at least quasi-
reversible. These observations lead to the proposition of an EqrCirEir mechanism. A
thorough mechanistic and chemical analysis of this reaction system would be necessary
to corroborate this hypothesis. These experiments were not carried out, because such
investigations would lie beyond the scope of this work.
Petersen and Evans [44] reported a standard potential for 2,3-butanedione of
— 1.718 V vs. Ag/AgN03 in acetonitrile with 0.1 M TEAP (tetraethylammoniumPerchlorate) measured with a mercury working electrode. The corresponding value for
2,2,5,5-tetramethylcyclohexane-l,3-dione is —2.903 V. The standard potential of 1,2-
cyclohexanedione lies in between these two values. However, even after the addition
of a stoichiometric amount of BEMP no peaks in the negative potential area down to
—2.3 V could be seen. The reason for the absence of any oxidation peak for adipoinor reduction peak for 1,2-cyclohexanedione is not yet known. It is feasible to assume
that adipoin as reducing agent is deactivated by the formation of either the carbino-
lammonium addition products with the amino groups of BEMP or by the formation of
its cyclic dimer [2].
O O"il I
-C-C-H
HU
o o
-OH
HO' H
O OHll l
-—C-C-H =s=*
"O OH\ /
c=c
HO OHl l
—C-C-Hl I
HO'
HO OH\_
/
+ H+ ==*- C-C
/ \ .
/ \
\_o o-
C=C +2H+
/ \
Ox^
Red+
O Oil ll
-c-c-
HO Ol ll
—c-c-I
HO
HO OHl l
—c-c—I I
HO OH
Figure 3.13: Hydratisation, enolisation and dimerisation of a-hydroxyketones and
their oxidation products m aqueous solvents.
3.5 Hydratisation Effects
The lack of agreement of the cyclovoltammetric experimental data with the model sim¬
ulations (see section 3.4-5) might also be due to the fact that in the mechanistic model
used for the simulations the following hydratisation and dimerisation pre-equilibria [4]
54 Digital Simulation in Cyclic Voltammetry
shown in figure 3.13 have been neglected.
Figure 3.13 clearly shows that such side-equilibria might very well influence the
effective concentration of the active reducing species and thus the rates of the redox
as well as the comproportionation steps. It seems that for future investigations these
side-equilibria have to be taken into account.
Chapter 4
Oscillating Reactions
4.1 Introduction
The interdisciplinary field of self-organising processes covers terms like oscillating sys¬
tems, non-linearity and chaos theory. Current investigations in this area have their
origins in biology, chemistry, economics, mathematics, medicine and physics. No mat¬
ter whether the scientists try to predict a sudden heart failure or the developmentof a share at the stock exchange, the simulation models are mostly based upon the
theory of self-organisation. Many phenomena in our life seem to be influenced or even
determined by such a process.
In an earlier study [8] we have been able to demonstrate that the redox system with
a-hydroxyketones investigated in the present work is also fulfilling all requirements for
the occurrence of an oscillatory reaction behaviour.
4.2 A Brief Historical Overview
In the I920ies, Totka suggested the existence of oscillating chemical reactions on a
theoretical basis [45],[46]. In 192f, Bray and Tiebhafsky discovered the decompositionof hydrogen peroxide catalysed by iodate ions, which is now known as Bray-Tiebhafskyreaction [47],[48]. This was the first description of an oscillating chemical reaction in
liquid phase. Unfortunately, at that time nobody was really interested in these results.
Thirty years later, in 1951, Belousov created a reaction system intending to mimic
the behaviour of the famous Krebs cycle (citric acid cycle). By replacing the catalytic
enzyme with cerium and NAD with inorganic bromate in sulfuric acid he finally ob¬
tained a citric acid solution which rhythmically changed its colour from colourless to yel¬low and back. It took another six years until his work gained public attention, because
Belousov had published his results in an unknown journal in 1958. Zhabotmsky [49]made several refinements to Belousov's first combination of compounds. According to
this contribution this class of oscillating oxidations is now called Belousov-Zhabotmsky
(BZ) reaction.
Field, Koros and Noyes elucidated the detailed mechanism (FKN mechanism) of the
55
56 Oscillating Reactions
BZ reaction [50]. The occurrence of chemical waves in BZ solutions was first reported
by Zaikin and Zhabotmsky [51] and explained in terms of the FKN mechanism by Field
and Noyes [52],[53]. The FKN mechanism will be explained shortly in section 4-3-2.At the same time, Field and Noyes were also reducing the complexity of their own
mechanism while still retaining the essential characteristics of the BZ chemistry. The
final version of the simplified mechanism is usually referred to as the Oregonator [9].The Oregonator will be described in more detail in section 4-4-
Several scientists gained deeper insight into chaotic behaviour by means of special
experimental conditions in which reactant concentrations were chosen that differed con¬
siderably from the corresponding equilibrium concentrations. The use of a Continuous
Stirred Tank Reactor (CSTR) allowed the observation of chaos [54] and the systematic
design of chemical oscillators [55].
4.3 The Belousov-Zhabotinsky Reaction
4.3.1 General Aspects
Actually, the Belousov-Zhabotmsky reaction (BZ reaction) represents a whole class of
reactions, i.e. the catalytic oxidation of an organic compound that is brominated easily
by bromate ions in a strongly acidic aqueous medium. Metal ions such as Fe(III)/Fe(II)and Ce(IV) /Ce(III) are used as catalysts. The overall chemical reaction with malonic
acid as the organic compound can be written as
3CH2(COOH)2 + 2BrO^ + 2H+ = 2BrCH(COOH)2 + 4H20 + 3C02 (4+)
This equation only contains the stoichiometricalfy significant species. The concen¬
trations of the intermediates and the catalyst are several orders of magnitude lower
than those of the species in equation (4-1)-
4.3.2 The Mechanism of the BZ Reaction
Field, Koros and Noyes made the most notable proposition for the mechanism of the
BZ reaction [52] (see also section 4-%)- The FKN mechanism readily unveils the originof the oscillations in the BZ reaction. A detailed discussion of this mechanism lies
outside the scope of this thesis. However, the most important facts are summarised in
the following general survey.
The FKN mechanism consists of three main processes, namely A, B and C. Process
A is dominating during one oscillating state and process B during the other. Process
C acts as a "feedback" reaction which switches the system control from process B back
to process A. If Fe(II) (ferroin) is used as the catalyst, the reaction mechanism can be
written as:
4.4 The Oregonator 57
Process A
Br" + BrO^ + 2H+ ^ HBr02 + HOBr (4.2)
Br" + HBr02 + H+ -»• 2HOBr (4.3)
Br" + HOBr+H+ ^ Br2 + H20 (4.4)
Br2 + CH2(COOH)2 - BrCH(COOH)2 + Br" + H+ (4.5)
Process B
HBr02 + BrO^ + H+ ^ 2Br02 + H20 (4.6)
Br02 + Fe(II) + H+ ^ HBr02 + Fe(III) (4.7)
2HBr02 ^ BrO^ + HOBr+H+ (4.8)
Process C
Fe(III) + BrCH(COOH)2 - Fe(II) + Br" + • • • (4.9)
The bromination of malonic acid (see equation (4-5)) and the removal of the bro¬
mide ion are the main aspects of process A. As soon as the concentration of Br~ is
decreased to a certain value, the reaction rate of the oxidation in equation (4-6) be¬
comes comparable to the one in equation (4-3). Thus, reaction control is handed over
to process B, and the red ferroin is oxidised to the blue ferric form. The accumulation
of BrCH(COOH)2, Fe(III) and HOBr during the two processes subsequently acceler¬
ates process C, which in turn poisons process B by producing Br~. Hence, process A
becomes dominant again, and the colour changes from blue to red, because Fe(III) is
reduced to Fe(II). The first round of the oscillation cycle is finished and the system is
ready to start the second one. The dots in process C stand for the by-products of this
reaction, which actually is the top of the oxidative cascade of malonic acid to C02.
Györgyi et al. [56] discussed this type of reaction in detail.
4.4 The Oregonator
Field and Noyes were able to create a less complex model of the BZ reaction, while still
retaining all the characteristics of the FKN mechanism. This model is now referred to
as the Oregonator [9]. The mechanism still consists of process A (equations (4-10) and
(4-11)), process B (equations (4-12) and (4-13)) and process C (equation (4-14))'-
(4-10)
(4.11)
(4.12)
(4.13)
(4.14)
All the steps involved are irreversible. The autocatalytic production of X in equation
(4-12) comprises one of the most noteworthy parts in this mechanism. Currently known
A + Y --> X + P
X + Y --»• 2P
A + X --»• 2X + Z
2X --»• A + P
Z -- #Y
58 Oscillating Reactions
and well understood chemical oscillators contain a similar step. Transformation of this
scheme into the chemical reaction mechanism leads to the following equations:
BrO^ + Br" -»• HBr02 + HOBr (4.15)
HBr02 + Br" -»• 2HOBr (4.16)
BrO^ + HBr02 -»• 2HBr02 + 2Fe(III) (4.17)
2HBr02 -»• BrO^ + HOBr (4.18)
2Fe(III) -»• gBr~ (4.19)
The stoichiometric factor g controls how many Br~ appear for each Fe(III) that
disappears. Br~ and HOBr2 are the key compounds in terms of the determination
of the dominant process. The feedback reaction (equation (4-19)) plays a major role
in handing over the system control from one process to the other.
4.5 Electrochemical Oscillators
The oscillating systems mentioned above include one or several redox reactions. Elec¬
troanalytical techniques provide a powerful tool to elucidate this kind of mechanism.
Wojtowicz's review article [57] describes several different types of electrochemical oscil¬
lations. Tamy et al. [58] listed over 30 oscillating electrocatalytic systems. Their survey
is focusing on the oxidation of small organic molecules. Tamy distinguishes between
oscillations being based upon the passivation of oxidisable metals in acid solutions and
the type of oscillations encountered in electrocatalysis. Furthermore, he describes the
cyclovoltammetric investigation of the formaldehyde oxidation on rhodium electrodes.
Raspel et al. [59] employed cyclic voltammetry to investigate the current oscillations
during the oxidation of formic acid on Pt(fOO). Schlegel and Paretti [60] presented a
new electrochemical oscillator: the mercury/chloropentammine Co(III) oscillator (seesection 4-6). Baier et al. [61] tried to model the oscillations and instabilities, respec¬
tively, of coupled electrochemical and biochemical reaction systems. Most notable is
the occurrence of various quinones as reaction partners and the combination of nor¬
mal chemical kinetics with Michaehs-Menten kinetics. Koper and Sluyters studied the
indium/thiocyanate oscillator [62] and developed a mathematical model for this and
other "cathodic" oscillators [63]. Strasser et al. [64] presented a classification scheme
of oscillatory electrochemical systems. They proposed four principal oscillator cate¬
gories and an experimental procedure for the corresponding systematic mechanistic
categorisation.
4.6 The Mercury/Chloropentammine Co(III) Os¬
cillator
This electrochemical oscillator shows a strong dependence on the concentration of chlo¬
ride ions and on the pH of the solution. Schlegel and Paretti [60] investigated the system
4.7 Oscillations in Quinone Systems 59
by means of monitoring the open circuit potential at a hanging mercury drop electrode
and by cyclic voltammetry. According to the experimental observations they proposedthe following mechanism:
Co(NH3)5Cl2++ Hg = Co(NH3)5ClHg2+ (4.20)
Co(NH3)5ClHg2+ + H20 = Co(NH3)5H202+ + HgCl (4.21)
Co(NH3)5H202+ = Co(NH3)4H202+ + NH3 (4.22)
NH3 + 2HgCl = HgNH2Cl + HCl+Hg (4.23)
If a certain amount of chloride ions is present initially, the induction period is
shortened. If the concentration of these ions reaches a critical value, the oscillations
are quenched. This is probably due to the occupation of mercury surface sites bythe chloride ions. These sites cannot be accessed by the complex ions any more, and
therefore the reaction eventually ceases. For an oscillatory behaviour to occur, chloride
ions as ligands of a cobalt central ion have to be present in solution.
4.7 Oscillations in Quinone Systems
The investigation of chemical waves and pattern formation in BZ systems [51],[65] over
a longer period of time is hindered by the fact that bubbles of carbon dioxide formed
during the reaction evolve from the solution. First attempts to develop a gas-freeversion of the BZ reaction were made by Swinney et al. [66]. Kurm-Csösrgei et al.
[67] finally found the bromate-f ,4-cyclohexanedione-ferroin system, which meets the
requirements, i.e. it is a gas-free oscillating reaction system. The results of these ex¬
periments represent the starting point for the proposition of an a-hydroxyketone redox
mechanism based on the electrochemical Oregonator (see chapter 5), mainly because
of the similarities between the reaction systems of a-hydroxyketones and quinones.
60 Oscillating Reactions
Chapter 5
An Electrochemical Oregonator
5.1 Design of an Electrochemical Oregonator
The inspection of a series of voltammograms run in alkaline aqueous solution led to the
assumption that the oxidation of the various acyloins may follow an oscillating reaction
mechanism, which could also explain the fact that those experiments were practically
irreproducible. In view of the complicated reaction path proposed by Jermini [4]for this reaction, it seems very difficult and time-consuming to corroborate a similar
mechanism by means of digital simulation. This hypothesis is further supported bya recent paper of Zhabotmsky et al. [67]. They studied a gas-free reaction system
containing bromate, f,4-cyclohexanedione and ferroin. If the ferroin concentration
exceeds 5 • 10~5 M, the system behaves like a typical BZ oscillator.
A simpler mechanism should be found for the theoretical evaluation of the cy¬
clovoltammetric behaviour of an oscillating system. The mercury/chloropentammineCo(III) oscillator described in section 4-6 contains four reactions, but the adsorptionand desorption processes play the most important role with regard to the oscillation.
The derivation of the orthogonal collocation model of an electrochemical reaction sys¬
tem containing specific adsorption and desorption of certain reactants is complicatedand prone to errors. In addition, for the adsorption and desorption several novel —
and most likely unknown — parameters have to be included, which will increase the
complexity of the simulation. Biological oscillators like the peroxidase-oxidase system
[68],[69],[70] are even more sophisticated, because the corresponding detailed reaction
mechanisms comprise more than 10 reactions.
The Oregonator mechanism requires one reaction more than the mercury/chloro¬pentammine Co (III) oscillator, but there are no explicit adsorption/desorption steps
involved. An additional problem occurs when the Oregonator has to be reformulated
for electrochemical purposes. The original mechanism (see section 4-4) obviously is set
up for homogeneous reactions and does not include heterogeneous electron transfers.
Speiser [71] proposed the introduction of a mediator reaction that connects the hetero¬
geneous electrode reaction to the homogeneous part. The corresponding mechanism is
schematically shown here:
61
62 An Electrochemical Oregonator
F ^
Z + U h F + Y
A +Y -I X + P
X + Y -% U + 2P
A + X -% 2X + Z
2X -% A + P
F stands for Fe(II) and U for the oxidised form of Y (Br-). To make the mechanism
slightly more general and the task of programming the CVSIM model easier, the symbolsfor the different species were changed to A (Fe(II)), B (Fe(III)), C (Br*), D (Br-), E
(Br03-), F (HBr02) and G (HOBr):
A ^ B
B + C h A + D
D + E -% F + G
D + F -% C + 2G
E + F -% B + 2F
2 F -% E + G
5.2 Mathematical Formulation
This section summarises the basic dimensionless equations that are needed for the
derivation of the orthogonal collocation model. The more detailed explanations con¬
cerning the EqrCEqr/COMP mechanism (see chapter 3) contain all the necessary math¬
ematical elements for the oscillating system as well. First of all, the diffusion equationswith reaction terms for the electrochemical Oregonator are derived from the mecha¬
nism:
H = ßC^ -
KiCBcc + Ktc*Ec*F (5.2)
5.2 Mathematical Formulation 63
d°C rp CC **,** /r n\
~dfi=
^'dX2~
B c K3°dCf t5'3)
dT>= ^~dX2 KlCßCc
~
K2°dce~
K3CDcF (5.4)
d°E r>d CE **, * 2 Ir r\
~dT=
^~dX2~
K2Cd°e~
K4°ecf + K5°F (5-5)
&T=
~dX2 K2Cd°e~
KsCdcf + k4cecf- 2k5cf (5.6)
G O G , **|0**l * 2 /r rt
dT=
^^X2 K2Cd°e + 2KaCDcF + k*cf . (5.7)
The initial and boundary conditions contain the two excess factors c*D and c*E :
0 < X < 1;T' = 0 :
cA(X,0) = f
c*D(X,0) = c°D/cA
cE(X,0) = cE/coA
cB(X, 0) = Cc(X, 0) = 4(X, 0) = cG(X, 0) = 0
X = 1;T'>0:
cA(X,0) = f
c^(X,0) = cV4
4(x,o) = 4/4
4(x, 0) = <£(X, 0) = 4(X, o) = 4(X, o) = o
X = 0;T'>0:
dc*A\ Va/B„ ,m^_fi_„ u-ft-a-/»)
dX
dX
ddç_dX
x=o
x=o
x=o
VßJx{1}<>i~>\7A/B
[cA(0,T')- cb(0,T')0a/eÄ(T')]
(dcB\
\dx)x=*
(dc*D\\dXjx=0 \dXy'
X=0
= 0
dc*F\ ( dcGdx )
x=0 V dX j x=0
64 An Electrochemical Oregonator
The dimensionless current function (see section 3.3) for the quasi-reversible electron
transfer equals
These equations are used to create the OC model for this EqrCCCCC mechanism.
The discretisation follows the steps described in section 3.3.3. The correspondingFORTRAN code was incorporated into CVSIM with the model number 37.
5.3 Oscillating Behaviour of the EqrCCCCC Mech¬
anism
As a first step towards the simulation of "oscillating cyclovoltammetric curves" the
right combination of reaction rate constants had to be found. It was mandatory that
the homogeneous form of the electrochemical oscillator, i.e., the electron transfer from
Fe(II) to Fe(III) taking place in the solution and not at the electrode, oscillates without
the influence of the voltammetrie scan. CVSIM cannot be used for this task, because it
was not originally designed for the simulation of purely homogeneous reaction systems.Geisshirt 's program kc (short form of yoke or Yet Another Kinetic Compiler [72])turned out to be a valuable tool for performing simulations quickly with many different
parameter combinations, kc contains a powerful input file parser and a code generator.One simulation run consists of three parts:
f. Input file parsing
2. Code generation and compilation
3. Simulation (stepwise integration of the ODEs)
The biggest advantage of this kinetic compiler is the flexibility regarding the im¬
plementation of different reaction models. If one would like to look at the simulated
response of each out of numerous possible mechanisms, a first answer can be obtained
within a few hours or even within minutes. The code generation and the subsequent
recompilation has to take place after every change of one or more parameter values.
This is probably the main disadvantage of kc, if only a few models have to be compared.Modern computers are so fast that this fact is almost negligible.
The input file for our simulations contained the following lines:
stime = 0;
dtime=1.0;
etime = 1000.0;
epsr= 1.0E-06;
epsa= 1.0E-06;
5.4 Simulating the Electrochemical Oregonator 65
1: A -> B
2: B + C -> A + D
3: D + E -> F + G
4: D + F -> C + G
5: E + F -> B + 2F
6: 2F -> E + G
[A](0) = 5.0E-4;
[B](0) = 0.0;
[C](0) = 0.0;
[D](0) = 3.0E-2;
[E](0) =0.1;
[F](0) = 0.0;
[G](0) = 0.0;
The stime, dtime and etime parameters define the starting time, the time interval
between two integration points and the end time, respectively, of the simulation, epsr
and epsa determine the relative (the r in epsr) and the absolute (the a in epsa) toler¬
ance of the integration. The different reactions and their corresponding rate constants
have to be specified as numbered lines. This system only has forward rate constants
(k>) whereas k< values for the backward reaction could also be given for at least partlyreversible reactions. The third part of the input file has been used for the definition of
the starting concentrations. It is possible to define concentration values which remain
constant throughout the whole simulation. In this case an input line like [E] =0.1
would be appropriate.
The starting values for the rate constants in the input file applied here were taken
from [54] and [73]. As a consequence of the simplified mechanism the system did
not oscillate with this parameter combination. After varying empirically all of the
rate constants, the concentration vs. time plots finally exhibited periodically returning
peaks for species A and F. Figure 5.1 shows the corresponding curves for this behaviour.
The simplifications made in the Oregonator mechanism are probably the main
reason for the non-perfect oscillating behaviour. Also the cycle period is not very
regular. Table 5.1 compares the cycle periods and the peak times of figure 5.1 and
figure 5.2.
The peaks of species F appear a few seconds before those of species A. This be¬
haviour is a prerequisite for oscillations (see section 4-3.2).
5.4 Simulating the Electrochemical Oregonator
5.4.1 General Aspects
The preliminary studies with the kinetic compiler revealed that the electrochemical
Oregonator still oscillates if the corresponding combination of the different reaction
rate constants is employed. The cyclovoltammetric simulation of this mechanism should
; k> = 0.1;
; k> =1.0E2;
; k> =0.5;
; k> =5.0E4;
; k> =3.0E2;
; k> =3.0E3;
66 An Electrochemical Oregonator
0.010
0.005
0.000
200 400 600 800 1000
t/[s]
Figure 5.1: Simulated concentration vs. time plot for species A.
0.0030
0.0020
o
0.0010
0.0000
200 400 600 800 1000
t/[s]
Figure 5.2: Simulated concentration vs. time plot for species F.
5.4 Simulating the Electrochemical Oregonator 67
Table 5.1: Comparison of the cycle periods (AtA for species A and AtF for species
F) with the peak times fta for species A and tp for species F) taken from
figure 5.1 and figure 5.2.
Peak number tA / N AtA / [s] tF j [s] AtF / [s]
f 213 — 210 —
2 343 130 338 128
3 480 137 475 137
4 621 141 615 140
5 736 115 728 113
6 864 128 856 128
yield similar results. The complexity of the model required special attention regarding
possible problems with numerical instabilities during the simulations. The integrationroutine in CVSIM could either completely leave the boundaries or oscillate itself.
5.4.2 Testing the CVSIM Model
A small test procedure for every new CVSIM model is mandatory. Typos and other
errors in the program code mostly lead to wrong results. One of the best indicators for
faulty parts in the code is the concentration profile. If the boundary conditions were not
defined correctly, the dimensionless concentration values often leave the allowed range.
Finding the faulty program lines can be very cumbersome and time-consuming. Duringthe first tests of the electrochemical Oregonator model the expanding simulation space
did not expand as far as expected. It took several days until some critical values were
found to be lost or overwritten between two calls to the corresponding subroutine.
Following Speiser's comparisons (see [39]) the reversible electron transfer served
as the most simple test case. The results from the simulations with CVSIM model I
(Er) were used as reference values. -Estart = 0-0 X, Eswltch = f.O V, AE = 0.0005 V,T = 298 K, E° = 0.5 V and A^ = 9 collocation points remained constant throughoutthe test series. The dimensionless output form of the results was chosen, so that no
values for the concentration of the electroactive species, the diffusion coefficient, the
electrode area and the sweep rate had to be given. The expanding simulation space
was employed for all simulation runs.
The first couple of tests revealed that the ß values for model f and 2 differed by one
order of magnitude from the one of model 37. The significant influence of this deviation
on the results of the simulation was evident when looking at a CVSIM on-line plot of
the concentration profile. The simulation space only expanded to X = 0.8. The CVSIM
array called STFTAG contains boolean values (true or false). At the beginning of every
simulation run all elements of this array are set to false. If the initial concentration
68 An Electrochemical Oregonator
of a species (CINITO(J)) is greater than zero and the absolute difference between the
current concentration at the electrode surface and the initial concentration is greaterthan 0.001 -CINITO(J), then STFTAG(J) is set to true and the simulation space beginsto expand. This occurs in modell/dcdt.f. In fact the content of the STFTAC array was
overwritten or changed between two calls to the DCDT routine. Hence, the expansion
of the simulation space stopped several times more than expected. After introducinga simple save STFTAC statement into modell/dcdt.f, the procedure worked correctly.
The simulations with model 37 and without spline collocation yielded no results,because the ddebdf integrator aborted the runs instantly due to a prohibited changeof the relative error tolerance.
Model 2 and 37 exactly match the reference, if tp' = 100, a = 0.2 and splinecollocation is employed (only model 37). According to the discussion in [39], pp. 138,and Nicholson's paper [74] cited therein, the variation of the transfer coefficient should
not influence the peak current (V^Xp) f°r V > 50. Except for test 2f the model 37
completely fulfils this expectation. The difference between the two ET2 — E° values of
test 20 and 21 is relatively large. However, the main cause for this deviation seems to
lie in the mathematics of the exponential transformation. The remaining four splinecollocation tests (16 to f9) behaved as expected.
5.4.3 Simulations with the Full Electrochemical Oregonator
The CVSIM simulations with the electrochemical Oregonator model should elucidate the
behaviour of a quasi-reversible electron transfer which is part of an oscillating reaction
system. The importance of the time window is increased due to the oscillations. The
sweep rate and the number of consecutive cycles are the target parameters for this part
of the study.These investigations should also demonstrate to which extent the oscillations will
be visible in the voltammogram. Expectations ranged from either additional spikes
superposed onto the normal peak as in [58] or individual peaks completely separatedfrom the ordinary voltammetrie peaks.
The aspect of numerical stability was another point of interest. Experiences with
other less complex CVSIM models and the word oscillation invoked a special awareness
of a possible artifact. Therefore, the numerical correctness of the results had to be
verified extremely carefully.The following parameters remained constant for all simulations:
• £start = 0.0 V; Eswltch = f .0 V; AE = O.OOf V
• T = 298 K
• cA = 5 • f0~4 M (initial concentration of the electroactive species)
• A = O.Of cm2 (electrode surface area)
• D = 10 5 cm2/s (diffusion coefficient)
5.4 Simulating the Electrochemical Oregonator 69
• E° = 0.5 V (formal potential of the electron transfer)
• excess factor of species D = 60; excess factor of species E = 200
• spline collocation, 9 collocation points (if not mentioned otherwise), ddebdf
integrator, expanding simulation space
Table 5.2: Comparison of the first peak of the reversible electron transfer m CVSIM
model 1 (Er), 2 (Eqr) and 37 (EqrCCCCC).
No. Model tf)' a Spline Urban Exponential K/2 ~ E° 1 [mV] \ßXv
f 1 — — — — — 28.5 0.4463
2 2 50 0.2 — — — 29.0 0.4464
3 2 50 0.5 — — — 29.0 0.4452
4 2 too 0.2 — — — 28.5 0.4463
5 2 too 0.5 — — — 28.5 0.4457
6 37 50 0.2 — —
7 37 50 0.5 — —
8 37 50 0.2 yes 29.0 0.4463
9 37 50 0.5 yes 29.0 0.4452
fO 37 50 0.2 yes yes 28.5 0.4452
ff 37 50 0.5 yes yes 29.0 0.4439
12 37 50 0.2 yes yes 28.5 0.4465
13 37 50 0.5 yes yes 29.0 0.4465
14 37 too 0.2 — —
15 37 too 0.5 — —
16 37 too 0.2 yes 28.5 0.4463
17 37 too 0.5 yes 28.5 0.4457
18 37 too 0.2 yes yes 28.0 0.4450
19 37 too 0.5 yes yes 28.0 0.4445
20 37 too 0.2 yes yes 28.5 0.4464
21 37 too 0.5 yes yes 30.5 0.4477
The sweep rate was increased from f 0 mV/s up to f00 V/s. The heterogeneous rate
constant for the electron transfer was set to either f0_1 cm/s or f0-3 cm/s or f0-6 cm/s
70 An Electrochemical Oregonator
in order to simulate reversible, quasi-reversible and irreversible electron transfers. The
transfer coefficient a was given three different values, 0.2, 0.5 and 0.8.
Plain spline collocation without any transformation technique was mostly employedfor these simulations. However, certain critical parameter combinations demanded
more sophisticated integrator capabilities.
5.4.4 Results
A single CVSIM run in the first series consisted of a voltammogram with six half-cycles.The electron transfer was either reversible, quasi-reversible or irreversible (see section
5.4-3). The transfer coefficient a remained constant at 0.5 and the sweep rates were set
to O.Of, 0.02, 0.05, O.fO, 0.20, 0.50, f.0, 2.0, 5.0, fO, 20, 50 and fOO V/s, respectively.The time window of the kc simulations is relatively large compared to the one of
cyclic voltammetry. Three cycles from 0.0 V to f .0 V and back each with a sweep rate
of f V/s take exactly 6 seconds to complete. Fitting this slice into figure 5.1 makes
it obvious that the oscillations will hardly interfere with the voltammetrie actions.
However, the oscillatory impact will be visible on the slow scans up to 50 mV/s or
even fOO mV/s. In order to impose any kind of "chaotic distortions" onto the faster
scans the number of cycles has to be increased.
First of all, the one-cycle voltammograms of model 37 were compared with the
ones of model 2 over the whole range of sweep rates. The influence of the reversibilityof the electron transfer was investigated by altering the values for the heterogeneousrate constant and the transfer coefficient. The reversible case is shown in table 5.3, the
quasi-reversible one in table 5.4 and the irreversible one in table 5.5.
The slow scans (up to 0.5 V/s) show the biggest deviations between the two models.
The peak values are almost identical for sweep rates higher than 2 V/s. The first
three points for model 37 in the reversible and the quasi-reversible case are either far
away from their model 2 counterparts or there were no peaks at all. In this case the
oscillating nature of this reaction system exhibits its strongest impact. The integratorbecomes instable and the corresponding concentration profiles show irregular waves
with sometimes even negative values. The electron transfer is perceptive for the impactof the oscillations, if the first half-cycle takes more than 10 seconds to complete.
To prolong the interactions between the electron transfer and the oscillations, the
number of cycles was increased to 20 for a scan made with 50 mV/s. The cyclovoltam¬metric simulation then reaches a duration of 800 seconds, approximately the same
time scale as the electrochemical Oregonator itself. To simulate the voltammograms
depicted in figure 5.3, two variables were set to different values: &het = f0~6 cm/s and
a = 0.5. The irregular behaviour of the current vs. time curve is evident, especially if
figure 5.3 is compared with figure 5.4-
The series of anodic peaks in figure 5.4 shows a steadily decreasing peak current,
whereas the peak heights in figure 5.3 periodically decrease and reincrease. The time
period between one and the next increased peak is about f20 seconds. This value fits
quite well into the range of oscillation periods simulated with the kinetic compiler and
listed in table 5.1. Figure 5.5 compares the anodic peak times from figure 5.3 with the
5.4 Simulating the Electrochemical Oregonator 71
100
50
1 0
-50
-100
0 200 400
t/[s]
600 800
Figure 5.3: Simulated voltammogram with 20 consecutive cycles made with CVSIM
model 37. The sweep rate was 50 mV/s, khet = 10~6 cm/s and a = 0.5.
<=3.
1.U-
0.5 llll III u il:
0.0 \ \r VVWW ir \ ir u VW vW in
0 5
" I II I | | I I I I I
0 200 400
t/[s]
600 800
Figure 5.4: Simulated voltammogram with 20 consecutive cycles made with CVSIM
model 2. The sweep rate was 50 mV/s, khet = 10~6 cm/s and a = 0.5.
72 An Electrochemical Oregonator
Table 5.3: Comparison of the first peak of the reversible electron transfer m CVSIM
model 2 (Eqr) with model 37 (EqrCCCCC). The sweep rate is the onlyvariable m this comparison.
v 1 [V/s] khet / [cm/s ] a Ep(2) ip j [M] (2) Ep j [mV] (37) ip j [M] (37)
O.Of fO"1 0.2 529 0.4249 757 f6.8476
0.02 fO"1 0.2 529 0.6010 — —
0.05 fO"1 0.2 529 0.9503 — —
O.fO fO"1 0.2 530 1.3441 544 1.9355
0.20 fO"1 0.2 530 1.9010 534 2.1711
0.50 fO"1 0.2 531 3.0066 532 3.1250
f.00 fO"1 0.2 532 4.2528 533 4.3251
2.00 fO"1 0.2 534 6.0159 534 6.0625
5.00 fO"1 0.2 537 9.5150 537 9.5429
fO.OO fO"1 0.2 540 f3.4586 540 13.4773
20.00 fO"1 0.2 544 f9.0326 543 19.0451
50.00 fO"1 0.2 550 30.08f2 550 30.0868
fOO.OO fO"1 0.2 556 42.5095 556 42.5119
peak times of compound A from figure 5.1. The oscillation peaks of the kc simulation
always match the last anodic peak before the peak current increases again. Therefore,it can be assumed that the CVSIM model 37 is oscillating like the corresponding kc
model.
The first cycles in voltammograms made with model 37 and v = 200 mV/s are
already very similar to those made with model 2 and the same sweep rate (see figures 5.6
and 5.7). However, the anodic peaks in the consecutive cycles with the electrochemical
Oregonator steadily increase with every new cycle, whereas the peaks in the quasi-
reversible electron transfer model slightly decrease.
Increasing the sweep rate to 10 V/s clearly reduces the influence of the oscillatingreaction system. Figures 5.8 and 5.9 look almost identical at a first glance. The peakcurrents finally are in the same range and the consecutive cycles show small deviations.
The third cycles of figures 5.8 and 5.9 are compared in figure 5.10.
Increasing the sweep rate to even higher values does not decrease the relative dif¬
ference between the peak heights. It remains in the range of the one shown in figure5.10.
The simulations with the quasi-reversible and the irreversible electron transfer yieldsimilar results. Regarding the form and the height of the peaks, the irreversible case is
5.4 Simulating the Electrochemical Oregonator 73
<
100
80
60
40
20
0 200 400
t/[s]
^<
600 800
Figure 5.5: Comparison of the anodic peak times from figure 5.3 with the peak times
from figure 5.1.
fst cycle
2nd cycle
3rd cycle
0 200 400 600 800 1000
E/[mV]
Figure 5.6: Simulated voltammogram with 3 consecutive cycles made with CVSIM
model 37, v = 200 mV/s, khet = 10-1 cm/s and a = 0.5.
74 An Electrochemical Oregonator
200 400 600 800 1000
E/[mX]
Figure 5.7: Simulated voltammogram with 3 consecutive cycles made with CVSIM
model 2, v = 200 mV/s, khet = 10-1 cm/s and a = 0.5.
400 600
E/[mV]
1000
Figure 5.8: Simulated voltammogram with 3 consecutive cycles made with CVSIM
model 37, v = 10 V/s, khet = 10-1 cm/s and a = 0.5.
5 4 Simulating the Electrochemical Oregonator 75
15
1 0
1st cycle
2nd cycle
3rd cycle
400 600
E/[mV]
1000
Figure 5.9: Simulated voltammogram with 3 consecutive cycles made with CVSIM
model 2, v = 10 V/s, khet = 10-1 cm/s and a = 0 5
400 600
E/[mV]
1000
Figure 5.10: Comparison of the third consecutive cycles taken from figures 5 8 and
5 9, v = 10 V/s, khet = 10-1 cm/s and a = 0 5
76 An Electrochemical Oregonator
Table 5.4: Comparison of the first peak of the reversible electron transfer m CVSIM
model 2 (Eqr) with model 37 (EqrCCCCC). The sweep rate is the onlyvariable.
v 1 [V/s] khet / [cm/s ] a Ep(2) ip j [M] (2) Ep j [mV] (37) ip j [M] (37)
O.Of fO"3 0.5 569 0.3686 654 18.7022
0.02 fO"3 0.5 582 0.5083 — —
0.05 fO"3 0.5 60 f 0.78f2 — —
O.fO fO"3 0.5 618 f.0872 685 1.7858
0.20 fO"3 0.5 635 f.5204 647 1.7886
0.50 fO"3 0.5 658 2.3820 660 2.4697
f.00 fO"3 0.5 676 3.3556 676 3.3982
2.00 fO"3 0.5 693 4.7345 693 4.7570
5.00 fO"3 0.5 717 7.4734 717 7.4839
fO.OO fO"3 0.5 735 f0.5626 735 f0.5676
20.00 fO"3 0.5 752 f4.9326 752 f4.9348
50.00 fO"3 0.5 776 23.6053 776 23.6044
fOO.OO fO"3 0.5 — — 794 33.3768
not interesting for a more detailed comparison, because the peaks are located almost
completely outside of our potential window. However, the quasi-reversible case sub¬
stantiates the observations made with the reversible one. Figures 5.11 and 5.12 clearlydemonstrate the similarities between the reversible and the quasi-reversible case.
5.5 Numerical Instabilities
There are (electro-)chemical as well as digital oscillations. Although the latter are
directly due to the numerical instability of the integrator, the real reason is hidden
beneath the chemical internals of the mathematical model. The oscillation peaks de¬
picted in figures 5.1 and 5.2 ascend very quickly and follow a steep slope back to the
baseline. The CVSIM integrator tries to cope with these extreme gradients. It usuallyfulfils the requirements, if the time window of the voltammogram is much narrower
than the one of the oscillation, i.e. the sweep rate has to be higher than 50 mV/s in
the case of the electrochemical Oregonator. The faster the scan the easier the slopeand hence the more stably the integrator will run.
The slowest scans made with 10 mV/s, 20 mV/s or 50 mV/s are the only ones that
are prone to numerical instabilities as long as the number of voltammetrie cycles is
5.5 Numerical Instabilities 77
3rd cycle
A _
____ 2nd cycle
1st cycle
0 200 400 600 800
E/[mX]
1000
Figure 5.11: Simulated voltammogram with 3 consecutive cycles made with CVSIM
model 37, v = 200 mV/s, khet = 10~3 cm/s and a = 0.5.
0 200 400 600
E/[mX]
800 1000
Figure 5.12: Simulated voltammogram with 3 consecutive cycles made with CVSIM
model 2, v = 200 mV/s, khet = 10~3 cm/s and a = 0.5.
78 An Electrochemical Oregonator
Table 5.5: Comparison of the first peak of the irreversible electron transfer in CVSIM
model 2 (Eqr) with model 37 (EqrGGGGG). The sweep rate is the onlyvariable m this comparison.
v 1 [V/s] khet / [cm/s ] a Ep(2) ip 1 [M] (2) Ep j [mV] (37) ip 1 [M] (37)
O.Of fO"6 0.8 fOOO 0.0224 fOOO 0.9574
0.02 fO"6 0.8 fOOO 0.0228 fOOO 0.0927
0.05 fO"6 0.8 fOOO 0.023f fOOO 0.0453
O.fO fO"6 0.8 fOOO 0.0233 fOOO 0.0338
0.20 fO"6 0.8 fOOO 0.0234 fOOO 0.0270
0.50 fO"6 0.8 fOOO 0.0235 fOOO 0.0240
f.00 fO"6 0.8 fOOO 0.0236 fOOO 0.0237
2.00 fO"6 0.8 fOOO 0.0236 fOOO 0.0236
5.00 fO"6 0.8 fOOO 0.0237 fOOO 0.0236
fO.OO fO"6 0.8 fOOO 0.0237 fOOO 0.0237
20.00 fO"6 0.8 fOOO 0.0237 fOOO 0.0237
50.00 fO"6 0.8 fOOO 0.0237 fOOO 0.0237
fOO.OO fO"6 0.8 fOOO 0.0237 fOOO 0.0237
restricted to 3. Thereby, the heterogeneous rate constant of the single electron transfer
plays an important role. Sometimes even a change of a can lead to instabilities or to
the complete interruption of the simulation.
The influence of the numerical stability can be visualised by varying the number of
collocation points and comparing the resulting voltammograms.The voltammetrie cycles in figures 5.13 and 5.14 significantly differ from those in
figures 5.15 and 5.16. The first oscillation at about 600 mV determines the further
course of the cycle. The number of collocation points is directly related to the capabilityof the model to follow the strong gradients.
The big cathodic peak at the end of the voltammetrie cycle in figure 5.14 is
somewhat surprising, because it does not appear in any of the other voltammograms.These two parameter combinations react most sensitively upon a small disturbance
that is probably smoothed out by the integrator in the case of N = 6 and can be
accurately simulated in the case of N = 10 and N = 15.
5.5 Numerical Instabilities 79
20
<^ -20 h
-40
-60
0 200 400 600 800 1000
E/[mX]
Figure 5.13: Simulated voltammogram made with model 37, v = 10 mV/s, khet
lO'1 cm/s, a = 0.2 and N = 6.
20
0
<
200 400 600 800 1000
E/[mX]
Figure 5.14: Simulated voltammogram made with model 37, v = 10 mV/s, khet
lO'1 cm/s, a= 0.2 andN = 8.
80 An Electrochemical Oregonator
201 i i i i
0 f = ^
-20 -\
^ -40
^-
-60
-so- y
-1001, i , i , i , i ,
0 200 400 600 800 1000
E/[mX]
Figure 5.15: Simulated voltammogram made with model 37, v = 10 mV/s, khet =
lO'1 cm/s, a = 0.2 and N = 10.
0 200 400 600 800 1000
E/[mX]
Figure 5.16: Simulated voltammogram made with model 37, v = 10 mV/s, khet =
10'1 cm/s, a = 0.2 and N = 15.
Chapter 6
Outlook
This work has shown that the oxidation mechanism of a-hydroxyketones in an aqueous
alkaline solution cannot easily be analysed only by means of cyclic voltammetry. The
computer simulations of the reaction did not verify unambiguously the mechanism
proposed (EqrCEqr/COMP), despite of some similarities between the experimental and
the simulated data. From the results it is difficult to decide whether the mechanism
used for the simulations was not appropriate or whether merely the correct parameter
combination could not yet be found.
There is good reason to believe that the latter is the case and that the lack of
agreement between the cyclovoltammetric experimental data and the model simula¬
tions discussed in section 3.4 is not due to adsorption or other electrode effects, but
rather to the oscillating behaviour of the redox system. For oscillating systems it is
very difficult to find the appropriate set of parameters for simulating the experimentaldata. Therefore, future investigations should take into account the procedure described
recently by Strasser et al. [64]. There, several identification tests build the basis to
place the unknown system in one of four principal oscillator categories, each of which
corresponds to a specific class of kinetic prototype models. The combination of these
models with the results from preliminary studies should lead to an electrochemical
oscillator which can be simulated by CVSIM.
A recent detailed analysis of the vatting redox system with a-hydroxyketones [4]has revealed a reaction mechanism which is depicted in figure 1.1 and defined by the
following reaction steps:
Reaction Model equations Rate constants
Ci RH2 + OH- -»• RH" + H20 k = 0.0005 s"1
Ei RH~ + Eox - RH* + Ered k = 0.003 l/(mole-s)
c2 RH* + OH- -»• R- + H20 k = 0.3 s"1
E2 R*~ + Eox - R + Ered k = 12000 l/(mole-s)
COMP R + RH- -»• R— + RH* k = 10000 l/(mole-s)
c3 R -»• p k = 0.4 l/(mole-s)
81
82 Outlook
This slightly modified mechanism differs from the EqrCEqr/COMP mechanism used
in section 3.4 to simulate the measured cyclovoltammograms by the general base catal¬
ysed deprotonation Ci and the only sink reaction, the aldol condensation C3 of the
diketone.
The kc simulations with this modified mechanism finally reveal a damped oscillation
(see figure 6.1) for the rate constants given above and for the following concentration
values:
• crh2 = 0.003 mole/1
• coh- = 1-0 mole/1
• cEox = 0.003 mole/1
• Crh- = CRH. = Cr- = Cr = Cp = CEred = 0.0 mole/1
0.50
0.40
& 0.20-
0.10
0.00
0 100 200 300 400
t/[s]
Figure 6.1: Simulated concentration vs. time plot for species R'~.
This somewhat surprising, although predictable, new result may lead to a more ba¬
sic understanding of the experimental cyclovoltammograms by future electroanalytical
investigations of the vatting redox system with a-hydroxyketones.
Chapter 7
Experimental Section
7.1 Instrumental Setup
Several combinations of measurement hardware and software were employed duringthe whole thesis. The combination for the experiments in aqueous solutions consisted
of the amel SYSTEM 5000 and the amel software EASYSCAN on an IBM INTEL 80286
personal computer under WINDOWS 3.1. Especially the program suffered from the very
archaic Graphical User Interface (GUI) and the extremely bad memory management.
Only the last cycle in a series of multiple consecutive scans could be saved to a file.
Therefore, the comparison of such a set of voltammograms was impossible in this way.
In order to circumvent the problem the corresponding number of single scans was
carried out one after another as fast as possible. It took several seconds until the next
run could be started. If one makes fast scans (> 10 V/s) and the system contains
relatively rapid chemical follow-up reactions, then the changes between two cycles are
likely to be too large.The experiments in acetonitrile were first carried out with the amel system 5000
combined with the new software CORRWARE from SCRIBNER. The potentiostat was
connected to a COMPAQ INTEL PENTIUM PRO computer under WINDOWS95 via a Gen¬
eral Purpose Interface Bus (GPIB) card from national instruments. Although the
CORRWARE program provided a comfortable GUI with a big variety of scan forms, the
communication part between the computer and the potentiostat did not work prop¬
erly. Scan rates over 10 V/s led to sudden halts of the run execution. After several
unsuccessful attempts to fix this problem the decision was made to switch over to a
new device.
The next combination consisted of the radiometer VoltaLab32 station and the
radiometer VoltaLab software installed on an IBM INTEL 80486 computer under
WINDOWS 3.1. This device provided the possibility to make experiments with scan
rates up to 100 V/s and an almost unlimited number of different cycle parts. None of
the aforementioned problems with the hardware and/or the software was encountered.
Due to iR drop problems mainly with the reference electrode the BAS (BioAnalyticalSystems) 100B/W electrochemical workstation together with the BAS 100W software
on an Olivetti INTEL 80386 computer under WINDOWS 3.1 was used for another
83
84 Experimental Section
series of experiments. The computer controlled measurement and compensation of the
iR drop was very helpful to determine and reduce the resistance of the whole system.
7.2 Reference and Working Electrodes
7.2.1 General Remarks
Beginning from section 2.3.3 p. 20 the most important criteria and the correspondingreferences to choose a suitable working electrode and reference electrode combination
have been summarised.
7.2.2 Electrodes in Aqueous Media
The platinum working electrode used for the measurements in aqueous media was
bought from METROHM. The rather large active electrode surface was about 0.2 cm2
measured by means of the method described in [75] p. 74ff. A Ag/AgCl electrode
(3.0 M KCl, +0.2 V vs. SHE) served as the reference electrode.
7.2.3 Reference Electrode in Acetonitrile
Section 2.3.6 mentions that the design of a good reference electrode for organic solvents
is a rather sophisticated task. Usually water has to be completely excluded from the
electrochemical cell. Hence, one must either use aqueous reference electrodes togetherwith salt bridges or directly organic ones. Junction potentials and the low conductivityof organic solvents are the main interferences the electrochemist has to cope with.
The reference electrode proposed by Speiser et al. [76] was chosen for these series
of experiments. Our own glass-blowing shop has built all the various models. The first
version employed was similar to the left one (a) shown in figure 7.1. There are three
main glass parts and the Ag-wire (part A) that acts as the "real" reference electrode.
Part B is filled with acetonitrile (MeCN), O.f M tetrabutylammonium hexafluorophos-
phate (TBAHFP) and O.Of M AgC104 (fluka, puriss). The transport of the latter
substance down to the other compartments is strictly limited by the glass frit at the
lower end of B. Both C and D contain MeCN and O.f M TBAHFP. The glass frit in
C ensures that the contamination of the cell solution with Ag+ and C104~ ions is keptat an absolute minimum. The Luggin capillary tip at the end of D should be one to
two millimeters away from the tip of the working electrode in order to minimise the
iR drop between the two electrodes.
The dual reference electrode system requires a redesign of part D. A Pt-wire, sealed
in a glass capillary, is introduced in the upper half of D'. The two capillaries are tightlybonded to each other, but there is no contact between the wire and the electrolyte.The tip of this wire presents the single point of exposition to the cell solution. The
upper end of the Pt-wire is connected to the capacitor/potentiostat via a copper wire.
The Pt-wire/capacitor line lowers the response time during fast changes whereas the
7.3 Solvent and Reagent Preparation 85
high impedance Luggin capillary/Ag-wire part works best for situations close to DC
conditions.
Filling the reference electrode turned out to be a little challenging. In order to gainthe maximum electrical conductivity we have to make sure that the capillary does not
contain any gas bubbles and that the pores in the glass frits are completely filled with
electrolyte. Strange oscillations were observed during the first couple of experiments,because some loose connections between the different compartments of the reference
electrode existed. The part with the Luggin capillary was finally filled by sucking up
the electrolyte through the capillary by means of vacuum. The middle and the upper
compartment (with glass frits) can be filled from the top as usual.
7.2.4 Working Electrode in Acetonitrile
The previously employed platinum working electrode sometimes caused oscillations
during cyclovoltammetric scans. The active electrode surface seemed to be to big, and
the combination of spherical and cylindrical tip parts made the definition of the surface
area quite difficult. Hence, a platinum disk electrode (disk diameter about f mm) built
by our glass-blowing shop was used instead.
7.3 Solvent and Reagent Preparation
7.3.1 Procedure for the Aqueous Electrolyte
The background electrolyte always consisted of f M aqueous NaOH, which was made
from solid sodium hydroxide (merck, p. A.) and doubly-distilled water. There was
no further purification of the electrolyte.
7.3.2 Purification of the Organic Electrolyte
Purification and drying of organic solvents for electrochemical use can be very complex.Kiesele [77] developed an elaborate procedure to convert technical grade acetonitrile
to an ultrapure product. The only drawback of this method is the long time it takes
to go through the four steps. It is possible to overcome this disadvantage by simply
coordinating your experiments and the purification procedure, so that you will never
run out of pure solvent.
There are other methods to turn commercially available products into electrochem¬
ically pure solvents. Parker and Jensen [78] employed a procedure from Mann [79]to carry out their studies. The solvent is thereby dried and purified over a column
filled with activated aluminium oxide (iCN biomedicals, ICN Alumina N - Super
I). The preliminary activation of the aluminium oxide is performed under high vac¬
uum (< 0.001 Torr) at a temperature of 150 °C during 12 hours. The acetonitrile
(riedel-de HAËN, gradient grade or LAB-SCAN, super gradient grade) is passed twice
under argon over the column. After the first pass O.f M TBAHFP (fluka, puriss., for
electrochemical use) was added to the MeCN and this solution was used for the second
86 Experimental Section
Copper Wire-
7/15 Glass Joint
Silver Wire
u
7/15 Glass Joint! I
10/19 Glass Joint
Glass Frit
10/19 Glass Joint
14/23 Glass Joint
Glass Frit
14/23 Glass Joint
14/23 Glass Joint
B
D
Capacitor L-
Clamp >
Copper Wire
Soldered Joint i'
B
D"
=::®Pt-wire (sealed in
glass capillary)
Luggin capillary tip
magnified
Pt-wire tip
Figure 7.1: Schematic view of two different versions of a reference electrode for use
in organic solvents. Part a shows the single pieces that make up the elec¬
trode. The so-called dual reference-electrode system with an additional
Pt -wire is already assembled in part b.
7.3 Solvent and Reagent Preparation 87
run. This method also removes impurities and water from the TBAHFP. In spiteof the simplicity of this method the resulting purity of the solvent is sufficient for
electrochemical use. Therefore, this was the stock solution for all of our measurement
solutions.
7.3.3 Preparation of the Acyloins
Purification of the acyloins, i.e. adipoin (2-hydroxy-cyclohexanone, ALDRICH, puriss.)and acetoin (3-hydroxy-2-butanone, MERCK, puriss.), was a simple procedure. The
white powder was first melted (at ca. f fO °C) and subsequently distilled. The colour of
the acyloins changes from white to yellow when they are melted. After the distillation
adipoin was obtained as a colourless liquid and acetoin as a yellow one.
In order to determine the stability and the change of the composition of the
freshly distilled acyloins, they were dissolved in doubly distilled water, acetone (fluka,puriss.) and acetonitrile (riedel-de haen, gradient grade) respectively. GC-MS and
GC-FID analysis of these solutions were recorded over a time period of three days. The
subsequent comparison of the chromatograms and mass spectrograms revealed, that
the two peaks of interest at the beginning belong to the acyloin and its corresponding
diacetyl. There were not any new compounds in any of the test solutions by the end
of the tests. One sample was even heated up to 60 °C for a couple of hours, but no
changes could be observed.
The long term stability of the acyloins involved into these tests seems to be granted.It is possible to take e.g. adipoin from the same stock solution for at least three or four
consecutive days and still have an equally composed analyte solution.
88 Experimental Section
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Appendix A
Symbols
Symbol Description Units
F
RT v, variable proportional to the sweep rate
ai = 7—, activity of species I in solutionmo
A electroactive area of the electrode
Ah3 elements of the discretisation matrix of the first derivative
with respect to the space coordinate
BhJ elements of the discretisation matrix of the second deriva¬
tive with respect to the space coordinate
Ci/m = K(0,T') cl(l,T>) c*mi(0,T>) C(f,T')]', dimen-
sionless concentration subvector for the short representa¬
tion of the spline collocation boundary matrix equationc° starting concentration of a species /
--idimensionless concentration of a species /
c* dimensionless concentration of a species I in the inner ("re¬action") layer (only SC)
c* dimensionless concentration of a species I in the outer
(diffusion-only) layer (only SC)c*(X,T') dimensionless concentration of a species I with respect to
X and V
Q,oo bulk concentration of species /
Di diffusion coefficient of species /
E potential
.Eceii actual electrochemical cell potential, i.e. the potential dropbetween the electrodes including iR drop and overpoten¬
tials at the electrodes)
cm2]
mole/1]
mole/1]
cm2/s]
V]
V]
93
94 Symbols
Symbol Description Units
E,cell, rev
^end
po
'
EP
K
-'-'start
-C'switch
Ex
AE
AElp,l/m
f
F
G
ArG
i
ip
iR
Ja
Je
Jl
kA
kc
reversible electrochemical cell potential, i.e. the theoretical
potential drop between the electrodes
end potential of the sweep
standard equilibrium potential of the redox pair l/m
Equilibrium potential of the redox pair l/m
peak potential
peak potential of the first peak (peak I)
starting potential of the sweep
switching potential of the sweep
switching potential
potential step size of the output
= Ep — E°,, difference between the first peak potentialand the standard equilibrium potential of the redox couple
l/mgeneral function of a variable
= 96485 C/mole, Faraday constant
(total) free Energy
free energy change during a reaction
current
peak current
Potential drop through the solution between anode and
cathode
Current density at anode
current density at cathode
limiting exchange current density
rate constant for the anodic reaction
rate constant for the cathodic reaction
forward rate constant of a chemical reaction i, f st order or
2nd order
[V]
[V]
[V]
[V]
[V]
[V]
[V]
[V]
[V]
[V]
[V]
[C/mole]
[J/mole]
[J/mole]
[A]
[A]
[V]
[A/cm2]
[A/cm2]
[A/cm2]
[cm/s]
[cm/s]
[f/s] or
[l/(mole-s)
Symbols 95
Symbol Description Units
k—%
s,l/m
Kt
L
mi
m0
N
ni
Ql/m
R
Sx(T>)
t
tx
T
V
U
U
U'
ux
U's
backward rate constant of a chemical reaction i, I st order
or 2nd order
heterogeneous rate constant of the electron transfer reac¬
tion of a redox couple l/mequilibrium constant of a reaction i
distance from the electrode where the experiment has no
influence on the concentration
unknown concentration terms 4x4 submatrix of the splinecollocation boundary matrix equation
molality of species I
= f mole/kg, standard molality
number of collocation points without the boundary points
number of transferred electrons of species I
dimensionless right-hand side constant terms subvector of
the spline collocation boundary matrix equation= S\(T')9i/m, mathematical abbreviation for the short rep¬
resentation of the boundary matrix equation= 8.314 J/(mole-K), gas constant
dimensionless time-dependent potential function for cyclic
voltammetrytime
time, when the switching potential is reached
(absolute) temperature
dimensionless time
dimensionless time necessary to scan the potential range
between E°, and E\ with a constant sweep rate v (cyclic
voltammetry)nonlinear transformation of the space coordinate after Ur¬
ban and also space coordinate after the application of this
transformation
exponential space coordinate transformation and resulting
space coordinate
spline collocation coordinate after the U transformation
spline collocation coordinate after the U' transformation
1/s] or
l/(mole-s)
cm/s]
m
mole/kg]
J/(mole-K)]
K]
96 Symbols
Symbol Description Units
V sweep rate [V/s]
V spline collocation space coordinate of the outer layer after
the application of the U transformation
[-]
V spline collocation space coordinate of the inner layer after
the application of the U transformation
H
w spline collocation space coordinate of the outer layer after
the application of the U' transformation
H
w spline collocation space coordinate of the inner layer after
the application of the U' transformation
H
X distance from the electrode surface [m]
Xi mole fraction of species I [-]
xs distance of the spline point from the electrode [m]
X dimensionless space coordinate [-]
xt space coordinate of the collocation point i [-]
xs dimensionless distance of the spline point from the elec¬
trode
[-]
x, dimensionless distance in the inner layer (SC) [-]
x2 dimensionless distance in the outer layer (SC) [-]
al/m transfer coefficient of the backward reaction of the redox
couple l/m, 0 < cx°, < I
[-]
Aai/m partial increment of the transfer coefficient 0.°, in depen¬
dence of the potential E, actually used as Aai/m/AE
H
ß dimensionless diffusion coefficient of the outer layer (SC) [-]
ß' dimensionless diffusion coefficient of the inner layer (SC) [-]
H activity coefficient of species I in solution [-]
8N thickness of the diffusion layer [m]
Vc applied overpotential at the cathode [V]
VA applied overpotential at the anode [V]
Ol/m potential constant of the redox couple l/m after Nicholson
and Sham
H
K% dimensionless forward rate constant of a reaction 1 H
K-i dimensionless backward rate constant of a reaction 1 H
Symbols 97
Symbol Description Units
chemical potential of species I in solution [J/mole]
standard chemical potential of species I in solution [J/mole]
electrochemical potential of species / in solution [J/mole]
stoichiometric coefficient of species / [-]
Galvani potential or electrical potential of species / [V]
standard Galvani potential difference [V]
chemical reaction term [mole/(l-s)
dimensionless chemical reaction term [-]
timescale definition parameter of an electroanalytical ex- [1/s]périmentdimensionless heterogeneous rate constant of the redox cou- [-]pie l/mdimensionless current function, mostly multiplied with V^ H
ßi
P°i
fil
<Pi
At/?°
P
P*
T
Vl/m
X
V^xi dimensionless peak current function of the first peak (peakI)
98 Symbols
Appendix B
Abbreviations
BZ Belousov-Zhabotinsky
Cir irreversible homogeneous chemical reaction
Cqr quasi-reversible homogeneous chemical reaction
Cr reversible homogeneous chemical reaction
CE Counter Electrode
COMP comproportionation reaction
CPU Central Processing Unit, actually the main microprocessor of the computer
CSTR Continuous Stirred Tank Reactor
CV Cyclic Voltammetry, an electroanalytical method
CVSIM Cyclic Voltammetry SIMulation, a program for the simulation of cyclic
voltammetry experiments
DC Direct Current
DDEBDF a backward differentiation algorithm readily applicable to stiff differential
equation systems
DISP disproportionation reaction
DME Dropping Mercury Electrode
DMF Dimethylformamide
DMSO Dimethylsulfoxide
Eir irreversible electron transfer
E quasi-reversible electron transfer
Er reversible electron transfer
99
too Abbreviations
EASI ElectroAnalytical Simulation, Speiser 's program system for this purpose
EFD Explicit Finite Differences
ET Electron Transfer
ETR Electron Transfer Reaction, the same as ET
FD Finite Differences, includes IFD (Implicit Finite Differences) and EFD
(Explicit Finite Differences)
FID Flame Ionisation Detector
FKN Field, Koros and Noyes, the developers of a detailed reaction mechanism
for the Belousov-Zhabotmsky reaction
GC Gas Chromatography
GPIB General Purpose Interface Bus; this bus was specifically designed to con¬
nect computers, peripherals and laboratory instruments so that data and
control information could pass between them; it is also known as IEEE-488
GUI Graphical User Interface
IFD Implicit Finite Differences
IHP Inner Helmholtz Plane
MeCN Methylcyanide = acetonitrile
MS Mass Spectroscopy
NAD Nicotinamide Adenine Dinucleotide
NHE Normal Hydrogen Electrode (see also SHE)
OC Orthogonal Collocation, a mathematical method for the solution of partialdifferential equations by polynomial approximation
ODE Ordinary Differential Equation
OHP Outer Helmholtz Plane
PC Propylene Carbonate
PDE Partial Differential Equation
PZE Potential of Zero Charge
RE Reference Electrode
SC Spline Collocation, a special variant of orthogonal collocation which di¬
vides the simulation space into an inner and an outer layer; especiallyuseful for fast homogeneous chemical reactions
Abbreviations fOf
SCE Saturated Calomel Electrode
SHE Standard Hydrogen Electrode
TBAHFP Tetrabutylammonium hexafluorophosphate
WE Working Electrode
102 Abbreviations
Curriculum Vitae
Education
since 1995
f994
f990 - f994
f983 - f990
f977 - f983
Ph.D. thesis in the Department of Industrial and Engi¬
neering Chemistry, ETH Zürich, Switzerland
Supervisor: Prof. Dr. Paul Rys
Diploma thesis with Prof. Dr. Andreas Manz, Ciba-
Geigy Ltd. (Novartis), Corporate Analytical Research,
Basel, Switzerland, now Professor at Imperial College,
London, England
Chemical engineering studies, Department of Chemistry,ETH Zürich
High school, "Kantonsschule Zug"
Elementary school, Baar
Experience
since 1999
Sept. f998 - Dec. f998
May f997 - Sept. f998
Deputy of the group leader, Software & Licenses group,
Computing Services, ETH Zürich
Member of the Software & Licenses group, Computing
Services, ETH Zürich (part-time employment)
Responsible for Silicon Graphics support at the User
Support section of the Computing Services, ETH Zürich
(part-time employment)
Zürich, March 22, 2000 Signature:
103