Models of Liesegang pattern formation

In this article different mathematical models of the Liesegang phenomenon are exhibited. The main principles of modeling are discussed such as supersaturation theory, sol coagulation and phase separation, which describe the phenomenon using different steps and mechanism beyond the simple reaction scheme. We discuss whether the underlying numerical simulations are able to reproduce several empirical regularities and laws of the corresponding pattern structure. In all cases we highlight the meaning of the initial and boundary conditions in the corresponding mathematical formalism. Above the deterministic ones discrete stochastic approaches are also described. As a main tool for the control of pattern structure the effect of an external electric field is also discussed

A new universal law for the Liesegang pattern formation

Classical regularities describing the Liesegang phenomenon have been observed and extensively studied in laboratory experiments for a long time. These have been verified in the last two decades, both theoretically and using simulations. However, they are only applicable if the observed system is driven by reaction and diffusion. We suggest here a new universal law, which is also valid in the case of various transport dynamics (purely diffusive, purely advective, and diffusion-advection cases). We state that ptot~Xc, where ptot yields the total amount of the precipitate and Xc is the center of gravity. Besides the theoretical derivation experimental and numerical evidence for the universal law is provided. In contrast to the classical regularities, the introduced quantities are continuous functions of time

Összetett reakció-diffúzió rendszerek vizsgálata és modelljeik párhuzamosítása = Investigation of complex reaction-diffusion systems and paralellization of their models

Csapadékmintázatokat vizsgáltunk reakció-diffúzió rendszerekben: (i) Egyedüli önszerveződést figyeltünk meg csapadékrendszerekben, ahol spontán kialakuló spirálok megjelenését vettük észre a csapadékfront vékony rétegében; (ii) Egy új csapadékrendszerben egymás utáni frontokat és a frontok torzítását mutattuk meg; (iii) Egy új és univerzális törvényt javasoltunk a reguláris Liesegang mintázatok leírására, amely érvényes több transzport feltétel esetén is. Egy kémiai transzport és ülepedési modellt dolgoztunk ki és kapcsoltunk össze az ózonfluxusok jellemzésére Magyarország területére. Több kémiai alkalmazást is készítettünk (reakció-diffúzió rendszerek modellezése; passzív nyomanyagok légköri terjedésének szimulációja) a P-GRADE fejlesztői környezet és P-GRADE portál segítségével. | Pattern formation phenomena in precipitation systems were investigated: (i) A unique kind of self-organization, the spontaneous appearance of traveling waves, and spiral formation inside a precipitation front was reported; (ii) A new simple reaction-diffusion system was presented focusing on pattern formation phenomena as consecutive precipitation fronts and distortion of the precipitation front; (iii) A new universal law for the regular Liesegang phenomenon has been proposed, which is also valid in the case of various transport dynamics. A chemical transport model and a dry-deposition model have been coupled for the purpose of simulating ozone fluxes over Hungary. Some chemical applications (simulation reaction-diffusion equations; simulation passive tracer from a point source) have been developed using P-GRADE programming environment and P-GRADE portal

Simulation of Liesegang Patterns: Effect of Reversible Complex Formation of Precipitate

Formation of 1D Liesegang patterns was studied numerically by assuming precipitation and reversible complex formation. The Ostwald's supersaturation model reported by Büki, Kárpáti-Smidróczki, and Zrínyi (BKZ model) was developed and extended further. The position of the first and last bands measured from the junction point of the inner and outer electrolytes is linearly proportional to the square root of its formation time for the different initial concentrations of inner electrolyte ( o ). The propagation of patterns along the diffusion column is slower at higher o . The correlation between the distance of the first and last bands measured from the beginning of the diffusion column is strictly linear. The variation of the total number of bands with o has a maximum. The presented model reproduces the moving precipitate zones and nonlinear oscillation of the total number of bands in time due to the complex formation of the precipitate