Reaction-diffusion processes

Tabon, J. Morphological bifurcations involving reaction-diffusion processes during microtubule formation. Science 1994 264 245-248. 

Beside the diffusive modes, the chemical modes can also be constructed in models of reaction-diffusion processes [33-36]. 

Glade, N., Demongeot, J., and Tabony, J. (2004). Micrombules self-organization by reaction-diffusion processes causes collective transport and organization of cellular particles. BMC Cell Biol, 5, 5-23. 

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. 

The catalyst intraparticle reaction-diffusion process of parallel, equilibrium-restrained reactions for the methanation system was studied. The non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key components have been established, and solved using an orthogonal collocation method. The simulation values of the effectiveness factors for methanation reaction Ch4 and shift reaction Co2 are fairly in agreement with the experimental values. Ch4 is large, while Co2 is very small. The shift reaction takes place as direct and reverse reaction inside the catalyst pellet because of the interaction of methanation and shift reaction. For parallel, equilibrium-restrained reactions, effectiveness factors are not able to predict the catalyst internal-surface utilization accurately. Therefore, the intraparticle distributions of the temperature, the concentrations of species and so on should be taken into account. 

Thus, at the end of ninth second the system A-AB-B returns to a new initial state which differs from the previous one only by the lesser number of atoms (atomic planes in two dimensions) in initial phases A and B and the greater number of molecular AB units in the growing AB layer. The reaction-diffusion process is repeated again and again, with the diffusion path of the B atoms from interface 2 to interface 1 becoming progressively longer, until either A or B is consumed completely. 

In regards to the stationarity of the reaction-diffusion process, it should be emphasised that the number of the B atoms diffusing across the ApBq layer is always equal to their number combined by the A surface into the ApBq compound at interface 1, if the growth of this layer is not accompanied by the formation of other compounds or solid solutions. The case under consideration is characterised by a kind of forced stationarity due to (z) the impossibility of any build-up of atoms at interfaces between the solids, (z z) the limited number of diffusion paths in the ApBq layer for the B atoms to travel from interface 2 to interface 1 and (Hi) the finite value of the reactivity of the A surface towards the B atoms. The stationarity is only... 

At tA> t3, this basic act of the reaction-diffusion process is completed by the formation of an additional row of molecules AB and then is repeated with an AB layer thicker by one molecule AB, and so on. Its driving force is the difference in values of the chemical potential of component B in initial phases A and B. This constant difference exists until at least one of initial substances A or B is exhausted. 

In the great majority of cases, a line of the markers located in the zinc phase displaces a few micrometres aside from a line located in the other phases, indicative of the crack formation at the interface with zinc. To understand the further course of the reaction-diffusion process after the rupture of any reaction couple, it is necessary first to analyse the growth kinetics of the same compound layer in different reaction couples of a multiphase binary system. This will be done in the next chapter. 

The consideration of the reaction-diffusion process in binary heterogeneous systems, carried out in this book, is actually based upon the two simple and obvious assumptions ... 

Nevertheless, in this book the number of the theoretically substantiated kinetic equations, for the experimentalist to use in practice, appears to exceed that resulting from purely diffusional considerations. Whether the experimentalist will be pleased with such an abundance of equations is a wholly different question. Still, for many researchers in the field it is so tempting to employ the only parabolic relation and then to discuss in detail the reasons for (unavoidable and predictable) deviations from its course. Note that unlike diffusional considerations where each interface is assumed to move according to the square root of the time, in the framework of the physicochemical approach the layer-growth kinetics are not predetermined by any additional assumptions, except basic ones, but immediately follow in a natural way from the proposed mechanism of the reaction-diffusion process. 

I hope that the reader have understood how dangerous it is to ignore any of the two steps of the reaction-diffusion process (either reaction or diffusion). Irrespective of his educational basis, the researcher must realise that only chemistry, physics and mathematics, taken together, even in the rather moderate amount, are likely to result in a correct consideration of solid-state chemical kinetics. 

It appears relevant to note that many workers tend to overestimate the significance of thermodynamic predictions concerning the direction of the reaction-diffusion process. In fact, however, those only bear a likelihood character. Even if the free energy of formation of one compound from its constituents is -200 kJ mol-1, while that of the other is -20 kJ mol1, this does not necessarily mean, as often (tacitly or directly) assumed, that the former will occur first and the more so that its growth rate must be ten times greater than that of the latter. As exemplified with the growth rate of a compound layer in various diffusion couples of the same multiphase binary system, the opposite may well take place. 

Stundzia, A. and Lumsden, C., Stochastic simulation of coupled reaction-diffusion processes, Journal of Computational Physics, Vol. 127, No. 1, 1996, pp. 196-207. 

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