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 p(tot) proportional X(c), where p(tot) yields the total amount of the precipitate and X(c) 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.     

Liesegang rings of copper chromate in gelatin gel

Summary DistinctLiesegang Rings of Copper chromate in gelatin gel are not formed as the salt forms a very stable colloid in this medium. As this sol is coagulated by KH₂PO₄, well defined rings of Copper chromate in gelatin gel are formed in the presence of this electrolyte.

---

Zusammenfassung DeutlicheLiesegang-Ringe werden in Kupfer-Chromat und Gelatine-Gel nicht gebildet, da das Salz ein sehr stabiles Kolloid in diesem Medium darstellt. Wenn dieses Sol durch KH₂PO₄ koaguliert wird, entstehen definierte Ringe von Kupfer-Chromat im Gelatine-Gel bei Gegenwart dieses Elektrolyten.

     

Simulation of a crossover from the precipitation wave to moving Liesegang pattern formation

Model simulations to investigate the precipitation wave phenomenon and a crossover from the precipitation wave to moving Liesegang patterns were performed. The chemical scheme contains four chemical species via the interaction of precipitation and redissolution (complex formation), in which the precipitation reaction term was based on Ostwald's supersaturation theory. In this article, for the first time, all the features and behaviors of the heterogeneous traveling waves are reproduced, which were observed experimentally in the work of Zrínyi et al. (Zrínyi, et al. J. Phys. Chem. 1991, 95, 1618.). The detailed investigation of the pattern formation showed three possible states of the system, which depend on the initial concentration of the inner and outer electrolytes, respectively. These are the precipitation wave (single moving precipitation band), the moving Liesegang pattern (moving precipitation bands), and the state where these two patterns coexist.     

Kinetics of the formation of Liesegang rings

The kinetics of the formation of Liesegang rings is considered and they are classified. The relationship between the position of the propagating diffusion front and the moments of the formation of Liesegang rings is shown. In order to describe the formation processes of the Liesegang rings a generalized model based on the diffusion equation hierarchy is proposed. The kinetics of changes in the dispersed phase sizes at the initial stage of the formation of Liesegang rings is studied.     

Effect of gel network on pattern formation in the ferrocyanide-iodate-sulfite reaction

Stationary patterns have been researched experimentally since the discovery of the Turing pattern in the chlorite-iodide-malonic acid (CIMA) reaction and the self-replicating spot pattern in the ferrocyanide-iodate-sulfite (FIS) reaction. In this study, we reproduced the pattern formation in the FIS reaction by using poly(acrylamide) gels. Gels with different swelling ratios were prepared to use as a medium. The effect of the swelling ratio was compared with the effect of thickness. It was found that the swelling ratio greatly influenced pattern formation. Oscillating spot patterns appeared at high swelling ratios, and lamellar patterns appeared at a low swelling ratio. Self-replicating spot patterns appeared in between the two areas. The front velocities, which were observed in the initial stage of pattern formation, depended on the swelling ratio. Furthermore, this dependence obeys the free volume theory of diffusion. These results provide evidence that the change in front velocities is caused by a change in diffusion. Pattern formation can be controlled not only by thickness but also by swelling ratio, which may be useful for creating novel pattern templates.     

Justification for the colloid explanation of Liesegang ring formation

 The persuasive evidence for the role of colloid in the formation of Liesegang rings is nullified by the low diffusion constants (less than 2 × 10⁻¹¹ m² s⁻¹) of sol particles; however, those values were obtained for sols suspended in otherwise homogeneous solutions. The essential randomness of Brownian motion in such situations is absent in Liesegang experiments, where the large excess of outer electrolyte diffusing into the gel creates a bias in molecular bombardment resulting in sol particles moving a given distance in fewer steps, hence in a shorter time. From Einstein's equation (D=x ²/2t) values for D of 2–4 × 10⁻¹⁰ m² s⁻¹ have been calculated for Liesegang experiments in the literature. It is suggested that such values could well pertain to sol particles diffusing in the heterogeneous conditions existing in those experiments. Received: 13 April 1999 Accepted: 16 November 1999     

Liesegang图案

介绍了Liesegang图案的生长规律、实验研究方法、形成影响因素、理论进展、应用及存在的问题,同时对当前两个活跃的研究方向即复杂Liesegang环和Liesegang图案的微观形成机制进行了讨论.该领域研究的新突破有助于理解自然界中一些复杂图案的形成和控制材料制备中的自组装结构及动物代谢中的结石生长.     

The Liesegang phenomenon

Summary The present work demonstrates that the development of clearly defined rhythmic precipitation is intimately related to the stability of the sol prior to precipitation. The factors affecting the stability of ammonia peptised metal salt sols have been analysed; and a new technique developed for study of the Liesegang phenomenon. Excess of the reacting ions cause flocculation and blurred rings: on the other hand, the spacing coefficient is shown to increase in the presence of peptisers for the precipitated substance.

---

Zusammenfassung Die Ausbildung von klar definierten Ringen (rythmischen F?llungen) ist — wie gezeigt wird — eng mit der Stabilit?t des Soles vor F?llung verknüpft. Die einzelnen Faktoren, die diese Stabilit?t von NH₄-peptisierten Metallsalz-Solen beeinflussen, werden analysiert, ferner eine neue Beobachtungsteohnik für Liesegang-Ringe entwickelt. überschu? reagierender Ionen verursacht Floekung und verwaschene Ringe, andererseits w?chst der Ringabstand bei Gegenwart vom Peptisator für die zu f?llende Substanz.

     

Conformational transition of kappa-carrageenan

The calorimetry of kappa carrageenan using differential scanning calorimetry and calorimetry at constant temperature is presented. Both sets of experimental results confirmed the formation of a helical dimer in the absence of gel formation. When gel is formed, the formation of a helical dimer is the major effect, the contribution of gelation represents only 20% of the enthalpy recorded.

---

Zusammenfassung Es wird über die Kalorimetrie von Kappacarrageen mittels DSC und Kalorimetrie bei konstanter Temperatur berichtet. Die experimentellen Resultate bestätigen beim Ausbleiben der Gelbildung, daß ein helixartiges Dimer gebildet wird. Auch bei Gelbildung steuert die Bildung des helixartigen Dimers den Hauptanteil zum thermischen Effekt bei, der Beitrag der Gelierung stellt nur etwa 20% der registrierten Enthalpie dar.

---

- . . , , 20% .

     

Chemical pattern formation in gels

Precipitate forming chemical reactions have been studied in chemically crosslinked poly(vinyl alcohol) gel medium. One of the reactive components was incorporated into the gel, the other was allowed to diffuse into the swollen network. This reaction-diffusion process often results in pattern formation which can occur intermittently in terms of time and space or both. Experiments were carried out in order to investigate the effect of various factors (crosslinking density, swelling degree as well as the concentrations of the outer and inner electrolyte). The following precipitates were intensively studied: Mg(OH)₂, Cr(OH)₃, PbCl₂. Three basically different morphologies (Liesegang band formation, moving bands and tree-like structure) have been observed and investigated. One- and twodimensional computer simulations were carried out in order to establish the influence of various factors on the precipitation.     

Influence of kappa-carrageenan gel structures on the diffusion of probe molecules determined by transmission electron microscopy and NMR diffusometry

The influence of the microstructures of different kappa-carrageenan gels on the self-diffusion behavior of poly(ethylene glycol) (PEG) has been determined by nuclear magnetic resonance (NMR) diffusometry and transmission electron microscopy (TEM). It was found that the diffusion behavior was determined mainly by the void size, which in turn was defined by the state of aggregation of the kappa-carrageenan. The kappa-carrageenan concentration was held constant at 1 w/w%, and the aggregation was controlled by the amount of potassium and/or sodium chloride and, for samples containing potassium, also by the cooling rate. Gels containing potassium formed microstructures where kappa-carrageenan strands are rather evenly distributed over the image size, while sodium gels formed dense biopolymer clusters interspersed with large openings. In a gel with small void sizes, relatively slow diffusion was found for all PEG sizes investigated. Extended studies of the self-diffusion behavior of the 634 g mol(-)(1) PEG showed that there is a strong time dependence in the measured PEG diffusion. An asymptotic lower time limit of the diffusion coefficient was found in all gels when the diffusion observation time was increased. According to the ratio, D/D(0), where D(0) is the diffusion coefficient in D(2)O and D is the diffusion coefficient in the gels, the gels could be divided into three classes: small, medium, and large voids. For quenched kappa-carrageenan solutions with salt concentrations of 20 mM K(+), 100 mM K(+), or 20 mM K(+)/200 mM Na(+) as well as slowly cooled solutions with only 20 mM K(+), D/D(0) ratios between 0.18 and 0.29 were obtained. By quenching a kappa-carrageenan solution with 100 mM K(+), the D/D(0) was 0.5, while D/D(0) ratios between 0.9 and 1 were obtained in a quenched solution with 250 mM Na(+) and slowly cooled samples with 20 mM K(+)/200 mM Na(+) or 250 mM Na(+).     

Lysozyme pattern formation in evaporating drops

Liquid droplets containing suspended particles deposited on a solid, flat surface generally form ringlike structures due to the redistribution of solute during evaporation (the "coffee ring effect"). The forms of the deposited patterns depend on interactions between solute(s), solvent, and substrate. In this study, deposition patterns from droplets of a simplified model biological fluid (DI water + lysozyme) are examined by scanning probe and optical microscopy. The overall lysozyme residue morphology is complex (with both a perimeter "rim" and undulating interior) but varies little with concentration. However, the final packing of lysozyme molecules is strongly dependent on initial concentration.     

Precipitate pattern formation in fluctuating media

Simulation of the Liesegang pattern formation in low concentration gradient is presented using concentration perturbation in a deterministic model. The precipitation process is based on ion-product supersaturation theory (Ostwald's model). In the classical experiments with high initial concentration gradients, appearance time and locations of the band formation are well reproducible. Decreasing initial concentration gradients results in a more stochastic pattern structure; this means that the reproducibility of the experiments becomes worse. The presented model and the results of the simulations exhibit the same trend, which were demonstrated and investigated experimentally by Kai et al.     

Dendrimer pattern formation in evaporating drops

The redistribution of organic solutes during drop evaporation is a nanoscale self-assembly process with relevance to technologies ranging from inkjet printing of organic displays to synthesis of biosmart interfaces for sensing and screening. We have used solutions of dendrimer molecules with incrementally varying terminal site chemistry to explore whether the condensed dendrimer patterns resulting from microdroplet evaporation sensitively depend on, and are characteristic of, the surface chemistry of the solute molecules. This hypothesis has been experimentally confirmed by comparing the behavior of microdroplets of G4, G4-25%C12, and G4-50%C12 dendrimers dissolved in pentanol and deposited on mica substrates. For the dilute concentration studied here, the presence of periodically 'scalloped' dendrimer rings is ubiquitous. The instability wavelength of the scalloped rings is found to be proportional to the width of the ring, similar to observations of the rim instability in dewetting holes. The effect of dendrimer surface chemistry is obvious in the detailed structure of the self-assembled rings. G4 rings are diffuse and disordered with no evidence for layered growth. G4-25%C12 exhibits highly ordered ring structures and the onset of monomolecular terracing. G4-50%C12 exhibits highly periodic scallops and very distinct monomolecular height terraced growth of the rings with flat terraces and sharply defined steps. On the basis of these results, it is likely that the morphology of condensed molecule-based ring patterns formed by evaporation of microdroplets on surfaces can be used as a 'fingerprint' to identify, for example, solute molecule surface chemistry and concentration and function as a sensor for a variety of biochemical events.     

Turing pattern formation in anisotropic medium

Diffusion of reacting species in chemical and biochemical systems in anisotropic medium is markedly different from those occurring in isotropic medium, therefore approximating diffusion coefficients as constants may not be desirable as this has dynamical consequences. This paper is devoted to the analytical and numerical investigation of the development of spatial patterns in such systems. To this end we consider a general reaction–diffusion system with concentration-dependent diffusion and formulate a scheme to derive the general form of envelope equation for such systems. The theory is applied to the chlorite–iodide–malonic acid system, a standard paradigm for activator–inhibitor mechanism, to derive the instability condition in terms of the anisotropy parameters (\(\kappa _{i}, i = u, v\) that impart concentration-dependence to the diffusion coefficients) and identify the supercritical and subcritical Turing regions in the bifurcation diagram. The theoretical predictions are in good agreement with the numerical simulations.