Waves, chemical

Kapral R and Showalter K (eds) 1995 Chemical Waves and Patterns (Dordrecht Kluwer)... 

Multi-author volume surveying chemical wave and pattern formation, an up-to-date introduction for those entering the field. 

An example of the application of J2-weighted imaging is afforded by the imaging of the dynamics of chemical waves in the Belousov-Zhabotinsky reaction shown in figure B 1.14.5 [16]. In these images, bright... 

Figure Bl.14.5. J2-weighted images of the propagation of chemical waves in an Mn catalysed Belousov-Zhabotinsky reaction. The images were acquired in 40 s intervals (a) to (1) using a standard spin echo pulse sequence. The slice thickness is 2 nun. The diameter of the imaged pill box is 39 nun. The bright bands...

Tzalmona A, Armstrong R L, Menzinger M, Cross A and Lemaire C 1990 Detection of chemical waves by magnetic resonance imaging Chem. Rhys. Lett. 174 199-202... 

Table B2.5.1 Fonn of the chemical wave y (t) (equation B2.5.24)) for the various cases depicted in figure...

Another class of instabilities that are driven by differences in the diffusion coefficients of the chemical species detennines the shapes of propagating chemical wave and flame fronts [65, 66]. 

Kapral, R. and Showalter, K. (1994) Chemical Waves and Patterns, Kluwer, London. 

Figure 8.8 Advective propagation of a chemical wave of tracer i moving with a velocity v in a wetted porous solid at times t= 1,2, 3 for different values of d2Cwl7(dCUq )2. Breaking takes place at t = 3 in cases b and c.

Chadam, J. Ortoleva, P. (1984). Moving interfaces and their stability Applications to chemical waves and solidification. In Dynamics of Non-Linear Systems, ed. V. 

Raymond Kapral, Simulating Chemical Waves and Patterns. 

Reaction-diffusion systems have been studied for about 100 years, mostly in solutions of reactants, intermediates, and products of chemical reactions [1-3]. Such systems, if initially spatially homogeneous, may develop spatial structures, called Turing structures [4-7]. Chemical waves of various types, which are traveling concentrations profiles, may also exist in such systems [2, 3, 8]. There are biological examples of chemical waves, such as in parts of glycolysis, heart... 

A typical chemical system is the oxidative decarboxylation of malonic acid catalyzed by cerium ions and bromine, the so-called Zhabotinsky reaction this reaction in a given domain leads to the evolution of sustained oscillations and chemical waves. Furthermore, these states have been observed in a number of enzyme systems. The simplest case is the reaction catalyzed by the enzyme peroxidase. The reaction kinetics display either steady states, bistability, or oscillations. A more complex system is the ubiquitous process of glycolysis catalyzed by a sequence of coordinated enzyme reactions. In a given domain the process readily exhibits continuous oscillations of chemical concentrations and fluxes, which can be recorded by spectroscopic and electrometric techniques. The source of the periodicity is the enzyme phosphofructokinase, which catalyzes the phosphorylation of fructose-6-phosphate by ATP, resulting in the formation of fructose-1,6 biphosphate and ADP. The overall activity of the octameric enzyme is described by an allosteric model with fructose-6-phosphate, ATP, and AMP as controlling ligands. 

Fig. 19. 2-D MR image of an oscillating chemical reaction occurring within a bed of diameter 15 mm packed with 1-mm-diameter glass beads. In-plane resolution was 195 pm x 195 pm, and the image slice thickness was 1 mm. A single image was acquired in 1 s. Chemical waves are imaged as a result of the...

The previous chapters have discussed the behaviour of non-linear chemical systems in the two most familiar experimental contexts the well-stirred closed vessel and the well-stirred continuous-flow reactor. Now we turn to a number of other situations. First we introduce the plug-flow reactor, which has strong analogies with the classic closed vessel and which will also lead on to our investigation of chemical wave propagation in chapter 11. Then we relax the stirring condition. This allows diffusive processes to become important and to interact with the chemistry. In this chapter, we examine one form of the reaction-diffusion cell, whose behaviour can be readily understood by comparison with the responses observed in the CSTR. 

Tam, W. Y. (1987). Sustained chemical waves in an annular gel reactor a chemical pinwheel. Nature, 329, 619-20. 

Showalter, K. (1987). Chemical waves. In Kinetics of nonhomogeneous processes, (ed. G. R. Freeman). Wiley, New York. 

Winfree, A. T. (1985). Organizing centers for chemical waves in two and three dimensions. In Oscillations and traveling waves in chemical systems, (ed. R. J. Field and M. Burger), ch. 12, pp. 441-71. Wiley, New York. 

In 1977. Professor Ilya Prigogine of the Free University of Brussels. Belgium, was awarded Ihe Nobel Prize in chemistry for his central role in the advances made in irreversible thermodynamics over the last ihrec decades. Prigogine and his associates investigated Ihe properties of systems far from equilibrium where a variety of phenomena exist that are not possible near or al equilibrium. These include chemical systems with multiple stationary states, chemical hysteresis, nucleation processes which give rise to transitions between multiple stationary states, oscillatory systems, the formation of stable and oscillatory macroscopic spatial structures, chemical waves, and Lhe critical behavior of fluctuations. As pointed out by I. Procaccia and J. Ross (Science. 198, 716—717, 1977). the central question concerns Ihe conditions of instability of the thermodynamic branch. The theory of stability of ordinary differential equations is well established. The problem that confronted Prigogine and his collaborators was to develop a thermodynamic theory of stability that spans the whole range of equilibrium and nonequilibrium phenomena. 

Fig. 9. Successive compartmental lines predicted on the fate map6-7 of the blastoderm by the chemical wave model, with the binary combinatorial code assignment in each compartment generated by the successive lines. A, antenna E, eye Pb, proboscis P, prothorax W, wing and mesothorax H, haltere LI, L2, L3, first, second, and third legs Abd, abdominal segments G, genital.

TABLE I. Predicted Relative Transdetermination Frequencies Derived from the Chemical Wave Model Applied to the Blastoderm ... 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

These models consider the mechanisms of formation of oscillations a mechanism involving the phase transition of planes Pt(100) (hex) (lxl) and a mechanism with the formation of surface oxides Pd(l 10). The models demonstrate the oscillations of the rate of C02 formation and the concentrations of adsorbed reactants. These oscillations are accompanied by various wave processes on the lattice that models single crystalline surfaces. The effects of the size of the model lattice and the intensity of COads diffusion on the synchronization and the form of oscillations and surface waves are studied. It was shown that it is possible to obtain a wide spectrum of chemical waves (cellular and turbulent structures and spiral and ellipsoid waves) using the lattice models developed [283], Also, the influence of the internal parameters on the shapes of surface concentration waves obtained in simulations under the limited surface diffusion intensity conditions has been studied [284], The hysteresis in oscillatory behavior has been found under step-by-step variation of oxygen partial pressure. Two different oscillatory regimes could exist at one and the same parameters of the reaction. The parameters of oscillations (amplitude, period, and the... 

For such applications of classical optimization theory, the data on energy and gradients are so computationally expensive that only the most efficient optimization methods can be considered, no matter how elaborate. The number of quantum chemical wave function calculations must absolutely be minimized for overall efficiency. The computational cost of an update algorithm is always negligible in this context. Data from successive iterative steps should be saved, then used to reduce the total number of steps. Any algorithm dependent on line searches in the parameter hyperspace should be avoided. 

Induction of New Phenomena by Imposed Gradients. Studies on the effects of imposed electric fields on chemical waves (see below on signal propagation) show that phenomena can be induced in the system by the imposed gradient that do not exist in the field free medium. For example, it was found (l l) that for a system wherein only one type of wave existed in the field free case, two stable types of waves exist in the system subject to the field. This "induction of multiplicity" implies that beyond a critical value of the applied field strength new phenomena may set in that are not simple distortions of field free patterns. Another strictly imposed field effect is found in the case of a new two dimensional crescent shaped wave that occurs when a circular wave is subjected to a supracritical field (15). 

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