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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

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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.


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:


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)


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


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)


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.


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

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


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


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:


<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)


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)


All _

^Sxiryv-ê-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

Â/Bn-^-»A/B) -/JTÂ/BVß Â/B

Â/B(n-^-»A/B) AlflVß Â/B

0

0

/ c*A(0,T) \

4(0, T')

4(0, T)

V 4(0, T') /

<A/B inpA/lVß Â/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)


3.3.4 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


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:


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.


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^-ô-^^[cAl(0,T')-cBl(0,T')eA/BSx(T')}

The extraction of the unknown concentration terms leads to:

(3.53)

/L,_

Â/B Q(r'\-{l-aAIK)9~{l~aA/B)î.i rgj

Û ) 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)

'Â/B

-(l-aA/B)Â/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)


^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 /


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


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:


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)


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:


(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)


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


• 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.


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


<

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).


<

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).


<

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).


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


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.


3.4.6 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


<

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]


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


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.

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


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;


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;


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.


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


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)


• 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


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


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


model 2. The sweep rate was 50 mV/s, khet = 10~6 cm/s and a = 0.5.



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


<

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]


model 37, v = 200 mV/s, khet = 10-1 cm/s and a = 0.5.


200 400 600 800 1000

E/[mX]


model 2, v = 200 mV/s, khet = 10-1 cm/s and a = 0.5.

400 600

E/[mV]

1000


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


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



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


3rd cycle

A _

____ 2nd cycle

1st cycle

0 200 400 600 800

E/[mX]

1000


model 37, v = 200 mV/s, khet = 10~3 cm/s and a = 0.5.

0 200 400 600

E/[mX]

800 1000


model 2, v = 200 mV/s, khet = 10~3 cm/s and a = 0.5.


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.


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.


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


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


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.


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.

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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


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


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


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


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

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