Resonance Forcing in Catalytic Surface
Reactions
vorgelegt von
Master of Science (M.Sc.)
Prabha Kaira
von der Fakult¨at II - Mathematik - und Naturwissenschaften
der Technische Universit¨at Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaft
-Dr.rer.nat.-
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. E. Sedlmayr
Berichter: Prof. Dr. H. Engel
Berichter: Prof. Dr. H. H. Rotermund
Tag der wissenschaftlichen Aussprache: 03.02.2009
Berlin 2009
D 83
Kurzfassung
Die vorliegende Arbeit untersucht theoretisch und experimentell die raumzeitliche
Musterbildung der CO-Reaktion auf Pt(110) mit den Methoden der nichtlinearen Dy-
namik. Die Reaktion selbst ist eine gut erforschte und verstandene chemische nicht-
lineare Reaktion auf einer Einkristalloberfl¨ache. Es werden hier die statistischen und
nichtlinearen Eigenschaften (wie Chaos, Turbulenz, Defekte, Cluster) der Musterbil-
dung dieser chemischen Reaktion in Abh¨angigkeit von Druck und Druckmodulation
betrachtet.
Die Untersuchungen werden in einer UHV-Kammer von 10−10 mbar Basisdruck durchge-
f¨uhrt. Die Musterbildung der Reaktion auf der Pt(110)-Oberfl¨ache wird mit einem
Photoelektronenmikroskop (PEEM) beobachtet. Ein spezieller neu entwickelter Kom-
pressor erlaubt CO-Druckmodulationen großer Amplitude in der UHV-Kammer.
Die Reaktionen werden in subharmonischen 2:1, 3:1, und 4:1 Resonanzen des Verh¨altnis-
ses Anregungsfrequenz (forcing frequency) zur mittleren nat¨urlichen Fourier-Eigenfrequ-
enz (natural frequency) der Katalysatorfl¨ache betrieben. In der 2:1 Resonanz kann die
chemische Turbulenz durch die erzwungenen Schwingungen unterdr¨uckt werden. Mit
steigender Amplitude dieser Modulation kann man dann ¨uber eine Periodenverdopplung-
skaskade die Musterbildung der chemischen Reaktion ins Chaos treiben. Bei der
3:1 Resonanz werden zwei-, drei- und sechs-Phasencluster beobachtet, was mit einer
entsprechenden subharmonischen Synchronisation (entrainment) des System einhergeht.
Die 4:1 Resonanz wird im turbulenten und nicht-turbulenten Bereich untersucht. Im
turbulenten Bereich werden 4-Phasencluster gefunden, im nicht-turbulenten Bereich
2-Phasencluster.
i
ii
Die Experimente werden mit Simulationen des g¨angigen realistischen Krischer, Eiswirth,
und Ertl (KEE) Modells verglichen. Das Modell zeigt einen signifikanten Unter-
schied zwischen der Eigenfrequenz eines einzelnen (Punkt-)Oszillators und der mit-
tleren Eigenfrequenz des fl¨achig verteilten Systems im turbulenten Bereich. Die nat¨urli-
che Eigenfrequenz der Fl¨ache ist gr¨oßer als die des einzelnen-Punkt-Oszillators. Im
nicht-turbulenten Bereich tritt dieser Unterschied nicht auf, weil hier die diffusive
Kopplung ¨uber die Fl¨ache geringer ist. In den meisten F¨allen zeigen die Simulatio-
nen qualitativ dasselbe Verhalten wie die Experimente. In den Simulationen kon-
nten jedoch keine 2:1 Amplituden-Cluster reproduziert werden Umgekehrt konnten
im nicht turbulenten Bereich die theoretisch vorhergesagten 4-Phasen-Cluster nicht
experimentell beobachtet werden. Weiterhin wurden im Rahmen der vorliegenden Ar-
beit die statistischen Eigenschaften chemischer Turbulenz anhand der topologischen
Defekte untersucht. Bei steigendem CO-Druck konnte eine Erhhung der Defektanzahl
nachgewiesen werden.
Abstract
Pattern formation is a subfield of nonlinear science. In the last few decades pat-
tern forming processes in non-equilibrium systems have been extensively studied. A
well known example of pattern-forming non-equilibrium systems is CO oxidation on
Pt(110). The dynamics of the reaction are widely understood. Thus, CO oxidation
on Pt(110) is utilized as a well-suited model system for the analysis of spatial and
temporal pattern formation.
A large part of the present work is focused on the effects of periodic external forcing
on chemical turbulence in CO oxidation on Pt(110), investigated both experimentally
and theoretically.
Experiments are performed in an UHV chamber with a base pressure of 10−10 mbar.
Photoemission electron microscopy (PEEM) is used to obtain spatially resolved im-
ages of adsorbate patterns on the catalytic Pt(110) surface. A compressor driven
reactor which allows global gas-phase forcing for frequency modulations up to 4 Hz
was specifically designed.
Experiments are performed in different resonant forcing regimes such as 2:1, 3:1, and
4:1.
Under 2:1 forcing, experiments show that periodic forcing on chemical turbulence may
suppress spatial turbulence and could lead to a chaotic response of the system. The
path to chaos is given by a period doubling cascade, which could be experimentally
followed by the subsequent increase of the forcing amplitude. Two different types,
phase clusters and amplitude clusters, are found.
iii
iv
At 3:1 forcing, two, three, and six phase clusters are found at 2:1, 3:1, and 6:1 entrain-
ment respectively.
4:1 resonance forcing is performed in turbulent and nonturbulent regimes. In turbulent
regime, four phase clusters are observed while in nonturbulent regime, two phase
clusters are observed.
The experimental results are compared with numerical simulations by using the realis-
tic Krischer, Eiswirth, and Ertl (KEE) model. An analysis of the KEE model reveals
significant differences between the oscillation frequency of the single oscillator and the
mean frequency of the extended system, which appears to be higher in the turbulent
state.
Numerical simulations support the findings of experimental results with only small
deviations found. Under 2:1 forcing, only phase clusters are observed numerically,
while under 4:1 forcing in nonturbulent regime, the four phase patterns could not be
observed experimentally.
Thus, the results of this work demonstrate that by means of periodic forcing, tur-
bulence can be effectively controlled and manipulated. Furthermore, the statistical
properties of chemical turbulence are determined with increasing order of CO pressure
experimentally.
Contents
1 Introduction 1
2 Basic Concepts 7
2.1 NonlinearDynamics ............................ 8
2.1.1 Limit Sets, Stability, and Bifurcations . . . . . . . . . . . . . . . 8
2.1.2 Extended Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.3 Periodically Forced System . . . . . . . . . . . . . . . . . . . . . 15
2.2 CO Oxidation on Platinum Crystal . . . . . . . . . . . . . . . . . . . . 16
2.2.1 The Platinum(110) Surface . . . . . . . . . . . . . . . . . . . . . 17
2.2.2 Interaction of Adsorbates with Surface . . . . . . . . . . . . . . 18
2.3 Mechanism of the Reaction . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.1 Bistability ............................. 23
2.3.2 SpatialCoupling .......................... 25
2.4 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Pattern Formation in CO Oxidation on Pt(110) . . . . . . . . . . . . . 28
3 Methods 31
v
vi CONTENTS
3.1 ExperimentalSetup............................. 31
3.1.1 UHVChamber ........................... 31
3.1.2 Photoemission Electron Microscopy (PEEM) . . . . . . . . . . . 33
3.1.3 Implementation for Resonance Forcing . . . . . . . . . . . . . . 35
3.2 NumericalMethod ............................. 38
3.2.1 Implementation for Resonance Forcing . . . . . . . . . . . . . . 39
3.3 PatternAnalysis .............................. 40
4 Resonance Forcing: Experimental Results 43
4.1 Natural Frequency of the System . . . . . . . . . . . . . . . . . . . . . 43
4.2 2:1Forcing.................................. 45
4.2.1 PhaseClusters ........................... 46
4.2.2 Amplitude Clusters . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 3:1Forcing ................................. 54
4.3.1 2:1Entrainment .......................... 55
4.3.2 3:1Entrainment........................... 56
4.3.3 6:1Entrainment........................... 59
4.4 4:1Forcing ................................. 61
4.4.1 4:1 Forcing in Turbulent Regime . . . . . . . . . . . . . . . . . . 61
4.4.2 4:1 Forcing in a Nonturbulent Regime . . . . . . . . . . . . . . 64
4.5 Conclusion.................................. 66
5 Resonance Forcing: Theoretical Results 69
5.1 Natural Frequency of an Extended System . . . . . . . . . . . . . . . . 69
CONTENTS vii
5.2 2:1Forcing.................................. 71
5.3 3:1Forcing.................................. 76
5.3.1 3:1Entrainment........................... 76
5.3.2 6:1Entrainment........................... 79
5.4 4:1Forcing.................................. 81
5.4.1 TurbulentRegime.......................... 81
5.4.2 Nonturbulent Regime . . . . . . . . . . . . . . . . . . . . . . . . 89
6 Defect Mediated Turbulence 97
6.1 Method ...................................100
6.2 ExperimentalResults............................102
6.3 Conclusion .................................105
6.4 Appendix: PDF of Topological Defects . . . . . . . . . . . . . . . . . . 106
7 Summary and Outlook 109
viii CONTENTS
List of Figures
2.1 Phase space portrait of the stable limit cycle. . . . . . . . . . . . . . . 9
2.2 Fixed points in two-dimensional phase space. . . . . . . . . . . . . . . . 10
2.3 Phase portraits in the vicinity of a supercritical Hopf bifurcation. . . . 11
2.4 The amplitude |A|of the limit cycle is shown as a function of the control
parameters. ................................. 12
2.5 Schematic phase space diagrams. . . . . . . . . . . . . . . . . . . . . . 13
2.6 Face centered cubic (fcc) crystal structure and the (1×1) and (1×2)
structure of the Pt(110). . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.7 Schematic diagram of synergic bonding of CO to a metal. . . . . . . . . 19
2.8 Basic reaction mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.9 Reconstruction from 1 ×1 to 1 ×2..................... 23
2.10 Schematic diagram of the relation between the conditions for faceting
and the kinetics of catalytic CO oxidation on Pt(110). . . . . . . . . . . 24
2.11 The function f(u) for parameters u0= 0.35 and δu = 0.05 , and its
piecewiseoriginalform............................ 28
2.12 Snapshots of PEEM images displaying different patterns in CO oxida-
tiononPt(110) ............................... 29
ix
xLIST OF FIGURES
3.1 Schematic diagram of the ultrahigh vacuum (UHV) chamber with pump-
ing and gas supply system. . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Schematic diagram of the Photoemission electron microscope (PEEM). 33
3.3 Schematic drawing of the experimental setup with periodic forcing. . . 35
3.4 The CO pressure regulating system is represented as an electric circuit. 36
3.5 Design of the forcing compressor. . . . . . . . . . . . . . . . . . . . . . 37
3.6 Bode plots of the UHV chamber showing the compressor forcing fre-
quency vs. phase and amplitude of resulting oscillations inside the UHV
chamber.................................... 37
4.1 Time series of the averaged image intensity in an area of 10 ×10 pixels
and power spectrum of the data. . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Time series of the gray value of one single pixel and respective power
spectrum. .................................. 44
4.3 Fourier spectrogram showing the time evolution of the natural frequency
of the system without forcing. . . . . . . . . . . . . . . . . . . . . . . . 45
4.4 Mean Fourier spectra at different forcing amplitudes at 2:1 forcing. . . 47
4.5 Phase cluster at 2:1 entrainment and forcing. . . . . . . . . . . . . . . . 48
4.6 Phase and amplitude representation of phase cluster patterns at 2:1
entrainment.................................. 49
4.7 Phase and amplitude representation of the four phase cluster patterns
at4:1entrainment. ............................. 50
4.8 Mean Fourier spectra at different forcing amplitudes at 2:1 forcing. . . 51
4.9 Phase and amplitude of amplitude clusters at 2:1 entrainment. . . . . . 52
4.10 Phase and amplitude representation of the amplitude cluster patterns
at8:1entrainment. ............................. 53
LIST OF FIGURES xi
4.11 Two phase cluster and 2:1 entrainment in 3:1 forcing. . . . . . . . . . . 55
4.12 Phase and amplitude representation of the cluster patterns at 2:1 en-
trainment under 3:1 forcing. . . . . . . . . . . . . . . . . . . . . . . . . 56
4.13 Fourier spectra at different forcing amplitudes in 3:1 forcing. . . . . . . 57
4.14 Three phase cluster formation and entrainment at 3:1 resonant forcing. 58
4.15 Phase and amplitude representation of the cluster patterns at 3:1 en-
trainment and 3:1 forcing. . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.16 Spatially averaged Fourier spectrum at 6:1 entrainment. . . . . . . . . . 60
4.17 Phase and amplitude representation of the cluster patterns at 6:1 en-
trainment................................... 60
4.18 Fourier spectra at different forcing amplitudes at 4:1 resonant forcing. . 62
4.19 Four phase cluster formation and entrainment at 4:1 resonant forcing. . 63
4.20 Phase and amplitude representations of four cluster patterns at 4:1 res-
onantforcing ................................ 63
4.21 Fourier spectra at different forcing amplitudes at 4:1 forcing. . . . . . . 64
4.22 Two phase cluster formation and 2:1 entrainment at 4:1 resonant forcing. 65
4.23 Phase and amplitude representations of the cluster patterns at 2:1 en-
trainment but 4:1 forcing. . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1 Frequency of an extended system. . . . . . . . . . . . . . . . . . . . . . 70
5.2 Oscillation frequency of a single oscillator and mean oscillation fre-
quency of the extended system using the KEE model with different
p0within the turbulent regime. . . . . . . . . . . . . . . . . . . . . . . 70
5.3 Two phase cluster at 2:1 entrainment, KEE model. . . . . . . . . . . . 72
5.4 Stroboscopic space-time plot at 2:1 entrainment and forcing. . . . . . . 73
xii LIST OF FIGURES
5.5 Phase and amplitude representation of the cluster patterns at 2:1 en-
trainment................................... 74
5.6 Periodically forced KEE model. Low frequency part of Fourier spectra
at different forcing amplitudes. . . . . . . . . . . . . . . . . . . . . . . . 75
5.7 Simulated CO coverage for different entrainment states. . . . . . . . . . 75
5.8 Mean spatial cross-correlation of unforced turbulence and forced chaotic
oscillations. ................................. 76
5.9 Three phase cluster at 3:1 entrainment. . . . . . . . . . . . . . . . . . . 77
5.10 Space-time stroboscopic plot at 3:1 entrainment. . . . . . . . . . . . . . 77
5.11 Phase and amplitude representation of the cluster patterns at 3:1 en-
trainment................................... 78
5.12 Six-phase cluster and 6:1 entrainment at 3:1 forcing . . . . . . . . . . . 79
5.13 Phase and amplitude representation of the cluster patterns at 6:1 en-
trainment................................... 80
5.14 Transition from a three phase cluster to oscillation. . . . . . . . . . . . 81
5.15 Three phase cluster and 3:1 entrainment at 4:1 resonant forcing. . . . . 82
5.16 Phase and amplitude representations of three-phase cluster at 4:1 reso-
nantforcing.................................. 83
5.17 Four phase cluster and entrainment at 4:1 resonant forcing - I. . . . . . 84
5.18 Stroboscopic space-time plot at 4:1 forcing. . . . . . . . . . . . . . . . . 85
5.19 Phase and amplitude representation of four phase cluster at 4:1 resonant
forcing-I................................... 86
5.20 Four phase cluster and entrainment at 4:1 resonant forcing - II. . . . . 87
5.21 Phase and amplitude representation of Four-phase cluster at 4:1 reso-
nantforcing-II. .............................. 87
5.22 Transition from four phase cluster to oscillation 4:1 at resonant forcing. 88
LIST OF FIGURES xiii
5.23 Natural frequency of the system in nonturbulent regime. . . . . . . . . 89
5.24 Four phase cluster at 4:1 resonant forcing in nonturbulent regime. . . . 90
5.25 Space-time stroboscopic plot at 4:1 resonant forcing in nonturbulent
regime..................................... 91
5.26 Phase and amplitude representations of four phase cluster at 4:1 reso-
nantforcing.................................. 91
5.27 Two phase cluster at 4:1 resonant forcing in nonturbulent regime. . . . 92
5.28 Space-time stroboscopic plot of two phase cluster at 4:1 resonant forcing
innonturbulentregime............................ 93
5.29 Phase and amplitude representations of two phase cluster at 4:1 reso-
nant forcing in nonturbulent regime.. . . . . . . . . . . . . . . . . . . . 93
6.1 PEEM images of defect mediated turbulence with increasing order of
CO pressure and respective phase patterns. . . . . . . . . . . . . . . . . 100
6.2 Number of negatively charged defects N−as a function of time (sec.)
with increasing order of CO pressure . . . . . . . . . . . . . . . . . . . 101
6.3 Probability distribution function (PDF) of number of defects (N) com-
puted from the all time series N+,−(t) and the modified Poisson distri-
bution.....................................102
6.4 Creation rates with increasing order of CO pressure. . . . . . . . . . . . 103
6.5 Entering rates with increasing order of CO pressure. . . . . . . . . . . . 103
6.6 Decay rates with increasing order of CO pressure. . . . . . . . . . . . . 104
6.7 Leaving rates with increasing order of CO pressure. . . . . . . . . . . . 104
xiv LIST OF FIGURES
Chapter 1
Introduction
Spatiotemporal pattern formation in spatially extended systems out of equilibrium
maintained through the dissipation of energy which is continuously fed into the system
has been a rapidly growing field of research for several decades, due to its importance
in many fields such as biology, chemistry, and physics [1–8]. Pattern formation in these
systems can lead to coherent pattern formation, which is generated by the interplay
of the nonlinear components of the system.
The study of the dynamics of two-dimensional patterns often includes the observa-
tion of spatiotemporal disorder, sometimes called turbulence as an analogy to fluid
dynamics. Such pattern turbulence has been observed in a wide variety of spatially
extended experimental systems with different governing mechanisms [4]. The general
study of model equations for these systems has led to the delineation of the categories:
phase turbulence [8]; spatiotemporal intermittency [9]; and defect-mediated turbulence
[10, 11]. The transition from a simple regular pattern (for example stripes, hexagons,
or a spiral) to a time-dependent disorder often involves the spontaneous nucleation of
defects in the pattern, which can move through the system as individual entities, or
1
2CHAPTER 1. INTRODUCTION
coherent structures.
Theoretical work by Prigogine and coworkers in 1955 [12, 13] provided the basis for the
understanding of how order can emerge out of disorder in systems far from equilibrium.
They pointed out in that open systems, i.e., systems open to the exchange of matter
and/or energy with their surroundings, kept far from equilibrium could exhibit spon-
taneous self-organization by dissipating energy to the surroundings to compensate for
the entropy decrease in the system. The temporal or spatial structures that can arise
in this way are called dissipative structures. A closed system must reach equilibrium
and so can exhibit only transitory oscillations as it approaches equilibrium. Sustained
oscillations require an open system with a continuous flow of new reagents and removal
of waste products. The first chemical model was proposed by Prigogine and Lefever in
1968 and dubbed the “Brusselator” by Tyson in 1973. In 1977, Nicolis and Prigogine
summarized the work of the Brussels school in a book entitled Self-Organization in
Non-equilibrium Systems. For his work on non-equilibrium systems, Ilya Prigogine
was awarded the 1977 Nobel Prize in Chemistry.
Nonlinear phenomena are essential in surface chemical reactions. The mechanisms of
surface chemical reactions are often relatively simple. The most prominent examples
are the Belousov-Zhabotinsky (BZ) reaction and CO oxidation on Pt(110) [14–16].
A BZ reaction involves several reagents and various intermediate species; the central
reaction step is the oxidation of malonic acid by bromate, catalyzed by metal ions.
The first observations of kinetic oscillations in a continuously stirred BZ medium were
reported by Belousov in 1951 [17, 18]. Two decades later, Zhabotinsky and Winfree
observed traveling waves of chemical activity in an unstirred reactor [19, 20].
CO oxidation on Pt(110) has emerged as a fascinating interdisciplinary branch of the
natural sciences, since Langmuir’s pioneering studies [21], the oxidation of CO over
Pt is a classic example of a heterogeneous catalytic reaction. It is considered to be
generic due to its apparently simple mechanism, richness of spatiotemporal behavior,
and practical relevance [22–24]. Kinetic oscillations in this reaction were first found by
Hugo in 1970 on a supported catalyst [25]. This phenomenon was later observed for
other types of catalysts (polycrystalline wires and single crystals) both at ultrahigh
vacuum (UHV) and sub-atmospheric conditions [26].
3
In particular, Ertl and co-workers have demonstrated that on Pt(100) and (110), at
UHV, the oscillations result from the interplay between bistability and adsorbate-
induced surface reconstruction exposing patches with different O2sticking probabilities
[23, 26]. For his contribution on studies of the catalytic oxidation of CO on platinum,
Ertl was awarded the Nobel Prize in Chemistry 2007.
A Modern surface imaging technique, photoemission microscopy (PEEM, described in
Chapter 3) with high spatial resolution has provided real time pictures of propagating
fronts, spiral waves, target patterns, standing waves, and chemical turbulence. Before
these spatial features could be resolved, work function measurements had already
revealed that the reaction rate may become oscillatory and even chaotic.
Controlling deterministic chaos has become an active field in the study of nonlinear
dynamics over the past few decades. Since the pioneering work of Ott, Grebogi, and
Yorke [27], significant progress [28–31] has been achieved in controlling chaos in systems
with few degrees of freedom. These efforts have been naturally extended to control
spatiotemporal chaos [32] in spatially extended systems, due to it’s many potential
applications in many fields: plasma devices [33]; laser systems [34]; chemical reactions
[35]; and biological systems [36], where both spatial and temporal dependence need to
be considered.
Theoretically spatiotemporal chaos has been extensively studied in the complex Ginzburg-
Landau equation (CGLE) system [37], which describes universal dynamics features of
spatially extended systems near a supercritical Hopf bifurcation. It exhibits defect
mediated turbulence or spiral wave turbulence in a wide range of parameter regions.
In order to control spiral wave turbulence in spatially extended systems, global con-
trol methods are practical since local access to all system elements is often difficult
to achieve. Previous studies performed in the framework of abstract models have
theoretically investigated the effects of periodic forcing [38–41] and different schemes
of global feedback [42–44], suggesting that turbulence and pattern formation can be
successfully controlled in nonlinear systems.
As a recent theoretical contribution, Davidsen et. al. studied the dynamics of fronts
between phase-locked domains in resonantly forced catalytic CO oxidation on Pt(110)
4CHAPTER 1. INTRODUCTION
[45]. Their numerical investigations were carried out using the Krischer, Eiswirth,
and Ertl (KEE) model, a well-established realistic model of the CO oxidation reaction
[46]. Motivated by similar observations in the forced CGLE [47, 48], they focused
on explosion-type front instabilities that can be observed if the forcing amplitude is
decreased below a critical value.
In the 2:1 resonantly forced regime, this instability gave rise to a disordered state
of defect mediated turbulence. Depending on the forcing parameters, a cascade of
period doubling bifurcations was observed as the front instability was approached
with decreasing forcing amplitude.
In the case of 3:1 resonance forcing, a labyrinthine structure emerged. Interestingly,
resonance with a bistability between 3:1 and 2:1 locking to the external force can be
observed.
The present work is focused on spatiotemporal pattern formation in CO oxidation on
Pt(110). In particular, control of turbulence by resonance forcing is studied. Also, the
effect of resonance forcing is analyzed theoretically in extended systems by using the
KEE model. Furthermore, the effect of CO pressure on defect mediated turbulence is
presented and applied experimentally.
The outline of the thesis is as follows. In Chapter 2 the basic concept of nonlinear
dynamics and CO oxidation on Pt(110) is described in detail.
Chapter 3 deals with the experimental setup, numerical method, and the method used
for the pattern analysis. In the first section, the laboratory setup, Photoemission elec-
tron microscope (PEEM), implementation for periodic forcing, and the modification
is briefly discussed.
In the second section the numerical method and implementation for the resonance
forcing is explained. Finally in the last section the method used for the pattern
analysis is explained.
In Chapters 4 and 5, control of chemical turbulence by high frequency resonance forcing
is investigated in detail experimentally and theoretically, respectively.
In Chapter 6, the effect of CO pressure on defect mediated turbulence is considered
5
experimentally. Topological defects can be identified in the phase and amplitude
representation of the data and are characterized statistically.
Finally, Chapter 7 summaries the basic results presented in this thesis and gives pos-
sible perspectives for future research.
6CHAPTER 1. INTRODUCTION
Chapter 2
Basic Concepts
A reaction diffusion system is an extended nonlinear system. The field of nonlin-
ear systems is one that has been rapidly developing for the past 30 years. A nonlinear
system is defined as one which does not satisfy the superposition property. The sim-
plest form of nonlinear system is the static nonlinearity where the output depends only
on the current value of input but in a nonlinear manner, for example the mathematical
relationship.
y(t) = ax(t) + bx3(t) (2.1)
where the output is a linear plus cubed function of the input.
In the first section of this Chapter, we briefly discuss the basic concepts of nonlinear
dynamics and in the second section, the oxidation of carbon monoxide on Pt(110)
single crystal surface is introduced in detail.
7
8CHAPTER 2. BASIC CONCEPTS
2.1 Nonlinear Dynamics
The dynamic behavior of a single element with no diffusion coupling can be explained
in terms of a set of n coupled ordinary differential equations.
∂u
∂t =f(u,p) (2.2)
where the function f(u,p) represents the kinetics of the reaction and depends on a set
of time dependent concentrations of reacting species u= (u1, u2, u3, ....un) and system
parameters p= (p1, p2, p3, ...., pm). In a chemical context, urepresent concentrations
and pthe parameters (i.e., rate constants, temperature, reactant composition, flow
rate, etc.).
2.1.1 Limit Sets, Stability, and Bifurcations
The concepts of phase space and phase portrait are important tools for visualizing the
evolution of a system. Solving the dynamic system equation (2.2) for each variable
u1, u2, u3...., ungives a point in phase space. The trajectories in phase space are the
temporal evolution of a system with some initial conditions. The deterministic nature
of dynamics uniquely determines the function ffor a given initial condition. A phase
portrait is a geometric representation of the trajectories of a dynamical system in the
phase plane.
Subsets of phase space that are approached by the trajectories as t→ ±∞ are called
limit sets. The limit sets with t→+∞, are called attractors. A system may have
various attractors; they may correspond to stationary, periodic, quasi-periodic, or
chaotic dynamical states. Limit sets with t→ −∞ are called repellers.
Many features of system dynamics can be understood by the stability analysis of the
fixed points of the system.
The type of limit sets (fixed points) depends on the chemical kinetic term f(u;p) and
on the dimension n of phase space. While fixed points are the only possible attractor
2.1. NONLINEAR DYNAMICS 9
in one-dimensional phase spaces, another important type of attractor is possible in
two-dimensional phase spaces, namely the limit cycle. A limit cycle on a plane or a
two-dimensional manifold is a closed trajectory in phase space having the property
that at least one other trajectory spirals into it as time t→+∞. In cases where all of
the neighboring trajectories approach the limit cycle as time t→+∞, it is referred as
a stable limit cycle (see Fig. 2.1). Stable limit cycles imply self sustained oscillations.
Any small perturbation from the closed trajectory would cause the system to return
to the limit cycle, making the system stick to the limit cycle.
Figure 2.1: Phase space portrait of the stable limit cycle.
Stability
The stationary states or fixed points of system are denoted as usand satisfy the
condition ˙
u= 0. Consider any infinitesimal small perturbations δuon any orbit u0
leading to δu(t) = u(t)−us.
The difference vector δu(t) is inserted into equation (2.2) and fis expanded around
usin a Taylor series, where only the linear term is kept yielding
˙
δu=J(us)δuwhere Jij =∂fi
∂uj
10 CHAPTER 2. BASIC CONCEPTS
Figure 2.2: Fixed points in two-dimensional phase space. (a) Stable node, (b) saddle
point, (c) unstable node, (d) stable focus, and (e) unstable focus.
The eigenvalues λ1,λ2,...λnof the linear evolution matrix Jevaluated at a fixed point
usgovern its stability. The fixed point is stable if the real parts of all eigenvalues λi
are negative; it is unstable if the real part of at least one eigenvalue is positive. In
two-dimensional phase space, the eigenvalues λ1and λ2may either be real or com-
plex conjugated. Different types of fixed points for a two-dimensional vector field are
summarized in Fig. 2.2.
Bifurcations
The stability of a fixed point may changed when a system parameter is changed and
at least one of the eigenvalues of a fixed point changes its sign. This change is called
bifurcation.
If a control parameter µhaving a critical value µcis varied around the critical value,
this leads to non-stationary behavior. The simplest example of bifurcation leading
to non-stationary dynamical behavior is the supercritical Hopf bifurcation, see Fig.
2.1. NONLINEAR DYNAMICS 11
Figure 2.3: Phase portraits in the vicinity of a supercritical Hopf bifurcation.
2.3. In subcritical Hopf bifurcation the existing solution becomes stable and newly
emerging solutions are unstable. As an appropriate parameter µis varied beyond its
critical value µc, a stable focus becomes unstable and simultaneously a stable limit
cycle is born. Sufficiently close to the bifurcation point, the oscillations are harmonic
and amplitude follows a square root dependence A∼√µ−µc. Far from the Hopf
bifurcation, the amplitude may become large and strongly anharmonic, depending on
the properties of the system. Supercritical Hopf bifurcation does not depend on the
direction of the parameter change.
Further examples of local bifurcation include the subcritical variant of Hopf bifurca-
tion. Supercritical and subcritical Hopf bifurcations are displayed in Fig. 2.4. In
the subcritical case, the oscillations are born suddenly with finite amplitude at one
critical parameter value. Fig. 2.4(b) illustrates the situation when an unstable limit
cycle born in a subcritical Hopf bifurcation is stabilized in a saddle-node bifurcation
(a stable node and a saddle point appear at the bifurcation point) of limit cycles. In
subcritical Hopf bifurcation, when the parameter is scanned in the opposite direction,
12 CHAPTER 2. BASIC CONCEPTS
Figure 2.4: The amplitude |A|of the limit cycle is shown as a function of the control
parameter µin supercritical Hopf bifurcation (a), and subcritical Hopf bifurcation
with stabilized limit cycle (b). Solid (dashed) lines denote stable (unstable) states.
the oscillations disappear at another critical parameter value and hysteresis occurs.
Detailed information about limit sets and their stability can be found in [49].
2.1.2 Extended Dynamics
Pattern formation is a ubiquitous phenomenon in the dynamics of extended nonlinear
systems. Patterns in extended systems arise as a result of the interplay of many
factors including nonlinearities, external forcing and/or excitability of the medium,
spatial interactions, and internal dissipation.
In an extended or distributed system, the elements can be considered as being com-
posed of many individual components. Extended systems are commonly classified
according to the local dynamics of their individual elements [50]. Fig. 2.5 shows the
schematic diagram of monostable, bistable, excitable, and oscillatory systems.
2.1. NONLINEAR DYNAMICS 13
Figure 2.5: Schematic phase space diagrams. (a) monostable, (b) bistable, (c) ex-
citable, and (d) oscillatory dynamics.
Monostable System
In a monostable system (Fig. 2.5(a)) the dynamics is determined by the stable fixed
point. Under perturbation the system experienced damping and always returned to
the same stable steady state.
Bistable systems
A bistable system is characterized by the presence of two stable steady states separated
by a saddle point. In other words bistability refers to the situation in which two stable
steady states coexist. The nullclines (the line in phase space obeying ˙
u1= 0, ˙
u2= 0)
14 CHAPTER 2. BASIC CONCEPTS
of a bistable system possesses three intersection points, which correspond to two stable
fixed points separated by a saddle point (Fig. 2.5(b)).
For small perturbations the system remains in one of the stable states, while under
sufficiently strong perturbations transitions between the two states may occur. Bista-
bility often arises as a result of symmetry breaking the instabilities of uniform states.
Bistability comes from the fact that its free energy has three critical points with two
minima and one maximum. By default, the system state will be in either of the minima
states, because that corresponds to the state of lowest energy. The maximum can be
visualized as a barrier. A transition from one state of minimal free energy requires
some form of activation energy to penetrate the barrier. After the barrier has been
reached, the system will relax into the next state of lowest energy again. The time it
takes is usually attributed as the relaxation time.
Excitable Systems
Excitable systems are characterized by only one stable steady state corresponding to
a single intersection of nullclines (Fig. 2.5(c)). For small perturbations away from the
equilibrium, the return is monotonic, however, for perturbations beyond a threshold
value, the return is not monotonic, but undergoes a large excursion before settling
down.
An excitable system remains in a stable configuration in absence of (or in presence
of small) perturbations. If the perturbations surpass a threshold, the system per-
forms an excursion in phase space (in most cases independent on the strength of the
perturbation), returning back to the original state.
For the system in continuous or discrete media, it usually gives rise to some global
behaviors which are believed to be related to the realization of certain functions of the
system. A spiral wave is one of a typical global phenomenon of excitable medium, and
has been observed in various systems [51–57].
2.1. NONLINEAR DYNAMICS 15
Oscillatory Systems
Oscillatory systems are characterized by a stable limit cycle and an unstable fixed
point (Fig. 2.5(d)).
An oscillatory system, however, can be subjected to an external force which may alter
the nature of oscillation. For example, a system capable of oscillation can be set to
oscillate at a frequency other than a natural frequency. (The natural frequency is the
frequency at which a particular object or system vibrates when pushed by a single
force or impulse, and is not influenced by other external forces or by damping.)
Traveling pulses, target patterns [58], standing waves [59], and asymmetric target
patterns [60, 61] have been observed in the oscillatory systems and in some cases
reproduced by numerical simulations.
2.1.3 Periodically Forced System
A nonlinear dynamic system has four states: the fixed point, the periodic motion, the
quasi-periodic motion, and the chaotic motion. When the system is in the critical
state, a small perturbation of the system parameters may lead to the qualitative
change of the system state. Periodic forcing of nonlinear oscillators produces a rich
variety of dynamical responses, including frequency locking, quasi-periodic oscillations,
period doubling, and chaos [62]. The well known examples are physical, biological, and
chemical systems [63–68].
Single Oscillator
Most theoretical studies of periodically forced oscillatory systems have focused on
frequency locking phenomena and the onset of chaos in single oscillator models [45, 62].
Frequency locking refers to the property of a forced system to oscillate at a frequency
ωwhich is a rational fraction of the forcing frequency ωfin some range. These ranges
of resonant behavior get wider as the forcing strength is increased, and are commonly
refer to as Arnold tongues. The fractional frequencies a forced system can realize
follow the Farey rule: between the tongues ωf:ω= n : m, where n : m resonance is
denoted as sub-harmonic if n >m and super-harmonic if n <m.
16 CHAPTER 2. BASIC CONCEPTS
Extended System
The response of a spatially extended system to a periodic stimulus is more complicated
than that of a single oscillator since it is possible for individual elements to oscillate
with different amplitude and phase with respect to each other. Frequency locking in
spatially extended systems can be enhanced or suppressed by diffusive coupling. On
the other hand, close to the boundaries of Arnold’s tongues stable frequency locked
patterns may exist at forcing parameters where single oscillatory elements are not
locked. For example, the dominating contribution of the diffusion terms can prevent
frequency locking at small forcing amplitudes [41].
In spatially extended oscillatory systems, periodic forcing can change the nature of the
phase patterns from traveling waves or spiral waves in the unforced system to standing
wave labyrinths [69] or multiphase spirals [70].
Theoretical work on resonantly forced oscillators has focused on the complex Ginzburg-
Landau equation (CGLE) (a generic equation for oscillatory systems close to the
Hopf bifurcation), FitzHugh-Nagumo, Brusselator, and the Krisher, Eiswirth, and
Ertl (KEE) model [41, 45, 69, 71].
Frequency locking of extended oscillatory systems has been observed experimentally
in the light-sensitive Belousov-Zhabotinsky (BZ) reaction and in CO oxidation on
Pt(110). In a BZ reaction the uniform oscillations and spiral wave could be entrained,
however, the always stable and spontaneous development of turbulence is not known
[72–78], while in catalytic CO oxidation on Pt(110) periodic forcing was used to control
chemical turbulence [79, 80]. Catalytic CO oxidations on Pt(110) single crystal surface
show both stable oscillations as well as spiral wave turbulence [8, 14].
2.2 CO Oxidation on Platinum Crystal
CO oxidation on a platinum single crystal is one of the most studied heterogeneous
catalytic reactions due to its simplicity and it can be treated as model system for
the experimental and theoretical understanding of heterogeneous catalysis [16]. The
catalytic oxidation of CO on a platinum single crystal surface follows the Langmuir-
2.2. CO OXIDATION ON PLATINUM CRYSTAL 17
Hinshelwood mechanism where the two reacting species are chemisorbed on the cat-
alyst surface before the reaction takes place [21]. Under UHV condition with certain
sets of control parameters (partial pressure of CO, O2, and temperature (T)), different
spatiotemporal patterns can be observed [81, 82].
Different surface imaging techniques like photoemission electron microscopy (PEEM)
[14, 83–85], Reflection Anisotropy Microscopy (RAM) [22, 86], and Ellipsomicroscopy
for Surface Imaging (EMSI) [87] can be used to study the pattern formation on Pt
surface. In particular, PEEM is used as a powerful tool for real-time pattern formation
on catalytic surface due to its non-destructive imaging nature.
2.2.1 The Platinum(110) Surface
A clean Pt(110) single crystal surface is a face centered cubic (fcc) structure at room
temperature, having lattice constant a = 0.392 nm. The (110) surface of fcc metals
is the most open of the low Miller index surfaces therefore it has the lowest surface
atomic density and the highest surface energy.
Figure 2.6: Face centered cubic (fcc) crystal structure of Pt(110) (left) and the (1×1)
and (1×2) structure of the Pt(110) (right).
18 CHAPTER 2. BASIC CONCEPTS
In the non-reconstructed form, also known as (1 ×1), the platinum atoms on (110)
surface are arranged according to their bulk position, however, the clean Pt(110)
surface undergoes a reconstruction to lower its surface energy that leads to a missing
row structure characterized by (1 ×2) [88–92]. The (1 ×2) structure is composed of
alternating rows and troughs of Pt atoms in the [110] direction as seen in Fig. 2.6.
2.2.2 Interaction of Adsorbates with Surface
When atoms or molecules adsorb on ordered crystal surfaces, they usually form ordered
layer structures over a wide range of temperatures and surface coverage. The driving
force for ordering originates in mutual interatomic interactions. Here, an important
distinction must be made between adsorbate-adsorbate and adsorbate-substrate inter-
actions.
The adsorbate-adsorbate forces are usually small compared to the adsorbate-substrate
binding forces, so that the adsorbate locations are determined by an interplay between
their entropy-related accessibility and the optimum adsorbate substrate bonding. The
adsorbate-adsorbate interactions dominate the long-range ordering of the over-layer.
These interactions can be studied experimentally by examining, e.g. the changes in
the over-layer structure as a function of coverage, or by theoretical calculations. The
surface coverage is thus an important parameter in ordering.
The effects of a strong adsorbate-substrate bond on the surface structure of the sub-
strate can be large. The presence of an adsorbed over-layer often removes the recon-
struction of clean surfaces and the substrate surface atoms usually return to their bulk-
like equilibrium position. The thermodynamic driving force for adsorbate-induced
restructuring is the difference in strength of the adsorbate-substrate bonds for the
reconstructed and unreconstructed surfaces. More specifically, the loss in adsorption
energy is larger than the gain in energy associated with the reconstruction of the clean
surface. If massive diffusion-controlled atom transport along the surface is not needed,
adsorbate-induced restructuring can occur on the time scale of catalytic reactions (sec-
onds).
2.2. CO OXIDATION ON PLATINUM CRYSTAL 19
Carbon Monoxide Adsorption on Pt(110)
The adsorption of CO on Pt surfaces has attracted much attention because of the
many potential applications, such as in car exhaust catalysts where it promotes the
oxidation of CO to CO2. CO generally prefers binding at low coordination sites, such
as on-top of a Pt atom or bridging two Pt atoms.
Figure 2.7: Schematic diagram of synergic bonding of CO to a metal. In CO the
molecular orbital are 1σ22σ23σ24σ21π45σ22π∗. The 4σorbital is localized on the
oxygen atom while the 5σorbital is localized on the carbon atom, and both of these
orbitals are non-bonding. The empty 2π∗anti-bonding orbital is also available to take
parting the interaction with the surface. This combination of σand πorbital of CO
in the interaction with the surface is called synergic bonding. In the case of molecular
chemisorption of CO, a covalent bond is formed by donation of electrons from the 5σ
orbital to a vacant metal d orbital (a). At the same time, the full d orbital are able to
donate electron density into the vacant 2π∗orbital (b). On adsorption the situation
is analogous (c).
20 CHAPTER 2. BASIC CONCEPTS
The adsorption of CO on Pt(110) takes place in molecular form and induces a struc-
tural change in the surface. The CO bonding to metal surfaces is described in the
terms of the Blyholder model, which invokes a donor-acceptor mechanism [93, 94].
The 5σand the 2πfrontier molecular orbital (MO) of the CO molecule are substan-
tially modified by the presence of the metal surface.
A filled 5σ“lone pair” orbital interacts with the empty dσmetal orbital, leading to
a partial transfer of electron density to the metal. At the same time the filled metal
dπorbital overlap with the 2π∗antibonding molecular orbital of the CO (Fig. 2.7)
[22, 87, 95, 96]. Moreover, since the 5σand 2πMO of CO are localized mainly at the
C atom, the bonding occurs with the carbon atom facing the surface.
In the clean platinum surface (1×2) structure, the CO sticking probability on Pt(110),
sco is close to unity and remains almost constant for low coverage of CO [88, 97, 98]. For
higher coverage (u= 0.35 monolayers), the sticking coefficient decreases. According
to Gasser and Smith who described it in [99], sco =s0
co(1 −uq) (where q is a mobility
parameter between 3 and 4). The CO saturation coverage on Pt(110) is equal to unity
[97, 100, 101].
The adsorption of CO on Pt(110) induces a structural change in the surface. The 1×2
missing row reconstruction is lifted to 1×1 bulk phase.
The Adsorption of Oxygen on Pt(110)
The adsorption of oxygen take place dissociatively at temperature 240K. Dissociative
adsorption of oxygen was found to proceed via second order kinetics in the free sites
[102, 103]. The activation energies for oxygen diffusion are much higher than CO
diffusion, and depends on the crystallographic orientation. Diffusion of oxygen is
practically limited to the [110] direction and no transport occurs perpendicular to the
ridges of the missing row structure.
Oxygen desorbs only in molecular form. At about 800K recombinative molecular
desorption takes place. The dissociation of oxygen on metal surfaces has been modeled
by ab-initio fully quantum-dynamical simulations [102, 104].
2.3. MECHANISM OF THE REACTION 21
The initial sticking probability for oxygen on the (1×2) surface of the Pt(110) facet
is about 0.4 at room temperature [103, 105, 106]. With growing oxygen coverage, the
sticking coefficient decreases to 0.03 for a coverage greater than 0.35 ML [107].
Subsurface of Oxygen
Subsurface oxygen is defined as an atomic oxygen species located directly underneath
the uppermost metal crystal layer. The formation of subsurface oxygen can take place
only on the non-reconstructed (1×1) surface and subsurface oxygen tends to stabilized
(1 ×1) phase which effects the reaction dynamics.
Subsurface of oxygen is responsible for the drastic decrease of the local work function
of the Pt(110) surface.
2.3 Mechanism of the Reaction
In a catalytic process, the reaction occurs in a sequence of elementary steps. This se-
quence includes adsorption, surface diffusion, chemical rearrangement (bond breaking,
bond-forming, molecular rearrangement) of the adsorbed intermediates, and products
desorption.
The catalytic oxidation of CO on platinum follows the Langmuir-Hinshelwood mecha-
nism (i.e., CO and oxygen have to adsorb before the reaction to CO2can take place),
both reactants adsorb on the catalyst surface in order to yield the product [16].
The reaction steps are
2CO + 2⊕2COad
O2+ 2⊕ → 2Oad
2COad + 2Oad →2CO2↑+ 4⊗
22 CHAPTER 2. BASIC CONCEPTS
Figure 2.8: Basic reaction mechanism: adsorption of CO and O2molecules, CO
diffusion, and reaction.
where index ad denotes adsorbed molecules or atoms and ⊕stands for a free adsorp-
tion site. Due to a high energy barrier for spontaneous reaction of CO and O2in the
gas phase they have to adsorb before the reaction. The adsorption of oxygen is dis-
sociative. Adsorbed CO molecules are bound to the surface considerably less strongly
than oxygen atoms and hence may desorb as well as diffuse on the surface and such
processes are negligible for Oad under typical reaction conditions.
At temperatures above 300K, produced carbon dioxide almost immediately desorbs
into the gas phase, again leaving free space for adsorption. The reaction mechanism
is illustrated in Fig. 2.8.
2.3. MECHANISM OF THE REACTION 23
2.3.1 Bistability
In a certain range of parameters, the system exhibits bistability between a mainly
oxygen covered, reactive state, and a non-reactive CO covered state. This bistability
can be traced back to an asymmetric inhibition of adsorption. Adsorbed oxygen forms
an open structure where CO molecules can still adsorb and react, whereas a fully CO
covered surface completely inhibits the adsorption of oxygen, and hence poisons the
reaction.
Figure 2.9: Reconstruction from 1 ×1 to 1 ×2.
After the sputtering (Ar ion) and annealing a reconstructed (1 ×2) phase is observed,
as illustrated in Fig. 2.9. When CO is admitted in the reconstructed (1 ×2) phase
this reconstruction will be lifted and a phase transition to a non-reconstructed (1 ×1)
phase occurs. CO starts to lift the reconstruction of the surface at a CO coverage
of 0.2 ML and completes the reconstruction at a coverage of 0.5 ML. The activation
energy for this phase transition in 29 KJ/mol [108].
As already discussed in section 2.2, the sticking coefficient of the oxygen is higher in
24 CHAPTER 2. BASIC CONCEPTS
the non-reconstructed (1 ×1) phase compared to the reconstructed one by a factor of
1.5. As a consequence, the surface will transform in to an oxygen covered one. When
the surface is oxygen covered, with no CO species to lift the reconstruction, it will
reconstruct again. A reconstructed surface has low sticking coefficient for oxygen thus
enabling CO to take over. Now the surface become CO covered and the oscillation
cycle starts again.
Faceting
Faceting is a process which causes an initially flat, single-crystal surface to separate
into two (or three) other surface orientations [109, 110]. This process has been studied
intensively on Pt(110) [99, 109, 111].
Figure 2.10: Schematic diagram of the relation between the conditions for faceting
and the kinetics of catalytic CO oxidation on Pt(110).
Faceting takes place at temperatures below 530K, above this temperature a thermal
2.4. MATHEMATICAL MODELING 25
reordering process keeps the (110) surface flat. A CO covered 1×1 surface constitutes
the starting point of an oscillation cycle. On this surface the reaction rate is low and
the facets grow slowly. These facets have a high sticking coefficient for O2, and at a
certain point the surface becomes oxygen covered [23, 112].
The faceting of Pt(110) is associated with an increase in catalytic activity, which in a
reaction rate vs. pCO diagram, shows up as a shift of the rate maximum toward higher
pCO [23], as displayed in Fig. 2.10.
2.3.2 Spatial Coupling
Spatial coupling in CO oxidation on Pt(110) surface is provided by two different mech-
anisms: local coupling and global coupling. Surface diffusion of adsorbed CO molecules
gives rise to local coupling between neighbored sites.
Global coupling acts in the gas phase as a consequence of mass balance in the reaction.
Since the mean free path in the gas phase is typically large in comparison to the size
of the chamber, local partial pressure variations that result from the consumption of
the educts by the reaction quickly extend to affect the whole system. Therefore, the
gas-phase coupling is global.
The interplay between diffusion and gas-phase coupling can lead to phenomena such
as synchronous oscillations, standing waves, cellular structures, and spiral wave tur-
bulence [14, 83].
2.4 Mathematical Modeling
The mechanism of low-pressure CO oxidation on Pt(110) is described through a three
step reaction-diffusion type model knows as the Krischer-Eiswirth-Ertl (KEE) model
[46].
26 CHAPTER 2. BASIC CONCEPTS
KEE Model
The KEE model is based on decomposition of the entire reaction into elementary steps.
It consists of three coupled ordinary differential equations for the local dynamics,
taking into account the most significant physical processes.
∂tu=k1sCOpCO −k2u−k3uv (2.3)
∂tv=k4sOpO2−k3uv (2.4)
∂tw=k5(f(u)−w) (2.5)
where u, is the local CO coverage, i.e., the fraction of CO adsorption sites on the metal
surface that are occupied by adsorbed CO molecules. The second variable, v, is the
local oxygen coverage. The third variable, w, specifies the local fraction of the surface
area occupied by the non-reconstructed (1 ×1) structural phase.
The first term in equation (2.3) describes the process of CO adsorption. Here, k1is
the adsorption rate constant and sco is the sticking coefficient for CO molecules. As
mentioned above, the sticking coefficients are coverage dependent. For sCO a precursor
effect has to be considered and is modeled following Gasser and Smith [113].
Thus the sticking coefficient of CO is given by the expression,
sCO =s0
CO(1 −u3)
where s0
CO is the initial sticking probability of CO. The term (1−u3) for CO adsorption
describes a precursor effect.
pCO is the partial pressure of CO in the gas phase. The second and the third terms
in this equation describe desorption of CO and its reaction with adsorbed oxygen
molecules, where k2and k3are the desorption rate constant respectively.
The second equation (2.4) of the KEE model describes the kinetics of adsorbed oxy-
gen. The first term is the adsorption rate depending on the partial pressure oxygen
2.4. MATHEMATICAL MODELING 27
molecules, where k4,pO2, and sOare the impingement rate constants, partial pressure,
and sticking coefficient of oxygen molecules respectively.
sOgiven by the expression
sO= [s0
O,1×1w+s0
O,1×2(1 −w)](1 −u−v)2
where s0
O,1×1and s0
O,1×2denote the initial sticking probabilities of oxygen on the (1×1)
and (1 ×2) surface.
The last equation (2.5) of the KEE model is a phenomenological mean-field descrip-
tion of the phase transition kinetics. The surface free of CO molecules is in the
reconstructed (1×2) phase, while the surface completely covered by CO is in the non-
reconstructed (1 ×1) phase. At intermediate CO coverage, a mosaic of microscopic
domains of both structural phases occupy the surface. The characteristic sizes of such
domain are, however, on the nanometer scale and cannot be resolved in the above
mean-field micrometer scale description. Here, it is simply assumed that, at a fixed
CO coverage u, the local fraction wof the surface area in the non-reconstructed phase
tends to approach
f(u) = 1
1+exp[(u−u0)/δu]
The values of parameters u0and δsets the threshold above which the surface structure
is affected by the CO coverage and the steepness of the threshold [46, 114].
The partial pressure of the reactants (pCO and pO2) and temperature T can be changed,
which determines the rate constants k2,k3, and k5according to the Arrhenius activa-
tion law,
ki=νiexp[Ei/kT]
Depending upon the control parameters the model exhibits monostable, bistable, ex-
citable, and oscillatory behavior.
28 CHAPTER 2. BASIC CONCEPTS
Figure 2.11: The function f (u) for parameters u0= 0.35 and δu = 0.05 (solid line),
and its piecewise original form (dashed line).
2.5 Pattern Formation in CO Oxidation on Pt(110)
In the 1970s, the group of E. Wicke discovered oscillations of the reaction rate in
catalytic oxidation of carbon monoxide [115]. In 1982 Ertl et. al. observed the
oscillatory kinetics on single crystal surfaces, in CO oxidation on Pt(100) and later
in 1986, oscillations were also reported on Pt(110) where they showed rich behavior,
ranging from periodic and mixed-mode oscillations to deterministic chaos [26, 116].
The development of spatially resolving techniques such as PEEM [84, 117] has shifted
the focus from purely temporal phenomena to spatiotemporal pattern formation. Mea-
surements employing PEEM allow the display of the local work function, which is
changed by the adsorbates, across a surface area of about 500 µm in diameter. The
evolution of patterns on the catalytic surface can be followed in real time with a spa-
tial resolution of about 0.2 µm. Among surface chemical reactions, by far the richest
variety of spatiotemporal patterns has been found in CO oxidation on Pt(110). The
observed phenomena include rotating spiral waves, target patterns, standing waves,
2.5. PATTERN FORMATION IN CO OXIDATION ON PT(110) 29
Figure 2.12: Snapshots of PEEM images displaying different patterns in CO oxidation
on Pt(110). Dark areas in the images correspond to predominantly oxygen covered
regions, and bright areas indicate mainly CO covered regions. (a) Rotating spiral
waves, (b) Target patterns , (c) standing waves [122], and (d) chaos.
cellular structures, chemical turbulence, and solitary waves [14, 118–120]. Examples
of such patterns are displayed in Fig. 2.12.
30 CHAPTER 2. BASIC CONCEPTS
Chapter 3
Methods
This Chapter describes the experimental setup used, numerical method used, and the
method used to analyze the patterns obtained from the experiments and numerical
simulations.
3.1 Experimental Setup
This section explains the UHV, PEEM, implementation for periodic forcing, and mod-
ification for high periodic forcing.
3.1.1 UHV Chamber
The experiments presented in Chapters 4 and 6 are conducted in a stainless steel UHV
chamber with a volume of 60L and the pressure about 10−10 mbar in the chamber.
Attached to the UHV chamber are an automated dosing system that keeps the partial
31
32 CHAPTER 3. METHODS
pressures of gases constant within a maximal variation of 0.1, gas supplies for CO
and O2, and two pressure gauges that allow for controlled dosing of the reactants into
the UHV chamber [121]. With the combination of feedback-regulated gas dosing and
permanent pumping of the chamber the CO oxidation reaction can be observed under
constant flow conditions.
Figure 3.1: Schematic diagram of the ultrahigh vacuum (UHV) chamber with pumping
and gas supply system.
The chamber is equipped with a PEEM, a quadrupole mass spectrometer, low energy
electron diffraction (LEED), an Ar-ion sputtering gun, and resistive sample heating.
For imaging the spatiotemporal adsorbate patterns on the catalytic surface, the PEEM
was used, which operated under differential pumping. Platinum crystal is mounted
on a sample holder which allows controlled movement of the sample in x, y, and z-
directions as well as radial and azimuthal rotation by electric step. Preparation of
the clean platinum single crystal was performed by oxidizing at 750K (10−6mbar),
sputtering, and annealing cycles at up to 1100 K to image pattern formation on the
3.1. EXPERIMENTAL SETUP 33
Pt(110) surface, which produces images of the intensity distribution of electrons photo-
emitted from an area with a typical diameter of about 500 µm.
3.1.2 Photoemission Electron Microscopy (PEEM)
The PEEM is an excellent imaging technique for the real time observation of pattern
formation during catalytic reactions [14, 83–85]. The PEEM provides spatially resolved
information of reacting species on the surface, since the local work function at a given
point depends on the adsorbate coverage on the surface, therefore, images with different
brightnesses are obtained due to different values of local work function (∇φ).
Figure 3.2: Schematic diagram of the photoemission electron microscope (PEEM).
The clean Pt surface has the lowest work function and displays therefore the bright-
est image. Compared to the clean Pt(110) surface, a monolayer of oxygen coverage
increases the work function (∇φ) by 0.8eV, thereby strongly decreasing the brightness
34 CHAPTER 3. METHODS
of PEEM images. Full CO coverage also increases the work function but produces a
smaller effect (∇φ= 0.3eV) [22, 122].
Fig. 3.2 shows the systematic drawing of the PEEM, creating a complete picture of
the photoelectron distribution emitted from the imaged surface region. To induce
the emission of photoelectrons, the platinum sample is irradiated with ultraviolet light
from a 200 W deuterium discharge lamp. The ultraviolet light has a continuous spectral
intensity characteristic [123]. The angle of light incidence is about 750from the surface
normal. To capture as many photoelectrons as possible for the imaging, the distance
between the sample and the objective is small (d = 4 mm.). Within this distance, the
electrons are accelerated by a potential difference of about 20kV.
Besides the objective lenses, on the left side in Fig. 3.2, two more lenses are used to
create an image of the sample, an intermediate three electrode lens next to the objective
lens magnifies the electron image by a factor of 102to 103. The lens combination also
decelerates the fast electrons to energies for which the channel plate has its highest
sensitivity (about 1keV). The channel plate typically amplifies the electron distribution
by a factor of 103.
Finally, a phosphor screen converts the electron image into a photon image which then
recorded by a CCD camera.
An additional problem for investigating surface reactions with PEEM is the restric-
tion of the pressures below 10−6mbar. This is in part circumvented by the differential
pumping of the PEEM which allows its operation up to pressures of 10−3mbar even in
the presence of the oxygen around the sample. To maintain the three orders of mag-
nitude pressure difference, an aperture of 300 µm in diameter has to be incorporated
at the focus of the cathode lenses indicated in Fig. 3.2.
In our experiments, the PEEM instrument has been used to monitor a surface area of
500 µm in diameter. The spatial resolution of the images was about 1 µm. The PEEM
is combined with a CCD camera, and provides information on temporal evolution of
reacting species on the surface. A frame rate of 25 images per second gives a sufficiently
good temporal resolution of the PEEM recordings. The video pictures are stored on
a recorder or are used as input for the LABVIEW card controlling the feedback loop.
3.1. EXPERIMENTAL SETUP 35
3.1.3 Implementation for Resonance Forcing
Periodic forcing has been implemented experimentally in gas-phase. Partial pressure
variations affect the reaction conditions on the catalytic surface in a uniform way.
The automated gas inlet system allowed the controlled modulation of the CO partial
pressure in the chamber by changing the dosing rate of CO molecules. Resonance
forcing has been implemented by using a frequency generator to control the dosing
rate of CO molecules. The schematic diagram of periodic forcing is shown in Fig. 3.3.
Figure 3.3: Schematic drawing of the experimental setup with periodic forcing.
For resonant forcing, the carbon monoxide pressure pCO in the chamber is varied by:
pCO(t) = po[1 + Asin(2πωft)],(3.1)
where pois the base pressure of CO, A is the forcing amplitude, and ωfis the forcing
frequency. In this way, the CO partial pressure in the reaction chamber could be
periodically modulated with a nearly harmonic signal of amplitude A and frequency
36 CHAPTER 3. METHODS
ωf, while its temporal average powas kept constant. For forcing frequencies ωf≤
2Hz, an electronic valve was used to control the carbon monoxide flux. The valve is
connected to the computer and is regulated by an oscillating voltage signal generated
by the LABVIEW program. This setup provides the ability to scan over a predefined
range of forcing amplitudes and frequencies to measure the system’s response in a wide
range of the parameter space [124].
Modification: The above setup fails for higher forcing frequencies, as the forcing
amplitude is strongly damped.
Figure 3.4: The CO pressure regulating system is represented as an electric circuit.
Symbols, abbreviations, and indices: σ= conductivity, C = capacity, X1= pressure
in the UHV chamber, X2= regulated CO pressure in the gas dosing system, X3=
pressure in the pre-pressure system, DV = dosing valve, EV = exhaust valve, GDS =
gas dosing system, i = inner, IM = ionization manometer, L = leakage, M = manome-
ter, MOT = compressor, P0= pressure after manometer, PPP = pre-pressure pump,
PPS = prepressure system, PI = Pirani pressure sensor, PT = pressure transducer,
RV = regulating valve, TP = turbo pump, and UHV = ultrahigh vacuum.
3.1. EXPERIMENTAL SETUP 37
Figure 3.5: Design of the forcing compressor.
Figure 3.6: Bode plots of the UHV chamber showing the compressor (a) forcing fre-
quency vs. phase, and (b) amplitude of resulting oscillations inside the UHV chamber.
The experimental results are shown as (+), while the dashed lines indicate approxi-
mation fits with a first order low-pass filter function.
38 CHAPTER 3. METHODS
Based on the analogy between electrical and pneumatic circuits [125], the UHV cham-
ber and the CO pressure regulating system was analyzed in detail.
Analysis of the UHV chamber and the whole CO pressure regulating system reveals
its intrinsic low pass filtering characteristics, as could be deduced from the equivalent
circuit diagram, given in Fig. 3.4. It neglects chemical and thermal driven flows in
the system and it ignores the finite velocity of the gas.
In order to enlarge the oscillation amplitude at higher frequencies, a small self-built
compressor was implemented in the gas dosing system (Fig. 3.5). This compressor
basically consists of a piston which periodically draws CO from the gas line, compresses
and pumps it back to the pre-pressure line, resulting in an harmonic modification of
the pCO in the chamber. The forcing frequency is adjusted by the velocity of the
stepper motor driving the piston, while the amplitude can be regulated in a limited
range by an additional cylinder setting an offset gas volume.
The forcing frequency is adjusted by the rotational frequency of the stepper motor
driving the piston compressor. The other side of the piston is pumped by a rotary
pump to ease the movement of the piston, since the CO pressure is normally operated
between 50 and 100 mbar.
This new device allows the application of periodic forcing at frequencies up to 4Hz
and well-defined amplitudes. The frequency response of the UHV chamber on partial
pressure in the pre-pressure line is shown as Bode plot in Fig. 3.6. The measurement
of the pCO = 2 ×10−4mbar in the UHV chamber was performed by using a ionization
manometer (Leybold IM510 with VIG17-head) in linear scaling. The experimental
results are shown as (+). The solid line indicates the linear approximation fit with a
chamber’s time constant.
3.2 Numerical Method
Mathematical modeling of the experiments is performed using a realistic model of
catalytic CO oxidation on Pt(110) known as the KEE model, introduced in Chapter 2.
The model takes adsorption of CO and oxygen molecules, reaction rates, desorption of
3.2. NUMERICAL METHOD 39
CO molecules, the structural phase transition of the Pt(110) surface, surface diffusion
of adsorbed CO molecules, surface roughening into account while faceting, formation
of subsurface oxygen, and the effects of internal gas-phase coupling are not considered.
The differential equations describing a single element of the extended system are given
by:
ut=k1scopco(1 −u3)−k2u−k3uv +D∇2u(3.2)
vt=k4pO2[so,1×1w+so,1×2(1 −w)](1 −u−v)2−k3uv (3.3)
wt=k5(1
1 + exp[(u−u0)/δu]−w) (3.4)
Numerical simulations of the model were performed using a second-order finite differ-
ence scheme for the spatial discrimination with a grid resolution of dx = 4 µm. For the
temporal discrimination an explicit Euler scheme with a fixed time steps dt = 0.001s
is used. A system size of 400 µm2and no-flux boundary condition is taken.
3.2.1 Implementation for Resonance Forcing
Like in the experiments, resonance forcing is artificially introduced by means of con-
trolled variation of CO partial pressures. Therefore, to implement resonance forcing in
the CO oxidation model, it is assumed that the CO partial pressure pCO in equation
(3.2) is not constant but varies according to the equation (3.1).
All of the numerical simulations were carried out in a programming tool called Matlab.
In all cases, model equations were integrated in time by an Explicit Euler method. The
model parameters used for the numerical simulations are given in Tables 3.1 and 3.2.
40 CHAPTER 3. METHODS
3.3 Pattern Analysis
For further analysis, either the experimentally obtained PEEM image sequence or the
numerical values of the CO surface coverage (for model simulations) are used. The
course of time is visualized by space-time plots, showing the pattern evolution along a
chosen line within the two-dimensional data as a function of time. Oscillatory behavior
as well as motion of cluster boundaries cross-sectioning this line can be determined
from these plots. Furthermore, the average image intensity in a small region of the
platinum surface is shown in comparison to the forcing as a function of time. From
these plots, the state of entrainment can be easily determined.
To measure the temporal response of the patterns, a frequency demodulation technique
is used. At each image pixel the brightness of the image is recorded, giving an ensemble
of time series of the local system dynamics. The Fourier transform of these time series
is calculated, and the complex Fourier coefficients of the main frequency component
allow for amplitude and phase representations of the data. The spatial distribution
of amplitude and phase is analyzed as well as the overall distribution of phase states,
given as phase histogram and mapping into the complex plane.
3.3. PATTERN ANALYSIS 41
k13.14 ×105s−1mbar−1Impingement rate of CO
k210.23 s−1CO desorption rate
k3283.8 s−1Reaction rate
k45.86 ×105s−1mbar−1Impingement rate of O2
k51.610 s−1Phase transition rate
sCO 1.0 CO sticking coefficient
sO,1×10.6 Oxygen sticking coefficient on the 1×1 phase
sO,1×20.4 Oxygen sticking coefficient on the 1×2 phase
u0, δu 0.35,0.05 Parameters for the structural phase transition
D40 µ2s−1CO diffusion coefficient
pO21.2×10−4mbar O2partial pressure
pO4.6219548 ×10−5mbar Base CO partial pressure
Table 3.1: Parameters of the KEE model (Turbulent regime)
k13.14 ×105s−1mbar−1Impingement rate of CO
k210.21 s−1CO desorption rate
k3281.6 s−1Reaction rate
k45.86 ×105s−1mbar−1Impingement rate of O2
k51.60 s−1Phase transition rate
sCO 1.0 CO sticking coefficient
sO,1×10.6 Oxygen sticking coefficient on the 1×1 phase
sO,1×20.4 Oxygen sticking coefficient on the 1×2 phase
u0, δu 0.35,0.05 Parameters for the structural phase transition
D39.59 µ2s−1CO diffusion coefficient
pO21.2×10−4mbar O2partial pressure
pO4.75 ×10−5mbar Base CO partial pressure
Table 3.2: Parameters of the KEE model (Nonturbulent regime)
42 CHAPTER 3. METHODS
Chapter 4
Resonance Forcing: Experimental
Results
4.1 Natural Frequency of the System
The natural frequency of the system is defined as the main frequency of the Fourier
spectrum of the local PEEM intensity, i.e., the frequency with the highest amplitude
in the power spectrum of a local intensity time series. For a quick measurement of the
natural frequency during the experiments, a section of the PEEM image with a size
of 10 ×10 pixels is chosen, its mean intensity is calculated, and its time series Fourier
transformed (Fig. 4.1).
The natural frequency is determined from the maximum of the power spectrum. Even
though the frequency analysis is performed locally, its validity is assumed for the whole
sample. To prove this assumption, 2500 pixels, equally distributed over the region of
interest were chosen from the same video sequence. The time series of the one single
pixel is shown in Fig. 4.2.
However, on a time scale of several minutes the natural frequency slowly decreases al-
43
44 CHAPTER 4. RESONANCE FORCING: EXPERIMENTAL RESULTS
Figure 4.1: Time series of the averaged image intensity in an area of 10 ×10 pixels
(top panel) and power spectrum of the data (bottom panel). The 1Hz oscillations are
clearly seen in both the time series and the spectrum. The reaction parameters are T
= 515K, po2= 1.5×10−4mbar, and po= 7.5×10−5mbar.
Figure 4.2: Time series of the gray value of one single pixel (top panel) and respective
power spectrum (bottom panel). The reaction parameters are the same as in Fig. 4.1.
4.2. 2:1 FORCING 45
though the reaction parameters are kept constant (Fig. 4.3). This effect is presumably
caused by a faceting of the platinum surface, which is known to take place at the used
reaction conditions [99].
Figure 4.3: Fourier spectrogram showing the time evolution of the natural frequency of
the system without forcing. For each time moment an interval of 20.48s (512 samples
at 25 frames per second) is analyzed. The frequency has an initial value of about
1Hz. During the first 30 min it drops by approximately 0.2Hz. Broadening of the
frequency line is also observed, which indicates, that the system has developed into a
more turbulent state. Values of temperature (T = 515K), and partial pressures (po2
= 1.1×10−4mbar, po= 9.0×10−5mbar) were constant.
4.2 2:1 Forcing
The natural frequency of the system (ω0) is taken after the full development of spiral
wave turbulence. We apply the forcing ωf= 2ω0in this system. After starting the
forcing, the spiral wave turbulence is first replaced by the intermittent turbulence
characterized by the repeated emergence and disappearance of localized turbulent
46 CHAPTER 4. RESONANCE FORCING: EXPERIMENTAL RESULTS
bubbles on a background of locked uniform oscillations. In different reaction and
forcing parameters region, two types of clusters named phase and amplitude clusters
were found.
4.2.1 Phase Clusters
Phase clusters are characterized by equal oscillation amplitudes and a constant phase
shift between the cluster states. The oscillations in both cluster states correspond
to the same limit cycle, but are opposite in phase. The phase fronts that separate
different cluster domains exhibit rich behavior.
Fig. 4.4(a) shows the natural frequency of the system ω0= 0.68Hz. The amplitudes
of the Fourier coefficients are normalized to the maximum peak. We fixed the forcing
frequency ωf= 1.36Hz and changed the forcing amplitude. During the experiment
the reaction parameters were fixed.
At different forcing amplitudes two and four phase cluster patterns under 2:1 and 4:1
entrainment were found.
2:1 Entrainment
At forcing amplitude A = 0.22, a phase locked regime with stable cluster patterns
due to 2:1 entrained was observed. Fig. 4.4(b) shows the Fourier spectrum of 2:1
entrained phase clusters and Fig. 4.5(a) shows three snapshots of PEEM images at
time intervals of one forcing period between subsequent frames.
Due to the non-harmonicity of the CO oxidation, the size of these phase domains is
not fixed but changes in time undergoing enlargement-reduction (breathing like) cycles
with a periodicity that is again two times the forcing cycle.
The space-time plot (Fig. 4.5(b)) shows that the domain wall of the opposite phase is
not stationary, taken along the line AB shown in Fig. 4.5(a). Fig. 4.5(c) gives the time
course of the surface marked by the (solid line) AB in the first image of Fig. 4.5(a)
4.2. 2:1 FORCING 47
Figure 4.4: Mean Fourier spectra at different forcing amplitudes A. (a) 0, (b) 0.22, (c)
0.24. Other parameters are T = 529K, po2= 1 ×10−4mbar, po= 9.26 ×10−5mbar,
ω0= 0.68Hz, and ωf= 1.36Hz.
and the course of the forcing signal (dotted line), shows the sub-harmonic entrainment
of the system to the forcing frequency.
During two cycles of the forcing signal, the system performs one cycle of periodically
changing CO coverage, indicated by low and high PEEM intensity.
To further analyze the dynamics of oscillatory clusters, the frequency demodulation
technique (described detail in Chapter 3) was used which is useful for the characteriza-
tion of resonant patterns. On analyzing the patterns at ωf/2, the two phase patterns
differed by πwere obtained.
48 CHAPTER 4. RESONANCE FORCING: EXPERIMENTAL RESULTS
Figure 4.5: Phase cluster under 2:1 entrainment. (a) snapshots of PEEM images
300 ×300 µm2illustrating a phase locked regime, (b) space-time plot taken along the
AB line (top panel), and (c) intensity of the PEEM image averaged globally (solid
line) and the forcing signal (dotted line). The forcing amplitude A = 0.22 and other
parameters are the same as in Fig. 4.4.
In phase pattern Fig. 4.6(a), two phase states are clearly visible. Fig. 4.6(b) illustrates
that the oscillation amplitude is strongly reduced in the domain interfaces and that is
the same within the domains of the opposite phase.
Phase portrait Fig. 4.6(c), shows two spots of accumulating points corresponding to
the pixels located within the different domain. Phase histogram Fig. 4.6(d), corre-
sponds to two phases which are differ by π. The two phase states are not evenly
weighted.
4.2. 2:1 FORCING 49
Figure 4.6: Phase and amplitude representation of the cluster patterns shown in Fig.
4.5(a). (a) phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase
histogram.
4:1 Entrainment
At forcing amplitude A = 0.24, two phase cluster patterns due to 4:1 entrainment were
observed. Fig. 4.4(c) shows the Fourier spectrum under 4:1 entrainment and one can
see the main sharp peak at ωf/4 and ωf.
The phase and amplitude analysis of the cluster patterns at ωf/4 shows two phase
cluster which differ by πhave similar properties as 2:1 entrained cluster (data are not
shown).
Four Phase Clusters under 4:1 Entrainment:
The experimental protocol is modified in a way that the system is brought back to
50 CHAPTER 4. RESONANCE FORCING: EXPERIMENTAL RESULTS
Figure 4.7: Phase and amplitude representation of the four phase cluster patterns. (a)
phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase histogram.The
forcing amplitude A = 0.25 and other parameters are same as in Fig. 4.4.
the unforced state and is then forced with these differing initial conditions, four phase
clusters could be obtained, in analogous to resonant 4:1 forcing. Fig. 4.7 shows the
phase and amplitude representation, analyzed at ωf/4. (PEEM images are not shown.)
The cluster patterns appears to be smaller scaled.
The phase portrait, given in Fig. 4.7(c), shows a cross-shaped structure rather than the
line structure in the preceding images, indicating the presence of four distinct cluster
states. This is confirmed by the phase histogram, shown in Fig. 4.7(d), where four
maxima appear with a difference of π/2. However, two of the phase states, differing by
π, are more predominant. This may indicate that the two phase cluster configuration
might be more stable and the system tends to reach a two phase state.
4.2. 2:1 FORCING 51
4.2.2 Amplitude Clusters
In amplitude clusters, not only the oscillation phase but also the oscillation amplitude
is different in the regions occupied by the two different states. Thus, uniform oscilla-
tions within two different clusters correspond to different coexisting limit cycles of an
equal period. The phase shift between the oscillations in two cluster states is constant,
but depends on the controlling parameters ωfand A.
Figure 4.8: Mean Fourier spectra at different forcing amplitudes A. (a) 0, (b) 0.014,
(c) 0.064, (d) 0.079, and (e) 0.093. Other parameters are T = 546 K, pO2= 1.5×10−4
mbar, po= 6.22 ×10−5mbar, and ωf= 1.27Hz.
2:1 Entrainment
As above, forcing is applied after the full development of spiral wave turbulence. Fig.
4.8(a) shows the frequency spectrum of the unforced turbulent system. The natural
52 CHAPTER 4. RESONANCE FORCING: EXPERIMENTAL RESULTS
frequency, defined as the most prominent line, is found at ω0= (0.59±0.03)Hz. The
forcing frequency is set to ωf= 1.27Hz, which is slightly higher than twice the calcu-
lated natural frequency.
At A = 0.014, two phase frequency locked amplitude clusters were observed. It is
locked to the external stimulus with a phase shift of one forcing period between the
two states. Analysis of the phase and amplitude representation of amplitude cluster
patterns at ωf/2 is shown in Fig. 4.9.
In phase pattern (Fig. 4.9(a)), two phase states are clearly visible and the amplitude
pattern (Fig. 4.9(b)) indicates that the amplitude is different not only at the border
of the cluster but also in different domains of the cluster pattern. The phase portrait
Fig. 4.9(c) and histogram Fig. 4.9(d) shows the phase difference between two clusters
is π.
Figure 4.9: Phase and amplitude representation of amplitude clusters at 2:1 entrain-
ment. (a) phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase
histogram.
4.2. 2:1 FORCING 53
4:1 Entrainment
At slightly stronger forcing amplitude A = 0.064, the system’s oscillations are period
doubled, indicated by the appearance of the sub-harmonic line ω=ωf/4 and it’s
rational multiples 3/4ωf, 5/3ωf, as seen in Fig. 4.8(c). Two phase amplitude clusters
were again observed.
The phase and amplitude representation of the amplitude clusters at ωf/4 shows the
same properties as two phase cluster at 2:1 entrainment (data are not shown).
8:1 Entrainment
Figure 4.10: Phase and amplitude representation of the amplitude cluster patterns
at 8:1 entrainment. (a) phase pattern, (b) amplitude pattern, (c) phase portrait, and
(d) phase histogram.
A slight increase in the forcing amplitude to A = 0.0108 leads to the next bifurcation
within the period doubling cascade. The system shows 8:1 entrainment, locking to
54 CHAPTER 4. RESONANCE FORCING: EXPERIMENTAL RESULTS
a frequency of ωf/8, while the Fourier coefficient of ωf/4 nearly vanishes, the sub-
harmonic 3/8ωfis strongly pronounced as seen in Fig. 4.8(c))
In Fig. 4.10, the spatial amplitude, phase distribution, the phase portrait, and the
phase histogram are shown, obtained for the Fourier coefficients of ωf/8. Fig. 4.10(a)
shows the phase patterns, where two phase states are mainly observed. They appear
in clusters, which are separated by low amplitude boundaries and the amplitude is
different in two phase states as can be seen from the amplitude pattern in Fig. 4.10(b).
The phase portrait, given in Fig. 4.10(c), is mainly line-shaped, with two accumulation
points with opposite phase. The clustering into two phase states can clearly be seen
in the phase histogram, given in Fig. 4.10(d).
Transition to Chaos
Finally, applying a slightly higher forcing amplitude of A = 0.093, the oscillation is
no longer entrained. The Fourier spectrum given in Fig. 4.8(e), shows the absence
of the sub-harmonic line at ωf/8. Lines at ωf/4, ωf/2, and ωfare still present, but
additional frequency components appear in the sub-harmonic regime.
The strong peak between ωf/4 and ωf/2 might be related to 3/8ωfwithin the fre-
quency resolution, but the peak slightly above ωf/2 (determined to be at ωf= (0.73±
0.03)Hz) cannot be assigned to a rational multiple of the forcing frequency. There-
fore, we state a chaotic response of the system at sufficiently high forcing amplitude.
Regarding the spatial dynamics, the entrainment of the system is accompanied by the
suppression of chemical turbulence and cluster formation.
4.3 3:1 Forcing
Pattern formation under 3:1 resonance is explained in this section. Two phase, three
phase, and six phase patterns were observed at 2:1, 3:1, and 6:1 entrainment respec-
tively.
4.3. 3:1 FORCING 55
4.3.1 2:1 Entrainment
As above the system is forced with three times its natural frequency (ωf= 3ω0). The
natural frequency of the system is found at about 0.76Hz. The forcing parameters
were fixed at ωf= 2.3Hz and A = 0.12 and the phase locked regimes were observed
(Fig. 4.11). As can be seen in the space-time plot (Fig. 4.11(b)), the system largely
performs oscillations with a frequency of ω=ωf/2 (2:1 entrainment). However, it
seems that for this amplitude of 3:1 forcing the system is not fully entrained. At t =
5.5 s the pattern inverts in the space-time plot and a large amplitude oscillation is not
followed by a small shoulder but by another large amplitude oscillation (Fig. 4.11(c)).
Figure 4.11: Two phase cluster due to 2:1 entrainment in 3:1 forcing. (a) snapshots of
PEEM images (size 300×300 µm2), (b) space-time plot showing the pattern evolution
along the AB line (see top panel), and (c) averaged intensity (solid line), and the
forcing signal (dotted line). The reaction parameters are: T = 534 K, po2= 1.6×10−4
mbar, po= 7.6×10−5mbar, ωo= 0.76Hz, and the forcing parameters are A = 0.11,
and ωf= 2.3Hz.
56 CHAPTER 4. RESONANCE FORCING: EXPERIMENTAL RESULTS
Figure 4.12: Phase and amplitude representation of the cluster patterns shown in Fig.
4.11(a). (a) phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase
histogram.
After this transition, the oscillations of the system are again 2:1 entrained. These
more complex dynamics are also visible when phase and amplitude of the patterns are
analyzed (Fig. 4.12). Two distinct clusters with sharp boundaries can be observed
(Fig. 4.12(a)). The two phase locked cluster having different amplitude in different
domains indicate the observation of an amplitude cluster (Fig. 4.12(a)), which are not
exactly separated by a phase difference of π(Fig. 4.12(c) and (d)).
4.3.2 3:1 Entrainment
Another experiment begins with spiral wave turbulence, and without forcing the sys-
tem oscillates with a natural frequency ω0= 0.8Hz (Fig. 4.13(a)). The forcing fre-
quency is set to ωf= 2.4Hz, which again is three times of the calculated natural
4.3. 3:1 FORCING 57
frequency. At forcing amplitude A = 0.12, the system oscillates with one third of the
forcing frequency, and the system frequency locked in 3:1 entrainment. Fig. 4.13(b)
shows the Fourier spectrum at 3:1 entrainment, where one can see the peak at ωfand
ωf/3.
Figure 4.13: Fourier spectra at different forcing amplitudes A: (a) 0, and (b) 0.12.
The reaction parameters are: T = 534 K, po2= 1.6×10−4mbar, po= 7.66 ×10−5
mbar, ωo= 0.8Hz, and ωf= 2.4Hz.
Fig. 4.14(a) shows the snapshots of stripe like wave fronts that periodically appear on
the platinum surface under 3:1 entrainment. Every PEEM image is taken after one
forcing period. The temporal evolution of the system is represented as a space-time
plot in Fig. 4.14(b) taken along the AB shown in Fig 4.14(a).
In space-time plot one can see that the domain of opposite phase is not stationary.
Fig. 4.14(c) gives the time course of the surface along the line AB shown in the first
58 CHAPTER 4. RESONANCE FORCING: EXPERIMENTAL RESULTS
Figure 4.14: Three phase cluster formation and entrainment at 3:1 resonant forcing.
(a) snapshots of PEEM images (size 300 ×300 µm2), (b) space-time plot showing
the pattern evolution along the AB line indicated in first image (top panel), and
(c) averaged intensity (solid line) and the forcing signal (dotted line). The reaction
parameters are: T = 534 K, po2= 1.6×10−4mbar, po= 7.66 ×10−5mbar, ωo=
0.8Hz, and the forcing parameters are A= 0.12, and ωf= 2.4Hz.
image of Fig. 4.14(a) (solid line) and the course of the forcing signal (dotted line),
shows the sub-harmonic entrainment of the system to the forcing frequency.
Fig. 4.15 shows the phase and amplitude representation at ωf/3. Well-defined phase
fronts such as black, dark gray, and light gray areas are visible in phase pattern in
Fig. 4.15(a). The amplitude pattern in Fig. 4.15(b) shows that the amplitude is
approximately the same except for the border. Accordingly, the phase distribution,
given in Fig. 4.15(c) and Fig. 4.15(d), shows three maxima with a distance of 2π/3.
4.3. 3:1 FORCING 59
Figure 4.15: Phase and amplitude representation of the cluster patterns shown in Fig.
4.14(a). (a) phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase
histogram.
4.3.3 6:1 Entrainment
At forcing frequency ωf= 2.5Hz, a response with large spectral power at ω/ωf= 1/6
was observed as shown in the power spectrum Fig. 4.16. The power spectrum shows
the main peak at ωf/6, ωf/3, ωf, 1.95ωf.
The phase and amplitude representation at ωf/6 is shown in Fig. 4.17. In phase
pattern the six phases are clearly visible (Fig 4.17(a)). In the amplitude pattern (Fig
4.17(b)), the amplitude drops only at the border. The phase portrait (Fig 4.17(c))
shows the six fold symmetry corresponding to six stable uniform phases, which are
found at the point farthest from the center. The states are connected with traveling
fronts that shift the phase by π/3 and do not go through the origin. The scattering of
the point is due to the experimental noise. The phase histogram (Fig. 4.17(d)) shows
that the phase difference in each phase is approximately 2π/6.
60 CHAPTER 4. RESONANCE FORCING: EXPERIMENTAL RESULTS
Figure 4.16: Spatially averaged Fourier spectrum of the PEEM intensity at ωf=
2.5Hz. Other parameters are same as in Fig. 4.15.
Figure 4.17: Phase and amplitude representation of the cluster patterns at 6:1 en-
trainment. (a) phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase
histogram.
4.4. 4:1 FORCING 61
4.4 4:1 Forcing
In this section we explore pattern formation of the forced CO oxidation on Pt(110)
under 4:1 resonance in turbulent and nonturbulent regimes. Recent developments
enable the exploration of a wider range of forcing parameters, allowing frequencies of
about four times the natural frequency of the oscillatory system to be reached for the
first time.
4.4.1 4:1 Forcing in Turbulent Regime
Like above, the natural frequency of the system was measured after the full devel-
opment of the spiral wave turbulence. The natural frequency of the system is about
0.72Hz (see in Fig. 4.18(a)). At forcing amplitude of A = 0.068, the spiral-wave tur-
bulence develops into stripe like wave fronts that periodically appear on the platinum
surface (see Fig. 4.19(a) and 4.19(b)). Fig. 4.19(c), which gives the time course of
the surface marked by the square in the first image of Fig. 4.19(a) (solid line) and the
course of the forcing signal (dotted line), shows the sub-harmonic entrainment of the
system to the forcing frequency.
During four cycles of the forcing signal, the system performs one cycle of periodically
changing CO coverage. While small changes within the PEEM signal occur at all
phases of the forcing frequency, the sudden increase of the CO coverage is in phase
with the rising edge of the applied CO pressure.
The same could be observed for the strong decrease of the CO coverage occurring in
accordance with every fourth trailing edge of the forcing signal. The moment of high
and low CO coverage occurs at different phases of the sub-harmonic oscillation for
different places on the surface. It can best be seen in the space-time plot, given in
Fig. 4.19(b). It shows the intensity along the line indicated in the first image of Fig.
4.19(a). Spatially resolved analysis of the appearing pattern using the demodulation
technique shows phase clusters (see Fig. 4.20(a)). Along with a notable amount of
phase defects indicating that the system is still turbulent, especially in the upper right
area and the lower part of the analyzed surface, a regular four-phase pattern can be
62 CHAPTER 4. RESONANCE FORCING: EXPERIMENTAL RESULTS
observed.
The different phase states are visible in the phase pattern in Fig. 4.20(a) as black, dark-
gray, gray, and light-gray areas. These areas also show higher oscillation amplitude,
given as bright areas in Fig. 4.20(b), while the domain interfaces are visible as regions
with reduced amplitude, shown in gray. The regions with strongly reduced amplitude
near zero, seen in black, are located at the defect points.
The two areas of high amplitude and regular phase pattern are separated by a tur-
bulent regime, where the number of defects is larger and phase clusters cannot be
identified. The phase portrait and the phase histogram, given in Fig. 4.20(c) and Fig.
4.20(d), show the presence of four distinct phase states with a phase difference of π/2,
accompanied by an underlying random phase distribution due to the turbulent regime.
Figure 4.18: Fourier spectra at different forcing amplitudes A. (a) 0, (b) 0.068. The
reaction parameters are: T = 534 K, po2= 1.7×10−4mbar, po= 5.88 ×10−5mbar,
ωo= 0.72Hz, and ωf= 2.88Hz.
4.4. 4:1 FORCING 63
Figure 4.19: Four phase cluster at 4:1 resonant forcing. (a) snapshots of PEEM
images (size 300 ×300 µm2), (b) space-time plot, and (c) averaged intensity of a local
area of the surface (solid line), indicated by the square in (a), and the forcing signal
(dotted line). The reaction parameters are the same as in Fig. 4.18.
Figure 4.20: Phase and amplitude representations of the cluster patterns shown in
Fig. 4.19(a). (a) phase pattern, (b) amplitude pattern, (c) phase portrait, and (d)
phase histogram.
64 CHAPTER 4. RESONANCE FORCING: EXPERIMENTAL RESULTS
4.4.2 4:1 Forcing in a Nonturbulent Regime
Figure 4.21: Fourier spectra at different forcing amplitudes A. (a) 0, and (b) 0.12.
The parameters are: T = 534 K, po2= 1.2×10−4mbar, po= 5.8×10−5mbar, ωo=
0.5Hz, and ωf= 2Hz.
4:1 forcing is done experimentally in a nonturbulent regime in the ruthenium-catalyzed
BZ reaction forced by periodic illumination revealed, where the unforced pattern is a
rotating spiral wave of ruthenium catalyst concentration. However, four phase patterns
at low forcing amplitude were seen but the two phase standing wave patterns at high
forcing amplitude were not seen.
The system was forced after the full development of homogenous oscillation. Fig.
4.21(a) shows the natural frequency of the system around 0.5Hz. The system was
forced around four times its natural frequency.
At weak forcing amplitude A = 0.11, the system oscillates with half of the forcing
frequency; the system is frequency locked in 2:1 entrainment (see Fig. 4.21(b)). Fig.
4.22(a) shows the PEEM images of the phase clusters taken after one forcing period.
4.4. 4:1 FORCING 65
Figure 4.22: Two phase cluster formation and 2:1 entrainment at 4:1 resonant forcing.
(a) snapshots of PEEM images (size 300 ×300 µm2), (b) space-time plot showing the
pattern evolution along the AB line indicated in (a), and (c) averaged intensity of a
local area of the surface (solid line) and the forcing signal (dotted line). The parameters
are: T= 534 K, po2= 1.2×10−4mbar, po= 5.8×10−5mbar, ωo= 0.5Hz, ωf= 2Hz,
A = 0.12.
The space-time plot represents (Fig. 4.22(b)) that the domain of the opposite phase
are not stationary, taken along the line AB shown in Fig. 4.22(a). In Fig. 4.22(c)
the solid line shows the average PEEM intensity and the dotted line shows the forcing
signal of the system.
To get an idea about phase and amplitude frequency demodulation technique is used
again. Fig. 4.23 shows the phase and amplitude representation of two phase clusters.
In Fig. 4.23(b) one can see that the amplitude drops only at the border of the cluster
and it is the same between the opposite domains Fig. 4.23(a). Fig. 4.23(c) and Fig.
4.23(d) represent that the phase difference is π.
66 CHAPTER 4. RESONANCE FORCING: EXPERIMENTAL RESULTS
Figure 4.23: Phase and amplitude representations of the cluster patterns shown in
Fig. 4.22(a). (a) phase pattern, (b) amplitude pattern, (c) phase portrait, and (d)
phase histogram.
4.5 Conclusion
Resonance forcing is investigated in turbulent and nonturbulent regimes in CO oxida-
tion on Pt(110). A compressor driven reactor which allows global gas-phase forcing for
frequency modulations up to 4Hz was designed and built [80]. Experiments in differ-
ent resonant forcing regimes (2:1, 3:1, 4:1) were performed and the observed pattern
formation is discussed with respect to experimental studies.
In the case of 2:1 resonance forcing, a complete path from chemical turbulence to
entrainment and further to chaotic oscillations via a period doubling cascade was
obtained. A variety of patterns were observed under 2:1 forcing. Under 2:1 and
4:1 entrainment both types of cluster (phase and amplitude) differing by πwere ob-
tained. Four phase clusters are obtained under 4:1 entrainment. In the case of 8:1
sub-harmonic entrainment, only an amplitude clusters were obtained. Phase fronts
separating different homogeneous phase locked states were clearly observed during the
4.5. CONCLUSION 67
experiments. In addition, the theoretical work under 2:1 resonance forcing in a sin-
gle oscillator predicts [45] front explosions for decaying forcing amplitude, turbulent
interfacial zones were not observed under 2:1 resonance forcing.
In 3:1 resonance forcing, three phase and six phase moving clusters were observed
at 3:1 and 6:1 entrainment, while in BZ reaction three phase moving and six phase
stationary clusters were observed respectively [77].
Theoretically predicted labyrinth patterns in a single oscillator [45], could not be found
experimentally probably due to the high sensitivity of the system to parameter changes
and present technical limitations in the application of soft changes in amplitude A.
Under 4:1 resonance forcing, 4:1 entrainment and four-phase cluster patterns could
be observed [126]. However, the cluster formation takes place in finite regions of the
surface, while other parts appear not to be 4:1 entrained, but still show turbulent
behavior. This is one of the reasons why global coupling [46, 127] can be neglected
under 4:1 forcing. Global coupling can stabilize homogeneous oscillations in a large
surface area. This system, however, breaks up into a large number of rather small
clusters. Thus, the effect of global coupling is averaged out. Additionally, the applied
forcing amplitudes are comparably high, making an influence of global coupling even
more unlikely.
In the case of 4:1 resonance forcing in a nonturbulent regime, only two phase moving
phase clusters differing by πwere obtained, while the four phase patterns could not be
observed. Experimentally in BZ reaction in nonturbulent regime the four phase cluster
patterns could be observed but the two phase standing wave patterns (predicted in
CGLE equation, FitzHugh-Nagumo and Brusselator models) were not observed [71].
In summary, it was demonstrated that attempts to control chemical turbulence by
periodic forcing may suppress spatial turbulence, but could lead to chaotic response
of the system. The path to chaos was given by a periodic doubling cascade which was
followed by the subsequent increase of forcing amplitude. Further with the help of a
new compressor (described in Chapter 3) 3:1 and 4:1 resonance forcing regions were
reached for first time.
68 CHAPTER 4. RESONANCE FORCING: EXPERIMENTAL RESULTS
Chapter 5
Resonance Forcing: Theoretical
Results
Numerical simulations have been performed for comparison to the experiments. The
KEE model was used and its implementation for resonance forcing is explained in
Chapter 3.
Periodic forcing under 2:1 and 3:1 is explained by Davidson et. al. in a single oscillator
[45] but this is unknown in an extended system. As the experimental system used an
extended system, to compare the results the simulation was performed in an extended
system.
5.1 Natural Frequency of an Extended System
As has been already discussed, the natural frequency of the single oscillator in nontur-
bulent state is identical with the oscillation frequency of the extended system, while
69
70 CHAPTER 5. RESONANCE FORCING: THEORETICAL RESULTS
Figure 5.1: Frequency of an extended system. (a) spatial distribution, (b) histogram of
most prominent frequency in a numerical simulation of unforced turbulence. ω0,single
denotes the natural frequency of a single oscillator. The parameters are given in
Chapter 3, Table 3.1.
Figure 5.2: Oscillation frequency of a single oscillator (circles) and mean oscillation
frequency of the extended system (squares) using the KEE model with different p0
within the turbulent regime. The end points, where both frequencies coincide, indicate
the onset of homogeneous oscillations.
5.2. 2:1 FORCING 71
in the turbulent state the system oscillates at higher frequencies due to diffusive in-
teraction of the surface elements.
To get an overview over the oscillatory characteristics of an extended system, we
performed Fourier analysis of the free running system in a fully developed turbulent
state. The initial parameters of the partial pressures were chosen such that the un-
forced system oscillates and exhibits spontaneously spiral wave turbulence. The model
parameters are given in Chapter 3 in Table 3.1.
Fig. 5.1(a) shows the most prominent frequency for each surface element. The local
generic frequency variation spans a frequency range of about 0.4-0.6Hz. A certain
frequency range is expected because of the turbulent state, but the distribution of
oscillation frequencies found locally, shown as histogram in Fig. 5.1(b), shows that
the mean frequency of 0.51Hz is remarkably higher than the oscillation frequency of
the single oscillator, which is rarely found in the extended system.
Fig. 5.2 shows the course of both characteristic frequencies, the mean frequency of the
extended system (squares) and the natural frequency of the single oscillator (circles),
with increasing CO base pressure p0. At the borders of the turbulent regime, homo-
geneous oscillations are found in the extended system, where the system behaves as a
single oscillator. A maximum frequency difference of more than 0.1Hz appears in the
center of the turbulent regime, at p0= 4.63 ×10−5mbar. Defining the characteristic
frequency of the extended system ω0, extended as the mean local oscillation frequency,
it was obtained for the chosen parameter set ωo,ext = 0.51Hz ≈1.2ω0,single.
5.2 2:1 Forcing
The forcing frequency ωf= 0.98Hz was chosen to be near the second harmonic of the
most prominent frequency in the extended system in fully developed turbulence (ω0,ext
= 0.51Hz) (Fig. 5.1(b)), rather than twice the single oscillator’s natural frequency
(ω0,single = 0.42Hz).
Increasing the forcing amplitude, frequency locked 2:1 entrainment is obtained at A =
0.0078 (Fig. 5.3(a)). In the space-time plot (Fig. 5.3(b)), one can see that the domain
72 CHAPTER 5. RESONANCE FORCING: THEORETICAL RESULTS
Figure 5.3: Two phase cluster at 2:1entrainment, KEE model. (a) Snapshots of CO
coverage (size 400×400 µm2), (b) space-time plot along the AB line from Fig. 5.3(a),
and (c) CO coverage along the line AB (solid line) and forcing signal (dotted line).
Forcing parameters are ωf= 0.98Hz and A = 0.086.
between opposite phases are stationary.
It can be seen clearly in the space-time stroboscopic plot (Fig. 5.4), showing the
pattern evolution along the same AB line of Fig. 5.3(a), choosing one frame every two
forcing cycles. Non-equilibrium Bloch walls were observed as the borders between two
different entrained states (π-fronts).
The phase and amplitude presentations are shown in Fig. 5.5. In phase pattern
(Fig. 5.5(a)), the two phase states of black and dark gray are clearly visible. The
amplitude is the same in the different domain except the cluster boundary (as seen in
the amplitude pattern (in Fig. 5.5(b)). The phase difference between the two phase
cluster is π, shown in the phase portrait (Fig. 5.5(c)) and phase histogram (Fig.
5.5(d)), respectively. The existence of two stable entrained states differing by a phase
5.2. 2:1 FORCING 73
Figure 5.4: Stroboscopic space-time plot showing the pattern evolution along the line
AB shown in Fig. 5.3(a), choosing one frame every two forcing cycle.
shift of πis a property of the 2:1 resonance, distinguishing it from the 1:1 resonance
regime [128].
Period doubling to 4:1 entrainment takes place at A = 0.0102. A further period dou-
bling to 8:1 entrainment could be found at A = 0.0108, leading to chaotic oscillations,
similar to the experimental results (explained in section 4.2.2).
At the higher forcing amplitude A = 0.06, the chaotic regime is confined by an inverse
period doubling cascade to final 1:1 entrainment. The phase and amplitude represen-
tation of 4:1 and 8:1 entrained data always exhibit phase clusters which differ by πor
π/2 (data are not shown). An overview over the sub-harmonic frequency spectrum for
increasing forcing amplitude is given in Fig. 5.6.
At 8:1 entrainment, the labyrinthine patterns were found. An example is given in Fig.
5.7(c). The transition between two phase cluster states and the labyrinthine pattern is
induced by phase instabilities within the cluster boundary. However, transition times
74 CHAPTER 5. RESONANCE FORCING: THEORETICAL RESULTS
Figure 5.5: Phase and amplitude representation of the cluster patterns shown in Fig.
5.3(a). (a) phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase
histogram.
are longer than 300s and might not be fully covered within the experiment. In the
chaotic regime, where the oscillation is not entrained to the forcing signal, chemical tur-
bulence is nevertheless suppressed. Global coupling induced by the forcing is assumed
to lead to low-dimensional chaos, where the system is spatially correlated. Cluster for-
mations were observed similar to a 4:1 entrainment (see Fig. 5.7(b)), although phase
fluctuation within the clusters were observed (see Fig. 5.7(d)).
The spatial correlation is determined by the cross-correlation of the dynamics at 100
evenly distributed surface locations, averaged over their distance. The results for un-
forced and forced spatiotemporal chaos in both the experimental (explained in section
4.2.2) and the simulated system are given in Fig. 5.8. The cross-correlation is nor-
malized to the mean auto-correlation of the sample points, while the cross-correlation
of the forced experimental system is nearly independent of the distance; it decreases
strongly with distance in the unforced turbulent state. The numerical results show
5.2. 2:1 FORCING 75
Figure 5.6: Periodically forced KEE model. Low frequency part of Fourier spectra at
different forcing amplitudes.
Figure 5.7: Simulated CO coverage for different entrainment states. (a) unforced
turbulence, (b) 4:1 entrainment, (c) 8:1 entrainment, and (d) chaotic.
76 CHAPTER 5. RESONANCE FORCING: THEORETICAL RESULTS
the same qualitative behavior, but the difference between the two states is less pro-
nounced. The shorter correlation length in the unforced experiment compared with
the simulation can be explained by noise. On the other hand, the higher correlation
of the forced experiment might be induced by stronger forcing, as the numerical result
is obtained near the lower amplitude boundary of the chaotic regime.
Figure 5.8: Mean spatial cross-correlation of unforced turbulence and forced chaotic
oscillations. Results are given for experimental and numerical data.
5.3 3:1 Forcing
5.3.1 3:1 Entrainment
Resonant forcing was applied (ωfclose to 3ω0,ext ). At forcing amplitude A = 0.0865,
the phase locked clusters were observed. In Fig. 5.9(a), the snapshots of three phase
locked clusters after each forcing cycle are shown. The space-time plot (Fig. 5.9(b))
explains that the domain walls are not stationary. This is visible in the space-time stro-
5.3. 3:1 FORCING 77
Figure 5.9: Three phase cluster at 3:1 entrainment. (a) snapshots of CO coverage
(size 400 ×400 µm2), (b) space-time plot along the AB line from Fig. 5.9(a), and (c)
CO coverage along the line AB (solid line) and forcing signal (dotted line). Forcing
parameters are ωf= 1.53Hz and A = 0.0865.
Figure 5.10: Space-time stroboscopic plot showing the pattern evolution along the
same AB line of Fig. 5.9(a), choosing one frame every three forcing cycles.
78 CHAPTER 5. RESONANCE FORCING: THEORETICAL RESULTS
boscopic plot Fig. 5.10 taken along the line AB shown in Fig. 5.9(a), but choosing one
frame for every three forcing cycles. The amplitude is increased, moving three-phase
clusters are observed until the system changes to 1:1 entrainment with homogeneous
oscillations.
The three phase states are clearly visible in phase pattern Fig. 5.11(a) and the phase
portrait Fig. 5.11(c). All phase states appear with the same amplitude (Fig. 5.11(b)),
while only the phase fronts show decreased amplitude, and amplitude defects i.e.,
points where the amplitude is decreased to zero and the phase is undefined appear at
locations where all three phase states meet. The histogram Fig. 5.11(d) shows that
the phase difference between the cluster states is 2π/3.
Figure 5.11: Phase and amplitude representation of the cluster patterns shown in Fig.
5.9(a). (a) phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase
histogram.
5.3. 3:1 FORCING 79
5.3.2 6:1 Entrainment
When the forcing amplitude A was fixed and the forcing frequency was changed to
ωf= 1.54Hz, a phase locked regime with stable cluster patterns, 6:1 entrained, was
observed (see Fig. 5.12). The arising spatiotemporal patterns exhibit a periodicity
of six forcing cycles, which indicates that the system performs period doubled 6:1
entrained oscillations. The space-time plot (Fig. 5.12(b)) indicates that the domain
walls are moving slowly.
Figure 5.12: Six phase cluster under 3:1 forcing at 6:1 entrainment. (a) snapshots of
CO coverage (size 400 ×400 µm2), (b) space-time plot along the AB line from Fig.
5.12(a), and (c) CO coverage along the line AB (solid line) and forcing signal (dotted
line). Forcing parameters are ωf= 1.53Hz and A = 0.0865.
The six phase states are clearly visible in phase pattern Fig. 5.13(a). The amplitude
is different only at the boundaries of the clusters (Fig. 5.13(b)). At locations where
80 CHAPTER 5. RESONANCE FORCING: THEORETICAL RESULTS
Figure 5.13: Phase and amplitude representation of the cluster patterns shown in Fig.
5.12. (a) Phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase
histogram.
more than two phase clusters meet, amplitude defects are present. The phase portrait
Fig. 5.13(c) shows that the fronts between the six phases always traverse through
zero, indicating a standing-wave pattern. The histogram Fig. 5.13(d) represents the
high density of six different phases states where the maxima have a phase difference
of 2π/6. The six phases are not equally weighted.
On further increasing the forcing amplitude and frequency, transitions from 3:1 to 1:1
oscillation were always found. Like a single oscillator [45], at high forcing amplitude
two phase cluster could not be observed under 2:1 entrainment. The transition from
a three phase cluster to 1:1 oscillation is shown in Fig. 5.14.
5.4. 4:1 FORCING 81
Figure 5.14: Transition from a three phase cluster to oscillation. (a) Snapshots of CO
coverage (size 400×400 µm2), (b) space-time plot along the AB line from Fig. 5.14(a),
and (c) CO coverage along the line AB (solid line) and forcing signal (dotted line).
Forcing parameters are ωf= 1.32 Hz, and A = 0.093.
5.4 4:1 Forcing
5.4.1 Turbulent Regime
Simulations in the turbulent regime are performed for resonant forcing of the complex
Ginzburg-Landau equation (CGLE) in Benjamin Fair instability with 2:1 and 3:1 forc-
ing [47, 69, 129], but no simulations on 4:1 resonant forcing have been performed for
oscillatory systems in a turbulent state.
As 4:1 forcing is unknown in a single oscillator in turbulent regime, as a first attempt
82 CHAPTER 5. RESONANCE FORCING: THEORETICAL RESULTS
Figure 5.15: Three phase cluster and 3:1 entrainment at 4:1 resonant forcing. (a)
snapshots of CO coverage (size 400 ×400 µm2), (b) space-time plot along the AB line
(see top panel), and (c) local CO coverage at point A and forcing signal (dotted line).
Forcing parameters ωf= 1.567Hz and A = 0.065.
the 4th harmonic of the generic frequency was applied (ωf= 4ω0,single ), 3:1 entrainment
was found with three phase cluster patterns in a wide parameter range of forcing
amplitudes and frequency detuning.
Fig. 5.15 shows the snapshots of three phase clusters for A = 0.065, which repeat after
three cycles of the forcing period and the space-time plot indicates that the domain
walls are not stationary.
The phase and amplitude representations are given in Fig. 5.16. The three different
phase states can clearly be seen in the phase plot Fig. 5.16(a) and the phase portrait
Fig. 5.16(c). All phase states appear with the same amplitude in Fig. 5.16(b). These
5.4. 4:1 FORCING 83
Figure 5.16: Phase and amplitude representations of the cluster patterns shown in Fig.
5.15(a). (a) phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase
histogram.
findings explain the 3:1 entrainment when forcing with the 4th harmonic of the single
oscillator’s natural frequency and the absence of 4:1 entrainment.
Even if transient 4:1 entrainment was sometime observed, long-term simulations al-
ways ended up in 3:1 entrainment and three phase pattern or at high amplitude in
homogeneous oscillations with 1:1 entrainment to the forcing frequency. Four phase
entrainment under 4:1 resonance forcing was observed. A possible bistability between
3:1 and 4:1 locking, similar to the bistability of 3:1 and 2:1 near ωf= 3ω0found by
Davidson [45], could not be verified. A 4:1 regime within the vicinity of the ωf= 4ω0
could not be found.
Therefore, the system was forced by the 4th harmonic of the most prominent frequency
of the turbulent system which leads in four phase patterns with partial or full 4:1
entrainment, depending on the reaction parameters chosen. On applying A = 0.086,
84 CHAPTER 5. RESONANCE FORCING: THEORETICAL RESULTS
Figure 5.17: Four phase cluster and entrainment at 4:1 resonant forcing. (a) snapshots
of CO coverage (size 400 ×400 µm2), (b) space-time plot along the AB line (see top
panel), and(c) Local CO coverage at point A and forcing signal (dotted line). Forcing
parameters: ωf= 2.04Hz and A = 0.086.
ωf= 2.04Hz, the system is entrained and four phase clusters appear. Snapshots of the
pattern, the evolution along a line section, and the CO coverage for a sample point is
given in Fig. 5.17.
The snapshots show separated phase clusters, which repeat every four cycles of the
forcing period. The boundaries of the phase clusters seem to move from one forcing
cycle to the other. This is visible in the space-time plot (Fig. 5.17(b)), where the
bright clusters seem to shrink, but then recover to their original width within the next
cycle of the system’s oscillation (covering four cycles of the forcing signal, as can be
seen in Fig. 5.17(c)).
5.4. 4:1 FORCING 85
Figure 5.18: Stroboscopic space-time plot showing the pattern evolution along a ver-
tical line on the surface every 4th forcing cycle. Forcing parameters are the same as in
Fig. 5.17.
Fig. 5.18 shows a stroboscopic space-time plot, where the CO coverage is shown along
a vertical line of the surface every 4th forcing cycle. The movement of the cluster
boundaries is clearly visible. The phase front velocity varies, which might be due to
the varying orientation of the phase front’s normal to the intersection as well as due
to the different curvature of the phase front.
The phase representation again shows the prominence of four phase states, visible as
four different gray levels in Fig. 5.19(a) and as highly populated points in the phase
portrait Fig. 5.19(c). The histogram Fig. 5.19(d) shows the high density of four
different phase states where the maxima have a phase difference of π/2.
Fronts between the clusters are represented by lines of low amplitude in Fig. 5.19(b)
where most of the phase fronts are π/2. The πfronts appear at the broadened bound-
aries. At locations, where more than two phase clusters meet, amplitude defects are
86 CHAPTER 5. RESONANCE FORCING: THEORETICAL RESULTS
Figure 5.19: Phase and amplitude representations of the cluster patterns shown in Fig.
5.17(a). (a) phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase
histogram.
present. The phase fronts are not stationary but move slowly.
Due to local analysis and finite frequency resolution of 0.02Hz within the experiments,
the estimation of the natural frequency is subject to errors, which may result in a
slight frequency mismatching, therefore we simulated the response of the system under
moderate detuning. The system is driven with ωf= 2Hz, keeping the amplitude at
A = 0.086. Results are presented in the time domain (Fig 5.20) as well as in the
Fourier domain (Fig. 5.21)). The snapshots of the CO coverage (Fig. 5.20(a)) show
remarkable higher turbulent behavior. A few phase clusters with locally homogeneous
CO coverage are visible, mainly located at the borders of the surface. Large parts of
the surface show no sharp phase fronts, but a smooth change of CO coverage. They
exhibit quasi-periodic behaviors, as deviations from periodic behaviors are not visible
on small time scales of a few Tf.
5.4. 4:1 FORCING 87
Figure 5.20: Four phase cluster and entrainment at 4:1 resonant forcing. (a) snapshots
of CO coverage size 400 ×400 µm2, (b) space-time plot along the AB line (see top
panel), and (c) local CO coverage at point A and forcing signal (dotted line). Model
parameters: ωf= 2.04Hz and A = 0.086.
Figure 5.21: Phase and amplitude representations of the cluster patterns given in Fig.
5.20(a). (a) Phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase
histogram.
88 CHAPTER 5. RESONANCE FORCING: THEORETICAL RESULTS
Figure 5.22: Transition from four phase cluster to oscillation at 4:1 resonant forcing.
(a) snapshots of CO coverage (size 400 ×400 µm2), (b) space-time plot along the AB
line (see top panel), and (c) local CO coverage at point A and forcing signal (dotted
line). Model parameters: ωf= 2.04Hz, and A = 0.129.
Although the space-time plot and the local time series show no qualitative difference
from the system at resonant forcing (compared to Fig. 5.18), analysis of the phase
and amplitude representation (Fig. 5.21 (a) and (b)) reveal an increased number of
topological defects; large parts of the system feature a decreased oscillation amplitude
while the phase clusters appear as regions with high amplitude, visible as bright regions
in Fig. 5.21(b). This can also be seen in the phase portrait, shown in Fig. 5.21(c),
where the phase-amplitude distribution of the surface elements is less concentrated at
maximum amplitude. The phase histogram, presented in Fig. 5.21(d), still contains
four peaks with a phase difference of π/2.
Increasing the amplitude, four phase clusters are observed moving until the system
changes to 1:1 entrainment with homogeneous oscillations (Fig. 5.22).
5.4. 4:1 FORCING 89
5.4.2 Nonturbulent Regime
Most of the numerical studies are done in the nonturbulent regime. The well known
examples are the CGLE, FitzHugh-Nagumo (FHN) and Brusselator models [39, 71].
All of the models predict four phase moving clusters at low forcing amplitude and two
phase stationary clusters at high forcing amplitude.
Figure 5.23: Natural frequency of the system in nonturbulent regime. (a) spatial
distribution, and (b) histogram of most prominent frequency in a numerical simulation
of unforced turbulence.
Four phase stationary cluster were consistently found at low forcing amplitude and two
phase stationary clusters were found at high forcing amplitude. The model parameters
and the reaction parameters are given in Chapter 3, Table 3.2. The natural frequency
of the system is shown in Fig. 5.23, which is about 0.25Hz.
Four Phase Cluster
The forcing was applied at four times the natural frequency. At forcing amplitude A
= 0.042, the four phase patterns were found. Fig. 5.24 shows the snapshots of the four
90 CHAPTER 5. RESONANCE FORCING: THEORETICAL RESULTS
Figure 5.24: Four phase cluster at 4:1 resonant forcing. (a) snapshots of CO coverage
(size 400 ×400 µm2), (b) space-time plot along the AB line (see top panel), and (c)
CO coverage (solid line) and forcing signal (dotted line). Model parameters: ωf=
0.92Hz and A = 0.042.
phase clusters taken after one forcing cycle. The boundaries of the phase clusters are
stationary from one forcing cycle another, as seen in the space-time plot Fig. 5.24(b).
The stationary behavior of the cluster can be seen clearly in the stroboscopic space-
time plot Fig. 5.25, where the CO coverage is shown along a vertical line of the surface
every 4th forcing cycle.
Phase and amplitude representations are given in Fig. 5.26. In phase pattern Fig.
5.26(a), four phase states (black, gray, light gray, and dark gray) are visible as four
different gray levels. The amplitude drops only at the boundary of the cluster (Fig.
5.26(b)). The phase portrait Fig. 5.26(c) shows that the fronts between the four
phases always traverse through zero, indicating a standing-wave pattern and the high
density of four different phase states, where the maxima have a phase difference of
π/2, as shown in phase histogram Fig. 5.26(d).
5.4. 4:1 FORCING 91
Figure 5.25: Space-time stroboscopic plot showing the pattern evolution along the
same AB line of Fig. 5.24(a), choosing one frame every four forcing cycles.
Figure 5.26: Phase and amplitude representations of the cluster patterns shown in Fig.
5.24(a). (a) phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase
histogram.
92 CHAPTER 5. RESONANCE FORCING: THEORETICAL RESULTS
Two Phase Cluster
At forcing amplitude A = 0.074, the two phase stationary cluster was observed. Fig.
5.27(a) shows the snapshots of the two phase cluster, taken after each forcing cycle.
The boundary of the cluster is always stationary, as can be seen in space-time plot
Fig. 5.27(b). Fig. 5.28 shows a stroboscopic space-time plot, where CO coverage is
shown along a vertical line of the surface every 2nd forcing cycle.
Figure 5.27: Two phase cluster at 4:1 resonant forcing. (a) snapshots of CO coverage
(size 400 ×400 µm2), (b) space-time plot along the AB line (see top panel), and (c)
CO coverage (solid line) and forcing signal (dotted line). Model parameters: ωf=
0.92Hz and A = 0.074.
Phase and amplitude representations are given in Fig 5.29. In phase pattern Fig.
5.29(a), two phase states are visible as two different gray levels. The amplitude pattern
Fig. 5.29(b) shows that the amplitudes are approximately the same for the different
domains except at the cluster boundary. The phase portrait Fig. 5.29(c) and phase
histogram Fig. 5.29(d) shows the high density of four different phase states, where the
maxima have a phase difference of π/2.
5.4. 4:1 FORCING 93
Figure 5.28: Space-time stroboscopic plot showing the pattern evolution along the
same AB line of Fig. 5.27, choosing one frame every two forcing cycles.
Figure 5.29: Phase and amplitude representations of the cluster patterns shown in Fig.
5.27(a). (a) phase pattern, (b) amplitude pattern, (c) phase portrait, and (d) phase
histogram.
94 CHAPTER 5. RESONANCE FORCING: THEORETICAL RESULTS
Conclusion:
Pattern formations have been investigated in a CO oxidation on Pt(110), forced by an
external periodic perturbation in an extended system.
An analysis of the KEE model reveals significant differences between the oscillation
frequency of the single oscillator and the mean frequency on the surface, which appears
to be higher in the turbulent state due to diffusive interaction of the surface elements.
Interestingly, a similar frequency increase with respect to homogeneous oscillations
was found for the spiral-wave oscillation frequency in the BZ reaction [71]. There,
simulations showed entrainment when using the homogenous oscillation frequency and
quasi-periodic behavior when forcing with the higher spiral-wave frequency, which is
in contrast to the findings for the turbulent case. This is of major impact for the
attempt of controlling the system by resonant forcing.
Under 2:1 forcing the path to chaos is given by a period doubling cascade, which could
be followed by subsequent increase of the forcing amplitude is verified theoretically.
Experimentally, two types of clusters named phase and amplitude clusters were found,
however, theoretically only the phase clusters were observed.
In 3:1 resonance forcing, three phase moving clusters and six phase stationary clusters
were found under 3:1 and 6:1 entrainment respectively, like in the FitzHugh-Nagumo
(FHN) model [77].
The bistability between 2:1 and 3:1 could not be found even at high forcing amplitude
and frequency. At high forcing amplitude and frequency the transition from three
phase cluster to homogeneous oscillations takes place (see Fig. 5.14).
The application of 4:1 forcing with the 4th harmonic of the natural frequency of the
single oscillator leads to 3:1 entrained three phase patterns; 4:1 entrainment could be
obtained by forcing with the 4th harmonic of the mean oscillation frequency. Weak
detuning leads to the appearance of turbulent regions.
Numerical simulations of the KEE model in the 4:1 nonturbulent regime reproduces
standing four phase clusters as well as two phase clusters at a higher amplitude, in
contrast to CGLE and FHN which predict four phase moving and two phase stationary
5.4. 4:1 FORCING 95
clusters at low and high forcing amplitudes [39, 71].
The greater the difference between these two characteristic frequencies, the more diffi-
cult it is to entrain both turbulent and phase-clustered regions. A necessary condition
for entrainment of the phase clusters is that the forcing frequency lie within the re-
spective Arnold’s tongue of the single oscillator.
Further investigation of high-frequency forcing of turbulent reaction-diffusion systems
may give new insight into the nature of turbulence and may lead to new strategies for
controlling chaos.
96 CHAPTER 5. RESONANCE FORCING: THEORETICAL RESULTS
Chapter 6
Defect Mediated Turbulence
The most basic feature of the chemical turbulence (also known as defect mediated tur-
bulence), is that the spatial disorder is generated by so-called defects, which present
singularities in the field of the oscillation phase.
The spatiotemporal chaos originates from the spontaneous and persistent creation and
annihilation of topological defects.
A defect is characterized by its integer topological charge mtop, which is defined by
mtop =1
2πH∇φ(x, t).ds
where φ(x, t) is the local phase and the integral is taken along a closed curve surround-
ing the defect.
Experimentally defect turbulence has been found to be abundant in systems such as
autocatalytic chemical reactions [130, 131], fluid convection [132, 133], electro convec-
tion in liquid crystals [134], and in nonlinear optics [135].
Theories about defect mediated turbulence have been extensively studied in the CGLE
97
98 CHAPTER 6. DEFECT MEDIATED TURBULENCE
system [10, 10, 136], which describes universal dynamics features of spatially extended
systems near a supercritical Hopf bifurcation.
The first probabilistic characterization of defect turbulence was given by Gil et. al.
[136] for the regime of amplitude turbulence in the CGLE. They considered a system
with periodic boundary conditions and assumed a constant rate of creation for pairs
of topological defects, independent of the number of pairs min the system. The
rate of annihilation was taken proportional to m2since defects annihilate in pairs
of opposite topological charge. In this Chapter the statistical properties of defect
mediated turbulence in catalytic CO oxidation on Pt(110) are described by using the
probabilistic model with increasing order of CO pressure.
Recently Beta et. al. [131] have analyzed the statistical properties of chemical tur-
bulence in oscillatory catalytic CO oxidation on Pt(110) based on the experimental
data, where the two-dimensional experimental images are transformed into phase and
amplitude patterns. The defects are identified from phase images which are not related
to the surface heterogeneities.
By assuming that the defects are statistically independent, the shape of the probability
distribution function (PDF) can be explained in terms of a simple probabilistic model,
based on the gain and loss rates of defects in the observed area. By use of a nearest-
neighbor tracking algorithm, they follow defects between adjacent frames and identify
creation (C) and entering (E) events as well as defects decay (D) and leaving (L) the
area of observation.
The observed rates are approximately given by
C(N)) = C0(6.1)
E(N) = E0(6.2)
L(N) = L0N(6.3)
99
D(N) = D01N2+D02N(6.4)
In the statistically stationary state, the master equation for the probability p(N) of
finding a number of N defects in the observed area can be written in the simple
recursive relation.
∂tp(N, t) = k+(N−1)p(N−1, t) + k−(N+ 1)p(N+ 1, t)
−k+(N)p(N, t)−k−(N)p(N, t)(6.5)
where k+(N) and k−(N) are the gain and loss rates of the defects. In a asymptotic
regime, ∂tp(N, t) = 0, equation (6.5) yields a simple recursive relation for the proba-
bility p(N)
p(N) = k+(N−1)
k−(N)p(N−1) (6.6)
where k+(N) = C0+E0,
k−(N) = D01N2+ (D02 +L0)N
where N denotes the number of positive and negative defects N = N±.C0,E0,L0,D0
are the creation, entering, leaving, and decay rates respectively.
By performing the recursion and normalization the distribution, a modified distribu-
tion is found,
p(N) = γ(ν/2+N)
Iν(2√γ)Γ(1 + ν+N)N!(6.7)
where Iνis the modified bessel function , γ= (Co+E0)/D01, and ν= (D02 +L0)/D01.
100 CHAPTER 6. DEFECT MEDIATED TURBULENCE
Figure 6.1: Defect mediated turbulence with increasing order of CO pressure. (a) ,
(b), (c) are PEEM images of size 300×300 µm2and, (d), (e), and (f) are the respective
phase patterns. The reaction parameters are: T = 534 K, po2= 1.58 ×10−4mbar, pco
= (a) 5.85×10−5mbar, (b) 5.91×10−5mbar, and (c) 6.05×10−5mbar
.
6.1 Method
The snapshots of the PEEM images shown in Fig. 6.1 (a), (b), and (c) are 300 ×300
µm2in size. Prior to the actual characterization of turbulent dynamics, standard
image processing techniques were applied to enhance contrast and minimize the level of
experimental noise. The PEEM images were denoised by application of a 3×3 median
filter and finally, a Butterworth high pass filter of order seven with a frequency cutoff
at k = 1 was applied to eliminate large scale modulations in the illumination of the
PEEM images. The chemical turbulence in oscillatory system with increasing order of
CO pressure is analyzed based on a phase and amplitude method.
The phase and amplitude variables are computed from the experimental data by em-
6.1. METHOD 101
Figure 6.2: Number of negatively charged defects N−as a function of time (sec.) with
increasing order of CO pressure. The mean over all series N+,−(t) in (a) 4.58, (b)
5.62, and (c) 8.23.
ploying a variant of the analytic signal approach. This method is used to transform
sequences of experimental PEEM images into time-dependent spatial distributions of
phase and amplitude variables. For the local PEEM intensity I(x, t) at an observation
point x, its Hilbert transform,
ζ(x, t) = I(x, t) + i¯
I(x, t)
where
¯
I(x, t)=1
πR∞
−∞(t−t0)−1I(x, t0)dt0
is the Hilbert transform of I(x, t). The time-dependent spatial distributions of phase
φ(x, t) and amplitude R(x, t) were determined from the analytic signal. The phase
was directly computed as φ= arg ξ, thus representing the polar angle in the plane
spanned by the variables Iand ¯
I.
102 CHAPTER 6. DEFECT MEDIATED TURBULENCE
Figure 6.3: Probability distribution function (PDF) of number of defects (N) computed
from the all time series N+,−(t) and the modified Poisson distribution (open circles).
The amplitude was defined as R=ρ/ρref (φ), where ρis the standard definition of the
amplitude modulus within the analytic signal approach.
6.2 Experimental Results
Fig. 6.1 (d), (e), and (f) are the phase representations of the PEEM image, shown
in (a), (b), and (c), with increasing order CO pressure. In PEEM images and phase
patterns one can see that as the CO pressure increases the patterns became more
chaotic.
Fig. 6.2 (a), (b), and (c) show the number of negative defects as a function of time
with increasing pressure of CO. A steadily increasing average number of defects and
growing fluctuation indicate that as the CO pressure increases the system becomes
6.2. EXPERIMENTAL RESULTS 103
Figure 6.4: Creation rates averaged over the rates for both positive and negative
defects. The measured rates are fitted with (a) 2.42, (b) 3.52, and (c) 4.93.
Figure 6.5: Entering rates averaged over the rates for both positive and negative
defects with increasing order of CO pressure. The measured are fitted with (a) 0.066,
(b) 0.19, and (c) 0.29.
104 CHAPTER 6. DEFECT MEDIATED TURBULENCE
Figure 6.6: Decay rates averaged over the rates for both positive and negative defects
with increasing order of CO pressure the measured rates are fitted with (a) D01(N2)
=0.016, D02(N) = 0.45, (b) D01(N2) = 0.017, D02(N) = 0.46, (c) D01(N2) = 0.014,
D02(N) = 0.44.
Figure 6.7: Leaving rates averaged over the rates for both negative and positive
defects. The measured rates are fitted with (a) 0.024, (b) 0.025, and (c) 0.035
6.3. CONCLUSION 105
more turbulent. The mean value of Fig. 6.2 (a), (b), and (c) are 4.58, 5.62, 8.23
obtained for the number of positive/ negative defects respectively. Fig. 6.3 (a), (b),
and (c) display the PDF for the number of defects (N) in area of 50 ×50 µm2with
increasing order of CO pressure.
The resulting gain (creation and entering) and loss (decay and leaving) rates are shown
in Fig. 6.4, 6.5, 6.6, and 6.7 respectively. This is the effect due to the dependence of
topological defects. The CO pressure is anticipated to have increased the amount of
phase instability in the system which is responsible for the creation of defects. The
phase gradient under the action of stochastic fluctuations of CO pressure has led to a
faster pinching of the equiphases. The relaxation of the field leads thereby to a faster
creation of defects (Fig. 6.4).
The modified Poisson distribution shown in Fig. 6.3 (a), (b) , and (c) for the values
(open circle) with increasing order of the CO pressure. The modified Poisson distri-
bution approximately match with the experimental data (bars).The modified Poisson
distribution obtained by performing the recursion of equation (6.6), shows the well
agreement with experimental data from Fig. 6.4, 6.5, 6.6, and 6.7. The modified
Poisson distribution correctly captures the mean and width of the PDF. In Fig. 6.3
the modified Poisson distribution is shown for the values in (a) γ= 150.70, ν= 28.75,
(b) γ= 218.70, ν= 28.52, and (c) γ= 373.07, ν= 33.92 respectively.
6.3 Conclusion
Like in Beta et. al. [131], it was found that the creation and entering rates were
constant, leaving rates increase linearly, and the decay rates increase in linear and
quadratic way.
The creation and entering rates of defects, which is approximately a constant, is raised
to a higher level as the CO pressure increases. The decay and leaving rate of the defects
is also increased by the CO pressure.
Under the influence of CO pressure, the probability distributions of defects are flat-
tened more and more as the CO pressure increases, and can be fitted with the modified-
106 CHAPTER 6. DEFECT MEDIATED TURBULENCE
Poisson distribution also found theoretically in modified FHN model under the influ-
ence of noise, however they found that the decay rate depended on N linearly [137].
To summarize, it has been shown that the dynamics of defects in defect-mediated
turbulence driven by CO pressure can be described by a simple statistical model. Like
noise, CO pressure has the ability to create defects in the turbulent background with
a constant rate [138], while at the same time destroying the existing defects at a rate
that is best approximated by adding a linear contribution to the quadratic annihilation
term.
6.4 Appendix: PDF of Topological Defects
The following Appendix includes a detailed derivation of the PDF of the number of
topological defects. Based on the gain and loss rates of defects, k+(N) and k−(N), the
master equation for the probability p(N, t) reads
∂tp(N, t) = k+(N−1)p(N−1, t) + k−(N+ 1)p(N+ 1, t)
−k+(N)p(N, t)−k−(N)p(N, t)(6.8)
In the asymptotic regime, ∂tp(N, t) = 0 transforms into a recursive relation for the
probability p(N),
p(N) = k+(N−1)
k−(N)p(N−1) (6.9)
The gain and loss rates are approximated by the following expressions
k+(N) = C0+E0(6.10)
k−(N) = D01N2+ (D02 +l0)N(6.11)
6.4. APPENDIX: PDF OF TOPOLOGICAL DEFECTS 107
putting 6.10 and 6.11 in equation 6.9
p(N) = C0+E0
D01N2+ (D02 +l0)Np(N−1) (6.12)
p(N) = γ
N(N+νp(N−1) (6.13)
where γ=C0+E0
D01 and ν=D02+L0
D01
the above equation can be further expanded to
p(N) = p(0)γN
N!
N
Y
k=1
1
k+ν(6.14)
using product of the components of mathematic series
a1.a2.a3............aN=dNΓ(a1
d+N)
Γ(a1
d)(6.15)
where a1is first term of the series with total N terms and d is common difference,
equation 6.14 leads to
p(N) = γN
N!
Γ(1 + ν)
Γ(1 + ν+N)p(0) (6.16)
Since p(N) is a probability, normalization leads to
∞
X
N=0
p(N) = 1 (6.17)
gives
108 CHAPTER 6. DEFECT MEDIATED TURBULENCE
p(0) = 1/
∞
X
N=0
γNΓ(1 + ν)
Γ(1 + ν+N)(6.18)
using the value of p(0) and modified Bessel function of the first kind
Iν(z) = (z/2)ν
∞
X
N=0
(z2/4)ν
N!Γ(1 + ν+N)(6.19)
in equation 6.16
p(N) = γ(ν/2+N)
Iν(2√γ)Γ(1 + ν+N)N!(6.20)
Chapter 7
Summary and Outlook
Spatially extended systems are known to exhibit spatiotemporal pattern formation
including oscillations, spirals, chemical waves and turbulence. The present work is de-
voted to giving deeper insight into the nature of chemical turbulence in catalytic CO
oxidation on a Pt(110) single crystal surface. The reaction is a well known example
of an extended system.
A focus of the work is on periodic forcing in order to control defect mediated turbulence
which has been studied experimentally and numerically by using the KEE model.
It is found that resonance forcing allows turbulence in the considered system, and it
can be successfully used as a tool to produce various complex patterns, but depending
on the forcing frequency, period doubling cascades to chaos are also observed.
The mean frequency of the turbulent state may differ strongly from the system ex-
hibiting homogeneous oscillations. Due to experimental noise, homogeneous oscilla-
tions are unstable, but can be simulated numerically. In contrast to the nonturbulent
state, where the natural frequency of the single oscillator is identical with the oscilla-
tion frequency of the extended system, in the turbulent state the system oscillates at
109
110 CHAPTER 7. SUMMARY AND OUTLOOK
higher frequencies due to diffusive interaction of the surface elements.
This is important for the definition of resonance as the generic frequency, experimen-
tally determined as the mean oscillation frequency average over a certain area of the
surface, should not be set equal to the single oscillators generic frequency, as it is
sometimes found in theoretical publications [126]. Furthermore, it implies that the
generic frequency in the turbulent state is more of a statistical measure than a sharply
defined value. Therefore a resonance forcing frequency is used, which is in vicinity of
the measured generic frequency or a harmonic of it.
The present work covers the range of 2:1 to 4:1 harmonic forcing. The following main
results are found experimentally. Under 2:1 resonances forcing two different types
of cluster patterns have been identified: amplitude clusters and phase clusters. In
contrast to phase clusters, that just differ in their oscillation phase relative to the (sub
harmonic) forcing frequency and belong to the same limit cycle, amplitude clusters
indicate the coexistence of limit cycles of equal periods but different amplitudes. In
both cluster types, the phase difference between the two cluster domains was π. The
patterns did not show the property of phase balance, i.e., the total fractions of the
medium occupied by the domains of the different clusters were different. However,
phase balance is not expected in a small area of the surface as it is a global property.
Stationary cluster walls could not observed probably due to the non-harmonicity of
the CO oxidation. The path to chaos is given by a period doubling cascade, which
could be experimentally followed by a subsequent increase of the forcing amplitude.
With the help of the new compressor, the regimes of 3:1 and 4:1 resonance forcing
were successfully reached. However, the cluster formation takes place in finite regions
of the surface, while other parts appear not to be 3:1 and 4:1 entrained, but still show
turbulent behavior.
Numerical simulations of the KEE model support the experimental findings, and give
further insight into the nature of catalytic CO oxidation. The resonant forcing is
applied to the system according to the natural frequency of an extended system. Un-
der 2:1 resonance which is in contrast to the experiments, phase clusters were found
rather than amplitude clusters. Interestingly, labyrinthine patterns were found at 8:1
entrainment. The transition between two phase cluster states and the labyrinthine
111
pattern is induced by phase instabilities within the cluster boundary.
In 3:1 resonance forcing, the bistability between 2:1 and 3:1 could not be observed by
applying the forcing according to the extended system.
At 4:1 periodic forcing, forcing with the 4th harmonic of the natural frequency of the
single oscillator leads to 3:1 entrained three phase patterns, 4:1 entrainment could be
obtained by forcing with the 4th harmonic of the mean oscillation frequency. Weak
detuning leads to the appearance of turbulent regions, similar to the experimental
results.
However, it was found that the greater the difference between the two characteristic
frequencies (single and extended system), the more difficult it is to entrain both turbu-
lent and phase-clustered regions. A necessary condition for entrainment of the phase
clusters is that the forcing frequency lie within the respective Arnold tongue of the sin-
gle oscillator. As the Arnold’s tongues generally become smaller for higher harmonics,
the effect of differing characteristic frequencies became more and more pronounced by
forcing the system with higher frequencies. In 2:1 forcing with the natural frequency
of the single oscillator leads to 2:1 entrainment, 3:1 forcing resulted in bistability of
2:1 and 3:1 entrainment [45]. 4:1 forcing finally resulted in 3:1 entrainment, while 4:1
entrainment could not be obtained.
Further investigation of high-frequency forcing of turbulent reaction-diffusion systems
may give new insight into the nature of turbulence and may lead to new strategies for
controlling chaos.
Further studies on cluster patterns are performed on 4:1 forcing within the nonturbu-
lent system.
The 4:1 resonance forcing in nonturbulent regime shows the two phase traveling phase
cluster experimentally. Numerical simulations of the KEE model in the nonturbu-
lent regime reproduce stationary four phase clusters as well as two phase clusters at
higher amplitudes, while other models namely the CGLE, FHN show traveling four
phase clusters and stationary two phase clusters at low and high forcing amplitudes
respectively [39, 78].
Finally, the spiral wave turbulence in catalytic CO oxidation was statistically charac-
112 CHAPTER 7. SUMMARY AND OUTLOOK
terized, with increasing order of CO pressure only experimentally. By using a Hilbert
transform, the experimental data was translated into phase and amplitude variables.
In this representation, topological defects could be identified at higher CO pressure
defining a measure for the strength of turbulence.
The temporal fluctuations in the number of defects were characterized in terms of
statistical moments and probability distribution functions. On the basis of the gain
and loss rates of defects, a probabilistic model was derived that yields a good ap-
proximation of the experimental results. As the CO pressure increases the probability
distribution of the defect are flattened more and more and can be fitted with the mod-
ified distribution function also found in FHN and R¨ossler models as the noise intensity
increases.
In summary, once again catalytic CO oxidation turns out to be one of the most power-
ful nonlinear model systems, where many effects predicted by nonlinear theory can be
observed experimentally and reproduced numerically. The system allows for switching
between distinct spatiotemporal chaotic states by tuning an easily accessible experi-
mental parameter. Investigation of the complex nature of reaction-diffusion systems
- as this work is a part of - may lead to improved strategies for control of extended
nonlinear systems.
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Publications
•P. S. Bodega, P. Kaira, C. Beta, D. Krefting, D. Bauer, B. Mirwald-Schulz, C.
Punckt, and H. H. Rotermund. High frequency periodic forcing of the oscillatory
catalytic CO oxidation on Pt(110). New Journal of Physics 9 (2007) 61.
•P. Kaira, P. S. Bodega, C. Punckt, H. H. Rotermund, and D. Krefting. Pattern
formation in 4:1 resonance of the periodically forced CO oxidation on Pt(110).
Physical Review E 77 (2008) 046106.
•D. Krefting, P. Kaira, and H. H. Rotermund. Period doubling and spatiotem-
poral chaos in periodically forced CO oxidation on Pt(110). Physical Review
letter, submitted.
•P. Kaira, D. Krefting, C. Beta, and H. H. Rotermund. Effect of CO pressure
on defect-mediated turbulence in a catalytic surface reaction. New Journal of
Physics, in preparation.
•P. Kaira, D. Krefting, P. S. Bodega, C. Punckt, and H. H. Rotermund. Pat-
tern formation in 3:1 resonance of the periodically forced surface reaction. In
preparation.
•C. Punckt, P. S. Bodega, P. Kaira, and H. H. Rotermund. Forest fires in the lab:
experiments, simple modeling, and relation to other phenomena. In preparation.
127
128 BIBLIOGRAPHY
Acknowledgements
First of all, I would like to thank my supervisor Prof. Harm. H. Rotermund for giving
me an opportunity to make this work within his group at the Fritz-Haber-Institute,
for encouraging and supporting me throughout my PhD. His ingenuity, enthusiasm
and extreme kindness will always be in my memory. I am truly honoured that I could
work in his outstanding group.
I would also like to acknowledge Prof. Gerhard Ertl, who provided me ideal conditions
to carry out my PhD work in his department.
I want to thank Prof. Harald Engel, for taking on the responsibility of being my second
supervisor at the Technische Universit¨at Berlin.
I would also like to thank Dr. Dagmar Krefting for her support and countless discus-
sions in the frames of this research work and outside, and for spending many hours to
carefully read and correct my thesis.
Dieter Bauer, for his friendly help and assistance in the experiments, numerous tech-
nical improvements and many helpful comments.
Prof. Dr. Alexander S. Mikhailov, Dr. Sergio Alonso, Dr. Carsten Beta, and Dr.
Oliver Rudzick for helpful discussions.
I am very grateful to my colleagues and friends, especially to Pablo S´anchez Bodega,
Christian Punckt, Willi Krauss, Dai Zhang, Katrin Domke, Monika Dornhege, Pablo
Kaluza, Peter Klages, Yuichi Togashi, Mirwald-Schulz Birgit, Santiago Gil, Michael
Stich, and Waruno Mahdi.
This work would not have been possible without the support and encouragement from
129
130 BIBLIOGRAPHY
my husband, Pushkar Singh. He has been extremely patient, understanding, and
loving throughout my research.
Finally, I would like to thank my family for their unconditional support.
