Scientific Results of the Visits

supported by the REACTOR Programme

(2000-2004)

Most of the scientific results of the REACTOR Programme were achieved in the visits supported and/or initiated by this project. Thus the aim of this summary is to report the results of these visits.

1. Table of the various research fields within the Programme

indicating the sequential numbers of the visits in a particular field and the author’s name who wrote the present brief report about those visits

2. Summary of the results

A Spatiotemporal pattern formation and control in novel open reactors

A 1. Chemical patterns and control in novel reactors

A 1.1. Chemical waves

(Visits 3, 14, 21, 22, 28, 44, 64, 69, 72, 76, 101)

One of the main research topics of the REACTOR Programme was studying the dynamics of excitable media. The motivation for this interest originated from the observation that excitable or active media occur widely in biological systems. Important examples include cardiac muscle and neuronal tissue. Methods for studying biological systems in vivo are developing, but in many cases it is easier to determine generic features by investigating the much simpler reaction-diffusion systems, e.g. the widely used Belousov-Zhabotinsky (BZ)-reaction. Several groups in the REACTOR Programme carried out such studies both in experiments and modelling.

Excitation waves in homogeneous media

Dilao (Lisbon) and Volford (Budapest) extended the investigation of excitability to a new class of reaction-diffusion systems, namely, to a model with a Hopf and saddle-node homoclinic (SNH) bifurcations [2004_10]. The generic model presents all the known types of topologically distinct active waves observed in experiments: solitary pulses, travelling waves and chemical turbulence even for equal diffusion coefficients of activator and inhibitor species.

Experimental and numerical works have been initiated by Gáspár (Debrecen) and Scott & Taylor (Leeds) to analyse re-entrant waves in a ring or annulus of excitable BZ-media. The motivation for this research is the following. It is known that re-entrant waves (responsible for an irregular cardiac rhythm) can be terminated by rapid stimulation (pacing) from an implantable unit. However, the mechanism of such termination is poorly understood. A special reactor utilizing the light-sensitive BZ-medium has been built in Leeds and shall be used in later experiments.

Excitation waves under external forcing

Excitation waves carry chemical signal over different ranges of a spatially extended system. It has been known from earlier experiments that signal transmission can give rise to resonant patterns. Tóth (Debrecen) and Müller (Magdeburg) started to investigate the effect of external forcing on such signal transmission by using the light-sensitive BZ-medium.

Excitation waves in inhomogeneous media

Resonance patterns of signal transmission through one passive barrier have been studied by Sielewiesuk & Gorecki (Poland) and in multiple gap system by Armstrong et al. (Leeds) and Gáspár (Debrecen). In these studies, features such as critical gap width W_cr, critical spacing between gaps S_cr and frequency transformation of the passage of a train of waves across a gap with width W  W_cr have been observed and analysed [2004_3]. It has been also found that the probability of a wave successfully propagating through the entire domain decreases with increasing number of gaps.

Chemical waves in sub-excitable medium

At low excitability wave segments with open ends occur. Depending on the length of these segments on the excitability of the medium, the segments may shrink or grow at their open ends in tangential direction. Bonyhádi (Szeged) and Müller (Magdeburg) investigated the effect of temperature gradient on propagation of sub-excitable BZ-waves in two configurations: wave segments a) perpendicular and b) parallel to the temperature gradient. A special reactor with temperature controlled compartments were built and applied.

Excitation waves in 3D systems

Alonso (Barcelona) and Mikhailov (Berlin) numerically studied the dynamics of scroll waves and filaments in three dimensional medium of weak excitability [2003_10]. To control the shrinking and growing of the scroll ring, they introduced diverse perturbations: periodic forcing, parameter gradient, feedback and electric field. The experimental possibilities for the control of these structures for each forcing were discussed and evaluated.

Pattern formation in reaction-diffusion-convection systems

In a series of papers ([2001_4], [2001_6], [2002_6]), Scott et al. (Leeds) and Gáspár et al. (Debrecen) studied the dynamics of excitable chemical waves in plug-flow reactors. In these reaction-diffusion-convection systems, both stable patterns and moving wave fronts have been observed. They showed that flow-distributed patterns may appear with a so-called wave-splitting mechanism not observed before. Travnickova (Prague) and Merkin (Leeds) extended these studies to non-isothermal conditions involving exothermic reaction of first order [2005_11]. Attention has been focused on the appearance of oscillatory and complex (chaotic) patterns as a function of convection velocity.

Dewel (Brussel) and Sörensen (Copenhagen) studied the effects of various boundary perturbations on convectively unstable states in open flow reactors and discussed the relevance of these works to biological systems

A 1.2. Other chemical patterns:
Liesegang, Langmuir monolayer, mechanochemical and Turing patterns

(Visits 39, 54, 74, 78, 83, 96)

Investigations on various new complex systems supporting pattern formation were among the most important aims of the REACTOR Programme. These systems include precipitation reactions (Liesegang type phenomena), Langmuir monolayers of azobenzene on the water surface, and systems where (bio)chemical reactions cause mechanical deformations of the support medium.

The collaborative research of precipitation patterns, realized through the visit of Hantz (Budapest) to Ratcliffe (Bristol) was focused on the application of Voronoi diagrams to evolving patterns of precipitates and resulted into publication [2004_14]. Another type of Liesegang phenomenon was studied in AOT-microemulsion based gel by Parisi (Palermo) during his visit in Magdeburg.

The dynamics of Langmuir monolayers of an azobenzene derivative was studied by Brewster angle microscopy (BAM). The films, prepared by depositing drops of a spreading solution of a photostationary mixture of cis- and trans-isomer of this azobenzene derivative on the surface of pure water, were formed by circular domains rich in the trans-isomer embedded in an isotropic phase rich in cis-isomers after the evaporation of the solvent. Experiments performed during the visit of Bánsági (Szeged) to Sagues (Barcelona) studied how these patterns are affected by the concentration of various cations in the water phase.

The visit of Panfilov (Utrecht) to Mikhailov (Berlin) focused on developing new methods of mathematical modelling for pattern formation in chemical and biological media that can undergo, in response to (bio)chemical reactions, mechanical deformations which, in turn, can affect the pattern’s characteristics.

J. Tóth (Budapest) visited University of Leeds and City University London to apply numerical methods of repromodelling and neural networks for a variety of lumped problems as combustion, reactions in the atmosphere. Another aim of the visit was to discuss simulations of Turing structures.

A collaborative research on pattern formation in a novel, one-side-fed open gel reactor with bistable pH-activated reactions has been further developed via the visit of Szalai (Budapest) to De Kepper (Bordeaux). A manuscript on the effects of the reduced-mobility proton-binding agent on spatiotemporal dynamics in the chlorite-tetrathionate reaction was prepared [2005_14] and the further collaboration focused on iodate-sulfite reaction system designed.

A 1.2. Other chemical patterns: isothermal flame balls

(Visit 62)

In three spatial dimensions, steady spherically-symmetric wave fronts exist in flames called flame balls and under isothermal conditions as reaction balls for autocatalysis with order greater than six [2002_13]. The solutions are unstable and represent a threshold between front propagation and failure of front initiation. These unstable structures can be stabilized by autocatalyst or heat removal [2003_2]. In the cooperation between Leeds and Budapest, the effect of heat loss is investigated in flame quenching, while in that between Leeds and Szeged, the effect of the order of removal has been studied both theoretically and numerically. It has been shown that there is a critical decay rate above which there is no spherically-symmetric solution, below which there are two (isothermal) flame balls with different radii. The smaller balls are always unstable, while the larger ones may become stable depending on the ratio of the diffusion coefficients of the reactant and the autocatalyst making the experimental realization feasible.

A 1.2. Other chemical patterns:
Influence of oxygen on wave propagation in the BZ solution

(Visits 8, 31)

Propagation of chemical waves has been often studied in the ferroin-catalysed Belousov-Zhabotinsky reaction which is known to be sensitive to oxygen. Bamforth (Leeds) and Müller (Magdeburg) determined the vertical distribution of oxygen in a BZ solution that supported a rotating, spiral-shaped wave. Interestingly, the vertical oxygen gradient is not uniform, showing pronounced differences in oxygen content between the top 0.5 mm of the solution and the bulk of the BZ solution (ranging from 0.5 to 1.5 mm depth). These findings are interpreted as due to oxygen-induced hydrodynamic flows in the top layers of the liquid.

A subsequent study by Hauser (Magdeburg) on the transport of oxygen into shallow layers of diluted sulphuric acid (the "solvent" of the BZ reaction) confirms the involvement of hydrodynamic phenomena in these systems [S23].

A 1.3. Electric field effects

(Visits 29, 35, 48, 55)

The long term experience with experimental studies of electric field (EF) effects on chemical waves was further utilized through regular visits (both within and outside ESF Programme) of Sevcikova (Prague) to Müller (Magdeburg) where she supervised the doctorate student Pornprompanya. His novel research, focused on the EF effects on chemical waves in the system with two internally coupled autocatalytic pathways, resulted into two publications [2002_7, 2003_10] and the successful defence of the PhD thesis by Pornprompanya (on 28.1.2005).

The novelty of the results obtained includes, e.g., the saturation dependence of the wave velocity on the electric field intensity and the discontinuous range of electric field intensities causing the wave annihilation. These results document that the behaviour of the system with two, internally coupled autocatalytic pathways is not a simple sum of the two composing pathways and increases the nonlinearity of the system.

The modelling approach to reaction-diffusion-electrotransportation ionic systems, developed alongside the experimental work in Prague, has been disseminated to simulations of capillary electrochromatographic (CEC) processes through the visit of Paces (Prague) to Tallarek (Otto-von-Guericke University, Magdeburg) and Seidel-Morgenstern (Max Planck Inst., Magdeburg) [2003_11]. This month visit under ESF programme initiated a year stay of Paces (Prague) in Magdeburg (Otto-von-Guericke University) where he worked with Tallarek (Magdeburg) on the modelling of reaction-adsorption-transport processes in capillary electrochromatography [S12, S13].

Their modelling approach focused on including the interdependence of adsorption of charged analytes and interfacial electrokinetic phenomena into the mathematical description of CEC columns. They showed that their model describes more realistically the dynamic behaviour of analyte adsorption and migration in the CEC systems and that it applies to a broader range of experimental conditions. The model can be well utilized in optimization strategies in CEC.

A 1.4. Electrolyte diodes

(Visits 2, 37, 47, 57, 93)

Research in the field of the so-called electrolyte or acid-base diodes and transistors started a few years ago in Budapest (L. Hegedűs, N. Kirschner, M. Wittmann and Z. Noszticzius, J. Phys. Chem. A 1998, 102, 6491; L. Hegedűs, N. Kirschner, M. Wittmann, P. L. Simon, Z. Noszticzius, T. Amemiya, T. Ohmori and T. Yamaguchi, CHAOS, 1999, 9, 283). In the framework of REACTOR Programme a cooperation was developed with Leeds in mathematical modeling (publications [2000_3, 2000_4 and 2002_2]), with Prague in numerical modeling and experiments (publication [2002_7], manuscript [2005_6]) and in nonlinear theory with Marburg (publication [2004_ 9]).

The “building block” of these devices is an acid-base reaction-diffusion system embedded in a hydrogel medium. In the experiments polyvinyl alcohol based hydrogel cylinders were applied connecting aqueous solutions of potassium hydroxide and hydrochloric acid. In the forward biased diode potassium and chloride ions migrate into the gel under the influence of the applied electric field and form a well conducting KCl solution there. In the reverse direction the hydroxyl and hydrogen ions migrate into the gel cylinder where they recombine. As a result a thin layer of pure water is formed whose conductivity is very low.

Main results:

Experimental: a new method was developed to determine fixed ionizable groups in weakly charged hydrogels, various new electrolyte diodes were constructed, and recently it was shown that application of a weak acid and base (instead of strong ones) has major advantages.

Numerical: various new softwares (e.g. the IonLab in Prague) were developed to model reaction – diffusion – ionic migration systems, especially with narrow acid-base boundaries, where the condition of electro-neutrality fails and where ordinary numerical methods also fail.

Theoretical: new analytical formulas were developed which give good approximate solutions under various conditions. These results can help numerical calculations and especially to design new electrolyte devices.

A 2. Biological patterns

(Visits 7, 42, 100)

The research of biological patterns, namely the waves of chemoattractant cAMP in the aggregating population of the slime mold Dictyostelium discoideum, has been facilitated by the visits of Forstova and Sevcikova (Prague) to Müller and Mair (Magdeburg), and Hilgardt (Magdeburg) to Prague (funded from other sources).

During the first visit, Forstova learned to cultivate and store the specimen, to prepare samples of cell population for experiments and set-up the apparatus for the dark-field observation of propagating waves. The knowledge gained enabled to start own experimentation in Prague, which supplies the data for the own theoretical research based on application of nonlinear dynamics. Numerical simulations of appropriate mathematical models were used to explain and understand the experimentally observed phenomena, namely propagating fronts of vegetative cells on bacterial lawns, the time course of aggregation centers’ formation, and the formation of large aggregation nests and their disintegration [S14, S15].

Taylor (Leeds) visited to Prague to study and discuss the possibilities establishing biology oriented nonlinear dynamics lab. After a critical evaluation, however, it was found that the research of biological systems presently would be too demanding especially due to initial investments to the equipment because of the strict requirements of sterile work.

In 2002, Hilgardt (a new student in Magdeburg) visited to Prague to learn refined experimentation methods and microscopic observation techniques from Forstova. The research in both laboratories then focused on the ways of controlling cAMP waves propagation and cell aggregation by external forcing. Forcing by applied electric fields (performed in Prague) was found to affect not only the velocity of propagating cAMP waves but also the emergence of new centers of aggregation [S19]. The possible effects on the specimen development are the matter of further research. The control by chemical agents has been pursued in Magdeburg. Through the visit of Vankova (Prague) to Hilgardt (Magdeburg), this research lately centered around the effects of external cAMP. The exposure of cells to cAMP is widely used when the chemotactic competence of cells is to be investigated. However, the findings of the joint research show that the increased ability of cells to chemotax immensely changes the cAMP waves and aggregation patterns, which brings a new view on the developmental role of these patterns. The joint publication is currently under preparation.

A 3. Convective patterns I.

(Visits 16, 18, 30, 32, 38, 45, 53, 65, 84, 102, 103, 107)

Most studies of reaction-diffusion dynamics are performed in gels to avoid any perturbing hydrodynamic motions. In the framework of the REACTOR network, several groups have collaborated to unravel the new spatiotemporal dynamics which occur in solutions and result from the coupling between autocatalytic reactions, diffusion and hydrodynamic flows.

Fingering of chemical fronts

When the density of a solution changes in the course of an autocatalytic reaction, the system becomes hydrodynamically unstable due to a Rayleigh-Taylor instability (also called a “density fingering” instability) if the denser solution is placed on top of the lighter one in the gravity field. The density may vary in the course of the reaction because of the compositional change or of the thermal expansion of the solution. After numerous studies of the problem in capillary tubes in the ‘80s and ‘90s, recent developments have focused on experiments on fingering of chemical fronts in spatially extended Hele-Shaw cells (2 glass plates separated by a small gap width less than 1 to 2 mm). This quasi-2D geometry allows to have insight into the competition between numerous unstable spatial modes. The chemo-hydrodynamic coupling is then characterized by both linear stability analysis and by the nonlinear dynamics of the fingers. Two important autocatalytic reactions are being studied experimentally. i) In the iodate-arsenous acid (IAA) reaction –analyzed in Magdeburg– the density decreases in the course of the reaction. As a consequence, downward propagating planar fronts are stable while upward propagating planar fronts lose stability and develop chemo-hydrodynamic cellular patterns. ii) Opposite to that, in the chlorite-tetrathionate (CT) reaction –studied experimentally in Szeged– the products are heavier than the reactants resulting in a stable upward propagating planar front and a downward evolving fingering structure.

The REACTOR network has allowed to initiate close collaborations on this subject between various experimental (Magdeburg, Szeged, Prague) and theoretical (Brussels, Leeds) groups. First of all, workshops and visits supported by the reactor network have allowed people from Brussels and from Magdeburg (visits 103, 107) to discuss the properties of the IAA experimental system and to set up a theoretical model for the evolution of iodide coupled through advection to Darcy’s law for the evolution of the flow field. In parallel, visits between the Szeged and Brussels teams (visits 18, 84, 102) have been the building block for similar discussions around the CT reaction for which a two-variable model (tetrathionate ions and protons) coupled to Darcy’s law has been constructed in collaboration by the Leeds and Brussels groups (visits 16, 30, 45, 38, 53). Discussions and visits between the experimentalists (visit 32) have also allowed to compare the similarities and differences between the IAA and CT systems. Further links between the Leeds, Prague and Brussels groups (visits 53, 65) have generated recent studies of the influence of the electric field on the fingering of IAA fronts.

As a result of this network of exchanges, the fingering instability of chemical fronts has been successfully characterized experimentally and theoretically in isothermal conditions both for the IAA and CT systems by the dispersion relation describing the growth rate of a given spatial mode as a function of its wave number providing a quantitative description of the fingers emerging at early times [2002_3], [S7], [2003_1], [S17]. Nonlinear dynamics have next been characterized both for the IAA [S18], [S19] and CT [2002_3], [2004_13], [S22] reactions.

These reactions are exothermic with a heat of reaction for the CT system one order of magnitude greater than for the IAA reaction which suggests that heat can have a significant effect on the fingering pattern formation. It has been shown in the CT reaction that thermal effects may indeed destabilize both upward and downward propagating fronts and lead to a profound qualitative change in the dispersion relation describing the initial evolution of fingers [2003_3], [2004_4], [2004_6]. Current collaboration involves studies of the evolution of the flow field [S19], of heat removal through the plates and of heterogeneities in the gap width on the properties of the fingers.

Electric fields are known to affect the kinetics of autocatalytic fronts. As such, they affect the composition of the solutions and thus the density change across the front. Collaborative work performed in the REACTOR network has demonstrated experimentally and characterized theoretically the fact that electric fields can drastically affect the fingering pattern and in some cases destabilize fronts that are stable in the neutral case [2005_1].

A 3. Convective patterns II.

(Visits 17, 41)

It has been shown by the group in Sassari that the Belousov-Zhabotinsky reaction exhibits chaos in an unstirred batch reactor induced by convection. For the explanation, a collaboration between the groups in Sassari and Lyon has been initiated. The effect of convection is also significant in combustion reactions which are highly exothermic. The cooperation between Leeds and Lyon resulted in the detailed investigation of heat explosion with convection. The convection eliminates thermal explosion and stabilizes stationary structures because of the enhanced heat loss through the boundaries. Close to the onset of instability, however, oscillatory profiles, while for more exothermic reactions period doubling leading to chaotic oscillations are observed. At even stronger convection there exists a hysteresis between stationary and oscillatory regimes.

A 3. Convective patterns III.

(Visits 61, 63, 81)

The main scope of the above visits was to facilitate an exchange of ideas between the parties regarding research work on a novel class of chemical reactor systems that exhibit new and interesting pattern forming capabilities. The visits were initiated by a new pattern forming scenario suggested by Satnoianu and collaborators (see e.g. [2003_13]) termed FDS or flow and diffusion distributed waves. The 3 visits under this report have looked at 3 aspects under this topic.

i) Visit 61 of Satnoianu (London) to Schreiber (Prague). During this CONT, a numerical bifurcation package derived by Schreiber was used to simulate and numerically analyse a variety of pattern forming scenarios that appear in the class of reaction-diffusion-advection systems, which support FDS patterns.

ii) Visit 63 of Satnoianu to Munuzuri & Miguez (Santiago de Compostela). This visit concerned the possibility to devise a set of experiments for the experimental validation of the theoretically predicted FDS patterns. The visit resulted with a paper [2005_13], which is now under review at PRL.

iii) Visit 81 of Miguez (Santiago) to Satnoianu (London). The visit concerned further analysis and study of the experimental patterns seen from the experiments started in visit 63.

B Nonlinear kinetics and mechanisms: complex chemical and biochemical processes

B 1. Chemical mechanisms

B 1.1. BZ oscillators

(Visits 1, 10, 27, 46, 51, 85, 88)

Presently the classical version of the Belousov-Zhabotinsky (BZ) reaction, the cerium ion catalysed oxidation and bromination of malonic acid by acidic bromate, is the most important chemical oscillator. This is because of the rich nonlinear temporal and spatial phenomenology exhibited by this special system under various experimental conditions (see e.g. in Gray, P.; Scott, S. Chemical Oscillations and Instabilities. Nonlinear Chemical Kinetics. Clarendon: Oxford, 1994. Epstein, I. R.; Pojman, J. A. An Introduction to Nonlinear Chemical Dynamics. Oxford University Press: New York, 1998). In spite of this some important details of its mechanism were still not clear in the year of 2000. Thus in the framework of the REACTOR Programme an intense research was initiated to reveal these missing details. The first results were due to a REACTOR cooperation between Budapest and Marburg (publications [2000_1], [2000_2]). The Marburg-Budapest-Missoula (MBM) mechanism [2002_1] presented at the Faraday Discussions in 2001 incorporated all the information obtained until that time about the organic intermediates. Nevertheless some open questions still remained. One was the role of the oxalic acid in the mechanism, an intermediate found by the group in Budapest. In 2001 an additional REACTOR cooperation has started between Budapest and Palermo. First two alcoholic perturbants were applied, both small molecules like methanol and ethylene glycol [2002_8], then larger polymeric perturbants like polyethylene glycol [2002_9]. The aim of these studies was to test the MBM mechanism as the alcohol-bromate reaction generates bromous acid which is an autocatalytic intermediate of the BZ reaction. In a next step we studied similar perturbations by oxalic acid [2004_11] which also reacts with bromate to give bromous acid. These experiments revealed that oxalic acid plays an important role in the BZ mechanism thus our further research focused on a simplified BZ oscillator with oxalic acid as the only substrate ([2004_7], [2004_8]). One of the most important results is that we could identify a chain reaction in that system. Comparison of experimental and simulated curves suggests, however, that an important inorganic reaction is still missing from the mechanism. To find that reaction and to solve other problems of the mechanism research continues in the framework of the Budapest-Palermo cooperation initiated by the REACTOR Programme.

B 1.2. pH oscillators

(Visits 5, 9, 71)

Design and systematic study of pH-oscillators have been carried out and possibility of temperature-compensation of the oscillatory frequency has also been investigated at the University of Debrecen. in the frame of the REACTOR Programme.

Zagora (Prague) visited Debrecen and worked on the kinetics of the ferricyanide-sulfite reaction. This reaction appears to be an important composite reaction of some pH-oscillator systems. In order to be able to model those oscillators, it is necessary to know the kinetics of the ferricyanide-sulfite reaction. The work by Zagora has revealed that the reaction is a first order reaction with respect to both the sulfite and the bromate ions, and he found reciprocal dependence of the reaction rate on the concentration of the hydrogen ions. These results appear to be useful in modeling excitability of the bromate-sulfite-ferrocianide system [2002_5].

McIlwaine (Leeds) has spent 2 weeks in Debrecen. Her aim was to try and develop a new oscillatory system involving the formaldehyde-bisulfite reaction. During her stay she learnt how to use a CSTR and she was able to successfully reproduce oscillations in the iodate-sulfite-ferrocyanide and the iodate-thiosulfate-sulfite systems. Unfortunately, attempts at producing an oscillatory system involving the formaldehyde bisulfite reaction in a CSTR were unsuccessful in Debrecen. This work was continued in Leeds, where an oscillatory system was developed.

B 2. Fed-batch and cross-flow reactors

(Visits 5, 9, 71)

Fed-batch reactors

Prague and Leeds groups cooperated in research on the dynamics and control of oscillatory chemical systems in a novel reactor system – the fed-batch reactor (FBR) – through exchange visits 13, 40 and 43. The results were published in ref. [2004_11]. The prototype oscillating system, the Belousov-Zhabotinsky (BZ) was employed in this study. A spontaneously oscillatory BZ mixture in a well-stirred reactor is perturbed by replacement of part of the mixture with fresh reactants at regular intervals. This kind of discrete time periodic forcing defines a fed-batch reactor. Let R be the fraction of the reactor contents replaced by fresh feed solution at intervals of period T_f . The parameter plane T_f – R contains regions of mode-locked p/q regimes, where p oscillations occur within a time interval spanned by q replacements of fresh solution. Among others, 2/1, 5/2, 3/1 and 7/2 regimes were found and period doubling bifurcations indicated. In addition, quasiperiodic regimes were also found. In a different set of experiments the phase response method was used to determine phase transition curves (PTCs). These curves were then used to reconstruct dynamics of the periodically forced system.

By using a variety of modelling and continuation techniques, complex dynamic behaviour is predicted over a wide range of parameter values. It was found that the reversible Oregonator model (6 reactions, 4 variables) gives a fairly realistic description of experiments. The fed-batch regime is modelled as a pulsed system of ordinary differential equations. Periodic regimes implied by the forcing are traced to their boundaries of existence to determine a bifurcation diagram T_f – R. The calculated p/q oscillatory regimes correspond well to experiments including the period doubling bifurcation and follow patterns called Farey sequences. An independent method of predicting dynamics is obtained by the use of measured PTCs as the basis for a one-dimensional discrete-time dynamical system. The modelling predicts also chaotic dynamics, which was however not confirmed experimentally, probably due to narrow parameter ranges where it could be observed. The fed-batch operation is shown to have certain advantages when compared with a continuous operation.

Cross-flow reactors

Prague and Leeds groups also cooperated in research on spatiotemporal dynamics in a cross-flow reactor through exchange visit 64 with results to be published in 2005 [2005_10]. Effects of convection on stability of propagating waves and patterns in a reaction-diffusion-convection model involving exothermic reaction of first order are examined by means of analysis involving linearized stability, numerical continuation and direct integration of spatially discretized equations. The reaction takes place in a spatially one-dimensional cross-flow reactor with continuous supply through a semipermeable membrane. The model is formed by two equations for conversion and (dimensionless) temperature. They represent a nonadiabiatic heterogeneous catalytic reactor in the limit of a homogeneous distribution of reactant and temperature in the solid catalyst. Solutions of this model provide different types of spatiotemporal patterns depending on parameters. The main parameters are the dimensionless convection velocity and thermal diffusivity, heat and mass transport coefficients and Lewis number. The simplest types of observed patterns are pulse and front waves, but under certain conditions also complex dynamical regimes can develop such as zig-zag fronts and chaos. In the focus of the attention were oscillatory and complex spatiotemporal patterns and Lyapunov exponents and dimension of the corresponding attractor were determined. Explanation was provided also for the zig-zag fronts in terms of collision of stable and unstable front.

B 3. Biochemical mechanisms

(Visits 6, 11, 12, 19, 58, 70, 95 )

It has been known for some time that biochemical reaction systems display a multitude of nonlinear behaviours, ranging from bistability to oscillations to chaos and quasiperiodicity. Most of these behaviours have been observed in vitro, but more recently they have also been observed in vivo. One example is the oscillations of reactive oxygen species (ROS) such as superoxide, hydrogen peroxide and hypochlorous acid in neutrophils (white blood cells). Here oscillations in ROS are used to kill not only invading microorganisms, but also tumour cells. The mechanism underlying the inactivation of tumour cells may, in addition to ROS, involve calcium ions. The biochemical processes believed to be involved in the formation of ROS are glycolysis, the pentose phosphate shunt and the enzymes NADPH oxidase and myeloperoxidase. Most of the visits made in this category have been dealing with the mechanism of ROS formation, either through experimental investigations of the reactions catalysed by myeloperoxidase or by attempts to set up a biochemical model of the reaction network leading to ROS formation. A couple of visits have also been devoted to the study of oscillations of intracellular calcium concentration and their coupling to calcium signalling. The cooperation of more than six research groups has resulted in the formulation of a fairly detailed mechanistic model for the oscillating production of ROS in neutrophils. Furthermore, we may begin to understand the exchange of information between a neutrophil and a tumour cell. The model has resulted in several predictions that could be verified by experiments, e.g. that the pineal hormone melatonin is involved in the activation of ROS formation through interactions with myeloperoxidase, and that the metabolism of tumour cells may contribute to its own death. This dynamic model of tumour killing seems to be quite revolutionary in immunology and may lead to new strategies for cancer prevention and treatment. Publications: [2003_12], [2005_11], [2005_12], [S10], [S11], [S20].

B 4. Bioreactors

(Visits 23, 24, 36, 59, 60, 82, 83, 87)

Yeast is one of the industrially most exploited microorganisms, due to its use in the production of beer, wine and bread, in biocatalysis. Furthermore, commercially relevant products such as therapeutic proteins and enzymes can be expressed in yeast. Yet, it is still relatively little we know about the basic metabolism in yeast cells and its regulation. For example it is still an open question what controls sugar metabolism such as glycolysis in yeast cells and other microorganisms. It has been known for some time that glycolysis exhibits temporal oscillations in yeast cells fermenting glucose to ethanol. In a suspension of yeast cells provided with glucose under anaerobic conditions glycolysis in each cell oscillates and all cells in the suspension oscillate in phase! Thus, the cells must communicate by rapid exchange of one or more intermediates in glycolysis. Most of the exchange visits in the bioreactor category have been devoted to the study of the mechanism of glycolytic oscillations, the synchronization of the cells and the physiological relevance of the oscillations. It was found that oscillations in suspensions of cells are a collective property, and no evidence has yet been found for an oscillating glycolysis in a single isolated cell. This is probably due to a dilution of the synchronizing agent, for which the most likely candidate is acetaldehyde. It was found that adding pulses of acetaldehyde, at a frequency corresponding to the natural frequency of the glycolytic oscillations, to a suspension of non-oscillating yeast cells will restore the oscillations. Furthermore, it was found that glycolysis will oscillate under physiologically relevant conditions, but the purpose, if any, of the oscillations remains an unresolved question. These studies are very important for our understanding of the metabolism of yeast cells and may help to optimise the growth of these organisms in industrial reactors. A single visit was devoted to another important aspect of biotechnology, namely the modelling and optimisation of micro-bioreactors with immobilized enzymes. Such reactors may be used in small-scale industrial production of special pharmaceuticals. In order to optimise the production, mathematical models are very useful for obtaining guidelines for design and operating conditions.

B 5. Biomimetic systems

(Visits 20, 50, 56, 67, 68, 79, 90, 97, 108)

A further topic of research supported by the ESF programme REACTOR was the investigation of nonlinear dynamic behaviour in biomimetic systems, i.e. of systems that are composed of purely synthetic components which are specially designed to reproduce the most prominent mechanistic and stereochemical features of a given natural system. The biomimetic system studied in the frame of the REACTOR Programme is a reaction that mimics the natural, membrane-bound cytochrome P450 enzyme system.

The research visit (visit 20) of Hauser (Magdeburg) to the group of Turco Liveri (Palermo) marked the starting point of an intensive collaboration between the two groups in Palermo and Magdeburg. During this first visit, the location of the incorporation of the enzyme model compound into the phospholipid bilayer was determined experimentally. The lipophilic cytochrome P450 model compound was confirmed to be intercallated in the lipid domain of the membrane, however it is located in the proximity of the polar end of the phospholipids. This represents a refinement of the position of this key compound which previously was thought to be located deep in the lipid domain, close to the middle of the phospholipids bilayer.

Once the location of the main components was determined, the next task of the collaborative research consisted in elucidating the kinetics of the reactions underlying the biomimetic system. The visits of Guarrasi, Savani, and Turco-Liveri (Palermo – visits 50, 56, 79) to Magdeburg were devoted to this purpose, especially by making use of the diode-array spectrophotometer available in Magdeburg. The kinetics underlying the biomimetic cytochrome P450 was found to be complex. The most surprising result from the visits of Guarrasi and Savani to Magdeburg consisted in the observation that the phospholipid vesicles have a pronounced influence on the reaction kinetics: the kinetic laws for some of the elementary reactions differ in the presence and in the absence of phospholipids bilayers, suggesting that the membranes play a crucial role in the mechanism of the biomimetic system. These experiments provide evidence that the membranes act as catalytic surfaces, which may promote and enhance the reactions of the biomimetic system. First evidence points at the involvement of the phospholipid membrane in an autocatalytic elementary reaction, thus suggesting that the medium and/or the interfaces provided by membranes may be a potential source for the nonlinearity in the biomimetic cytochrome P450 system.

The visits of Turco Liveri (Palermo, visit 69, 108) to Magdeburg and that of Müller (Magdeburg) to Palermo were used for the discussion of the results obtained at the respective groups, as well as for the coordination of further research activities on the biomimetic cytochrome P450 system. The main results of these coordinating discussion visits are manifested by an on-going cooperation between the groups of Turco-Liveri (Palermo) and Hauser (Magdeburg). This is documented by a successful application for a bilateral travel grant for the period 2005 – 2006. This joint DAAD (Germany) – CRUI (Italy) grant provides support for visits aimed to continue the fruitful collaboration started under the ESF programme REACTOR.

Last but not least, the visits of Viscardi (Torino, visit 68) and Turco Liveri (Palermo, visit 69) to Magdeburg were devoted to a strategic discussion on the scope of the cytochrome P450 system. The main topic was a discussion in how far changes in the conformation of one of the key components may be responsible for the oscillatory dynamics of the biomimetic system, thus addressing the question whether changes in a microscopic (molecular) property is reflected by the macroscopic dynamics of the entire system. As an outcome of this discussion, the three groups (together with a group from Heidelberg) have recently prepared a proposal to study this issue, and submitted this major grant application to the Volkswagen Foundation.

Publications in preparation: [2005_2], [2005_3], [2005_15].

C Nonlinear processes at interfaces

C 1. Heterogeneous catalysis I.

(Visits 15, 52, 66, 89, 98, 104)

The description of phenomena occurring over a range of reactor length-scales, from the molecular (e.g., adsorption equilibrium, reaction kinetics) through the mesoscopic (e.g., agglomeration) to continuum description of transport phenomena (momentum, heat and mass transfer) up to unit operation and process system levels is one of the main subjects of interest of chemical engineering and process industries. Information obtained from models at one length-scale has to be passed to those at different hierarchical levels. For example, a fixed bed catalytic reactor model often requires the use of the effectiveness factor as one of its input parameters, which can be obtained as a solution of a diffusion problem at the length-scale of an individual catalyst pellet. The values of a diffusion coefficient or adsorption equilibrium constant, which are required as input to the particle-scale model can, in turn, be obtained from molecular-level simulations.

The ESF visit of PhD student Salejová (Prague) to Štěpánek (London) was devoted to modeling of effective diffusity in high and low porosity foams. The results of the modeling are subject of publication in preparation.

Catalytic reactors form the basis of chemical industry. Most catalytic processes use porous catalysts, with active components distributed within the porous structure. PhD student Kočí (Prague) in the course of his two visits at Štěpánek (London) worked on the development of modeling techniques for the description of reaction and transport processes in a porous heterogeneous catalyst using spatially 3D model of digitized porous structure. Microkinetics of the CO oxidation on Pt on alumina catalyst with an explicit consideration of surface deposited species was considered. The dependence of the overall CO conversion and the effectiveness factor on temperature and properties of porous structure has been modelled and discussed. The obtained results are described in the paper Kočí P., Štěpánek F., Kubíček M., Marek M. Modelling of CO oxidation in a digitally reconstructed porous Pt/γ-Al2O3 catalyst. Proc. of 16th International Congress on Chemical and Process Engineering. CHISA 16, 22nd-26th August 2004, Prague, Czech Republic., and in another paper which is now being prepared.

There are many situations where phenomena occurring at different length-scales are so integrated that their modelling cannot be simply de-coupled. Such a case occurs for example in the gas phase catalytic polymerization of olefins, where the resulting polymer properties depend simultaneously on phenomena at several length-scales: from polymer growth around a catalyst support fragment, through monomer diffusion and heat transfer within a growing polymer particle, to population dynamics and fluid-particle flow at the length-scale of the reactor. The visit of PhD student Grof (Prague) at Adler (Paris) was devoted to modelling of transport processes and fragmentation of growing polymer particles in catalytic polymerizations. The methods of reconstructed porous media developed by. Adler’s group were used. The results of research are contained in a paper [2003_14].

The short visit of Marek (Prague) at Scott (Leeds) was devoted to study of nonlinear dynamic effects in monolithic catalytic reactors. Lecture on the subject was delivered at the University of Leeds.

The short visit of Marek (Prague) at Štěpánek (London) dealt with the studies of problems of multiscale modeling.

The above series of visits helped to transfer the results of research of nonlinear dynamics performed in the course of the ESF REACTOR Program to wider chemical and process engineering audience.

C 1. Heterogeneous catalysis II.

(Visits 25, 26, 91)

Visits 25, 26 and 91 were performed in a framework of a collaboration between the Free University of Brussels and the Fritz Haber Institute in Berlin. The aim of this collaboration was to develop a theory of nonequilibrium pattern formation in surface chemical reactions in presence of promoter species. Promoters are chemical species which are present on a catalytic surface and, though they do not participate directly in a chemical reaction, can increase the reaction rate. Typically, such species have strong attractive interactions with one of the adsorbed reactants. The experiments by Imbihl (Hannover) and Kiskinova (Trieste) have shown that such reactions are accompanied by formation of stationary stripe patterns on catalytic surfaces. In our theory, this is explained as an effect of an interplay between reactions, diffusion and phase separation induced by energetic interactions between the promoter and one of the reaction species. The results have been jointly published in two articles in 2004 ([2004_2] and [2004_15]). The collaboration, which was initiated by visit 25, is currently continued and further publications are in preparation.

C 2. Electrochemistry

(Visit 4)

During this visit cyclic voltammograms of the hydrogen electrooxidation reaction on a rotating Pt ring electrode in diluted electrolytes were studied [2002_11]. These voltammograms are either aperiodic or possess a period which is a multiple of the period of the cycling voltage. The complex voltammetric responses are experimentally studied in two electrolytes with different compositions of the base electrolyte (and thus different conductivity) as a function of the positive turning point of the applied voltage. The behavior can be rationalized in terms of the interaction between the inhibition of H₂ oxidation by Pt-oxides, the roughening of the electrode due to oxide reduction and its subsequent smoothing as well as a nonnegligible uncompensated voltage drop through the electrolyte. The proposed mechanism is translated into a mathematical model consisting of a set of four ordinary differential equations. The model indeed captures the main experimental findings. Since the principle ideas do not depend on the specific kinetics of the oxidation reaction under investigation, the mechanism should be operative in a variety of electrochemical systems. In particular, we suggest that it may underlie the complex cyclic voltammograms observed during many oxidation reactions of small organic molecules.

D Special aspects

D 1. Stochastic phenomena: Complex networks and noise I.

(Visits 33, 34, 86, 94)

Chemical and biochemical pathways form a network the general properties of which were studied in a Finnish-Hungarian cooperation. Networks constitute the scaffold of complex systems thus their applicability ranges from chemistry through the internet until social systems. The topology of the networks is of crucial importance from the point of view of the dynamics. The local structure of unweighted networks can be characterized by the number of times a subgraph appears in the network. We generalized this approach for weighted networks. We introduced subgraph “intensity” as the geometric mean of its link weights and “coherence” as the ratio of the geometric to the corresponding arithmetic mean. Using these measures, motif scores and clustering coefficient can be generalized to weighted networks. To demonstrate these concepts, we applied them to financial and metabolic networks and found that inclusion of weights may considerably modify the conclusions obtained from the study of unweighted characteristics.

Networks of companies can be constructed by using return correlations. A crucial issue in this approach is to select the relevant correlations from the correlation matrix. In order to study this problem, we started from an empty graph with no edges where the vertices correspond to stocks. Then, one by one, we inserted edges between the vertices according to the rank of their correlation strength, resulting in a network called “asset graph”. We studied its properties, such as topologically different growth types, number and size of clusters and clustering coefficient. These properties, calculated from empirical data, were compared against those of a “random graph”. Differences in the mean clustering coefficient lead us to conclude that most information is contained roughly within 10 % of the edges. The growth of the graph could be classified according to the topological role of the newly inserted edge. We found that the type of growth which is responsible for creating cycles in the graph sets in much earlier for the empirical asset graph than for the random graph, and thus reflects the high degree of networking present in the market.

Results of the research were published in a series of papers [2002­_12], [2003_5], [2003_6], [S9], [S16].

D 1. Stochastic phenomena: Complex networks and noise II.

(Visits 49, 77, 105)

Chemical reactors have been a prominent tool for the study of microbial population dynamics. The control of the chemical fluxes allowed to gather important information about the metabolism of yeast cells, paving the way for a quantitative modelling of the intracellular nonlinear biochemical reactions. The measure of bulk properties poses however some fundamental theoretical problems on the relation between the dynamical properties of the population and those of the individual cells. The macroscopic behaviour mirrors the microscopic one when complete synchronisation takes place. Such a perfect entrainment, occurring typically for strong coupling among the cells, is the basis for deriving quantitative information from CSTR experiments.

The activities of the class D1 have focused on the effect of microscopic disorder onto the collective dynamics of populations of nonlinear dynamical systems. In particular, they dealt with the case of strong coupling that, although fundamental for the interpretation of the experimental measures, had been so far dedicated little theoretical effort. Two general classes of populations have been considered, reflecting the two ways in which disorder is traditionally model. The first class consists of ensembles of dynamical systems that differ by one parameter (e.g. the oscillation frequency) (visit 49). In the second class, the diversity among the population elements is modelled by independent stochastic terms (visits 77 and 105). Besides the numerical studies, a general analytical approach has been formulated that allows to determine under very general conditions how the macroscopic evolution is modified by the presence of disorder. The visits supported by the ESF programme REACTOR allowed the establishment of an active collaboration with Chaté (CEA, France), to publish two papers [2004_1, 2005_7] and to the production of a third paper [2005_8]. Moreover, two more papers have been initiated after discussions at the REACTOR meetings [S6, S8]. The application of the theoretical results to experiments with yeast cells in a CSTR has been as well initiated in the course of the visit (visit 82), whose results are being collected in a paper [2005_9].

D 2. Nanoreactors and nanoparticles

(Visits 73, 80, 83, 92, 106)

Visit 73. Gold clusters protected by organic ligands have attracted the researcher interest both from the scientific point of view and because of the potential applications due to their unusual chemical, optical and electronic properties. Functionalized gold nanoparticles are very convenient prepared by ligand exchange reactions, and the rate law found depends on the experimental conditions used. To obtain useful information to elucidate the mechanism of exchange reaction the groups from Bucharest and York undertook a kinetic study on the exchange reaction of diradical (DIS3) with Au particles protected by butanethiol or triphenylphosphine, using different radical/Au particle ratios. They have used a powerful technique, namely the EPR, which allows to enhance the understanding on the nanoscale reactivity. At higher radical/particle ratios the kinetics of the exchange follows a non-linear rate law. This result has been ascribed to the different reactivity of different binding sites on gold surface.

Visit 80. Recently the dry bis(2-ethylhexyl)sulfosuccinate (AOT)/heptane reversed micelles have attracted considerable interest mostly because they are used to synthesize nanoparticles via a solid-solid reaction. The group in Palermo has synthesized several cobalt(II)-iron(II) complexes (Co_x[Fe(CN)₆]) confined in dry AOT reversed micelles. Depending on the mixing ratio between the surfactants and the salts the kinetics of solid-solid complex formation follows a non-linear rate low. Thus, to characterize the formed nanoparticle a collaboration between Palermo and Bayreuth has been started. The physico-chemical studies have been carried out by means of TEM measurements. It has been demonstrated for the first time that salt nanoparticles can be confined in the surfactant matrix as crystalline solids, in particular the precursors form very well defined nanoparticles while the cobalt(II)-iron(II) complexes form very small nanoparticles or complexes molecularly dispersed in the AOT matrix.

Visits 92 and 106. Removing sparingly water-soluble organic compounds from contaminated soil and water reservoirs is an important issue for the quality of life in industrialised countries and also a challenging question from the scientific point of view, due to the complex nature of the systems to be considered. The aquifer remediation based on the use of surfactants is most promising and innovative technique for removal of hydrophobic compounds. In the laboratories of Palermo it has been observed that, at a given temperature, during the solubilization of organic molecules sparingly soluble in water into aqueous block copolymers solutions oscillations of the interfacial tension value at the organic/aqueous solution interface have been generated and this phenomenon has been ascribed to chemically driven Marangoni instability. Thus, to get an insight into the evolution of the nano-aggregates structure while the additive solubilization process takes place the groups from Palermo and Bayreuth have performed a systematic study by means of DLS and fluorescence measurements. The shape changes of the aggregated systems due to the solubilization of hydrophobic compounds have been monitored as a function of both external and internal parameters, i.e., temperature, architecture of the block-copolymers, composition and stirring conditions. It has been found that both the hydrophilic/hydrophobic ratio in the copolymer and the temperature play a relevant role while shape and size of the nano-structured systems are not dependent on the stirring conditions.

Visit 109. The formation of Liesegang patterns has attracted considerable interest both as a possible application and as a complex scientific problem. Possible applications are photographic processes, crystal growth in gels. One among the several theoretical problems is pattern formation in hard rocks like agate. Several theories have been proposed but up to now none of these really yielded a satisfactory solution. Thus, the get a deep insight on the nucleation phenomenon during the formation of Liesegang structures the groups of Palermo and Magdeburg have studied the Liesegang precipitation in vertical tubes in bis(2-ethylhexyl)sulfosuccinate (AOT)-microemulsion based gels having different AOT-water-gelatin composition ratios. It has to be highlighted that up to now no study of this phenomenon exists in this kind of nano-organized system. It has been found that presence of gel-based microemulsions slows down the pattern formation with respect to classical gelatine-aqueous gel, in addition the precipitation patterns depend on the microemulsions composition. In particular, the pattern formation is favoured at high W value and the nature of the salt plays a relevant role. It is also worth to mention that when the lead nitrate was used as diffusive agent in the gel-based microemulsion (containing the potassium iodine) upward rings are formed; while in the case of silver nitrate as diffusive agent in the gel-based microemulsion (containing the potassium dichromate) downward rings can be detected as in the classical Liesegang.