Reaction diffusion patterns

Nettesheim S, von Oertzen A, Rotermund FI FI and ErtI G 1993 Reaction diffusion patterns in the catalytic CO-oxidation on Pt(110) front propagation and spiral waves J. Chem. Rhys. 98 9977-85... 

Patterns Generated by the Outer Membrane. As in the single cell case, the outer membrane itself can generate electro-physiological patterns, even in the absence of reactions in the cell mass, R = 0. Such surface induced patterns were first discussed in the context of reaction-diffusion patterns in Ref. 

B. Fiedler and A. Scheel. Spatio-temporal dynamics of reaction-diffusion patterns. In M. Kirkilionis (ed.) et al.. Trends in nonlinear analysis. On the occasion of the 60th birthday of Willi Jdger. Berlin Springer, 23—152, 411-417, 2003. 

Holloway, D.M Harrison, L.G. (1999) Algal morphogenesis modelling interspecific variation in Micrasterias with reaction-diffusion patterned catalysis of cell surface growth. Philosophical Transactions of the Royal Society, London, series B, Vol.354, pp. 417-433. 

The experimental determination of such profiles presents numerous challenges already found in the study of reaction-diffusion patterns in inert gels. In active gels, the problem is even more acute as shown in the experimental Section 9.5. 

For the present article we have chosen the above two systems from the large variety of catalyst systems exhibiting phenomena of self-organization reported in the literature. In Section 2 we describe thermokinetic pattern formation on polycrystalline monolithic catalysts Section 3 is dedicated to isothermal single crystal studies. Although a number of metals and reactants have been used in the last case (see [11] for a recent exhaustive survey), we focus our attention on the system which exhibits so far the richest variety of spatiotemporal patterns, namely the CO oxidation on Pt. Involving only two reactants and a catalyst, this system appears conceptually very simple consequently a realistic but not too complicated model could be constructed, which successfully describes the experimentally observed reaction-diffusion patterns. 

The search for Turing patterns led to the introduction of several new types of chemical reactor for studying reaction-diffusion events in feedback systems. Coupled with huge advances in imaging and data analysis capabilities, it is now possible to make detailed quantitative measurements on complex spatiotemporal behaviour. A few of the reactor configurations of interest will be mentioned here. 

Lengyel I and Epstein I R 1992 A chemical approach to designing Turing patterns in reaction-diffusion systems Proc. Natl Acad. Sc/. 89 3977-9... 

Toth A, Lagzi I and Florvath D 1996 Pattern formation in reaction-diffusion systems cellular acidity fronts J. Rhys. Chem. 100 14 837-9... 

The extension of generic CA systems to two dimensions is significant for two reasons first, the extension brings with it the appearance of many new phenomena involving behaviors of the boundaries of, and interfaces between, two-dimensional patterns that have no simple analogs in one-dimension. Secondly, two-dimensional dynamics permits easier (sometimes direct) comparison to real physical systems. As we shall see in later sections, models for dendritic crystal growth, chemical reaction-diffusion systems and a direct simulation of turbulent fluid flow patterns are in fact specific instances of 2D CA rules and lattices. 

In this section we introduce several CA models of prototypical reaction-diffusion systems. Such systems, the first formal studies of which date back to Turing , often exhibit a variety of interesting spatial patterns that evolve in a self-organized fashion. 

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

In what follows we will discuss systems with internal surfaces, ordered surfaces, topological transformations, and dynamical scaling. In Section II we shall show specific examples of mesoscopic systems with special attention devoted to the surfaces in the system—that is, periodic surfaces in surfactant systems, periodic surfaces in diblock copolymers, bicontinuous disordered interfaces in spinodally decomposing blends, ordered charge density wave patterns in electron liquids, and dissipative structures in reaction-diffusion systems. In Section III we will present the detailed theory of morphological measures the Euler characteristic, the Gaussian and mean curvatures, and so on. In fact, Sections II and III can be read independently because Section II shows specific models while Section III is devoted to the numerical and analytical computations of the surface characteristics. In a sense, Section III is robust that is, the methods presented in Section III apply to a variety of systems, not only the systems shown as examples in Section II. Brief conclusions are presented in Section IV. 

The set of the reaction-diffusion equations (78) can be solved by different methods, including bifurcation analysis [185,189-191], cellular automata simulations [192,193], or numerical integration [194—197], Recently, two-dimensional Turing structures were also successfully studied by Mecke [198,199] within the framework of integral geometry. In his works he demonstrated that using morphological measures of patterns facilitates their classification and makes possible to describe the pattern transitions quantitatively. 

It is possible to show that when the different parts of a system are connected by nonlinear interactions, one can again obtain oscillation in concentrations, patterns of chemical substances in space, and wave propagation. These phenomena are important in some biological problems when the reaction-diffusion mechanisms cannot give an adequate description of the system. Morphogenetic fields and neural networks are examples of such systems. 

Theories that account for pattern formation in a morphogenetic field, as a result of reaction-diffusion processes, must assume the existence of at least two small diffusable molecules throughout the field. These hypotheses can be relaxed if one considers that the concentration of morphogenetic substances is altered in each cell via nonlinear interactions between cell surface receptors. 

Kaas-Petersen, C. and Scott, S. K. (1988). Stationary-state and Hopf bifurcation patterns in isothermal, autocatalytic reaction-diffusion equations. Chem. Eng. Sci., 43, 391-2. 

---