Rabih Sultan

Bifurcation of the Ostwald-Liesegang supersaturation-nucleation-depletion cycle

Abstract A new formulation of the Ostwald supersaturation-nucleation-depletion cycle is adopted to model precipitate banding in rocks. In a special limiting case, the system is shown to support a propagating steady deposition pulse. The limit of existence is determined analytically for a simple model analogue of a pyrite/goethite system. Using numerical simulations, an unsteady pulse of mineral deposition is obtained and is shown to develop a variety of patterns with an appropriate change in the system parameters-apparently showing that the steady pulse obtained analytically is unstable. The domains of existence of unsteady pulse, undulatory patterns and discrete bands are determined.

Propagating fronts in 2D Cr(OH) 3 precipitate systems in gelled media

Abstract Diffusion fronts propagate as two co-precipitate ions inter-diffuse in a gel medium. Liesegang bands of precipitate form periodically behind the diffusion front of an outer electrolyte. The precipitation of Cr(OH)3 from NaOH diffusing into a Cr3+ gel matrix is known to yield a single band that propagates in a one-dimensional (1D) tube — Cr(OH)3 dissolves in excess OH− forming Cr(OH)4−. We perform similar experiments on the Cr(OH)3 system in two dimensions (2D), wherein we obtain a perfectly circular Cr(OH)3 ring that grows larger and thicker as time advances. Using a specially designed Petri dish, ring propagation is monitored both in the absence and the presence of a constant electric field. The field is applied along a radial direction, and the front velocities with the field on are compared with the field-free case. When the field is applied against the direction of front propagation (“negative” field), wave saturation is obtained, characterized by a slight increase in the velocity of propagation with field strength, until it reaches a constant value as the field strength is further increased. In a positive field situation, the wave velocity increases with field strength and exhibits some other interesting features: (1) wave stopping indicated by a freeze in the ring position at a certain characteristic time; (2) annihilation of the ring formation above a critical value of the field strength. Electrical effects in 2D are also studied when electrodes with different potentials are planted at various locations in the electrolyte periphery. Interesting patterning structures including the distortion of the circular symmetry and the birth of multiple rings are reported.

Mechanism of revert spacing in a PbCrO4 Liesegang system.

Periodic precipitation of sparingly soluble salts yields parallel Liesegang bands in 1D whose spacings obey either one of two known trends. The overwhelming trend is an increase in spacing as we move away from the junction, while some systems display a decrease in spacing as the bands get further away from the interface. The latter trend is much less common and is known as the revert spacing law. Whereas the direct (normal) spacing law is generally well-understood, the revert spacing trend has not been explicitly and distinctly elucidated. In this paper, we propose a mechanism of revert spacing governed by the adsorption of the diffusing CrO4(2−) ions on the formed PbCrO4 Liesegang bands and carry out a set of experiments that support the suggested scenario. It is shown that this adsorption increases as the band number (n) increases in revert spacing systems, while it decreases as n increases in direct spacing systems. It is concluded that this correlation in opposite directions decisively reveals the role of adsorption in the mechanism. The attraction between the CrO4(2−) and Pb(2+) in the gel causes the bands to form gradually closer and closer. Secondary structure (thinner bands formed within the main ones) obtained under some conditions is discussed in view of the light sensitivity of the chromate ion and the stability of the lead chromate sol.

Propagating fronts in periodic precipitation systems with redissolution

When co-precipitate ions interdiffuse in a gel medium, the sparingly soluble salt may precipitate in a stratification of parallel bands, traditionally known as Liesegang bands. In some salt systems, the precipitate can redissolve in excess diffusing electrolyte due to complex formation. As a result, the whole pattern propagates via band formation due to precipitation and band disappearance due to redissolution driven by complex formation. The properties of such unusual migrating Liesegang patterns were studied in our laboratory with an emphasis on the Co(OH)2 system. The latter is soluble in excess NH4OH because of the formation of the Co(NH3)62+ complex ion. We review the main experiments and results on that system and discuss potential directions for the extension of this research. In the Co(OH)2 propagation, the number of bands exhibits chaotic oscillations with time. The variation of the velocity of propagation with the concentrations of both inner and outer electrolytes was investigated, and the correlation between dissolution and precipitation was found to approach linearity at long times. An electric field of variable strength was applied across the propagation medium and a profound effect on the pattern properties was demonstrated. The introduction of Ni2+ ions, which compete with Co2+ for complex formation, induced oscillations in the band locations (at a given time) with concentration of diffusing electrolyte. It was shown that the intermediate species NH4+ is a precursor of such oscillations. The system was modeled using the theory of Polezhaev and Muller, incorporating ion diffusion, nucleation and kinetics of particle growth augmented by the dissolution process in a special kinetic scheme. The calculations agree qualitatively with the experimental results.

Effect of an electric field on propagating Co(OH)2 Liesegang patterns

Abstract The effect of an applied DC electric field on the propagation of a Co(OH) 2 Liesegang pattern from Co 2+ and NH 4 OH is investigated. The field free pattern is known to propagate down the tube due to band dissolution at the top and band formation at the bottom. At a fixed concentration of Co 2+ (0.134 M), the front propagation is accelerated by the field which is applied in the direction of wave propagation. The pattern propagates faster under a higher voltage. The dependence of wave velocity on field strength is non-linear. When the concentration of Co 2+ is varied at constant voltage (6.00 V), two opposite trends are obtained. Below a characteristic time t c =1.7 days, the velocity of propagation increases with decreasing concentration and above t c , the velocity increases with increasing concentration. This latter behavior (above t c ) completely reverses the field free trend. The effect of the field on the morphological appearance of the bands is discussed.

Ring Morphology and pH Effects in 2D and 1D Co(OH)2 Liesegang Systems

We study the factors that affect the morphology of Co(OH)(2) Liesegang rings, in a way to obtain concentric rings with large spacing, upon an appropriate variation in the experimental conditions. Such well-resolved patterns are obtained under optimum conditions: decrease in the concentration of the outer electrolyte, increase in the concentration of both the inner electrolyte and the gelatin in the hosting gel medium, and increase in the strength of a constant radial electric field applied across the pattern domain. The effect of pH on the bands in a 1D Co(OH)(2) Liesegang pattern is also investigated. The initial pH of the diffusing solution plays a central role in altering the band morphology, because the outer electrolyte (NH(4)OH) is a base, strongly affected by the H(+) equilibrium. The number of bands decreases and the interband spacing increases with decreasing pH of the NH(4)OH solution. The pattern morphology in that case is controlled by the NH(4)Cl/NH(4)OH ratio.

Periodic trends in precipitate patterning schemes involving two salts

Competitive particle growth (CPG) theory and linear stability analysis are combined to study the potential patterning trends of two-salt Liesegang systems. The pattern formation is treated as a post-nucleation phenomenon. Experimental investigations show that some systems mainly give patterns where the bands of the two salts essentially alternate, while others (such as typically the PbI2–PbF2 system) yield patterns where the bands completely overlap. We report on observations in a variety of two-salt systems, where we find a spectrum of patterning trends which generally fall into either of two categories: “correlated” patterns (with band overlap) or “anti-correlated” (with band alternation). The systems considered are Co(OH)2–Mg(OH)2, Ni(OH)2–Mg(OH)2, PbI2–PbF2, Ag2Cr2O7–PbCr2O7, MnS–CuS and MnS–CdS. The analysis of the reaction–diffusion equations shows that the simple precipitation scheme would predominantly yield an alternation of bands of the two salts. To resolve this disagreement between theory and experiment in the PbI2–PbF2 case (and some other cases), another scenario involving the complex ions PbI+ and PbF+ was also considered. The analysis here yields a dominant overlap between the PbI2 and PbF2 bands conformly to the experimental result. This study on two-precipitate pattern formation gives clues on the possible reaction schemes actually involved in the precipitation process and suggests interesting experiments for eventual elucidations. The results and conclusions are revisited in the Discussion section.

Density oscillations in precipitate domains of a propagating Cr(OH)3 ring

Abstract A propagating Cr(OH)3 band in 1D and a circular ring in 2D were observed and studied in PVA/GDA gels. In a recent simulation (Al-Ghoul and Sultan, J. Phys. Chem. A 107 (2003) 1095) using the model of Muller and Polezhaev, the precipitate density of the 1D pulse exhibited temporal oscillations at early times. We present here new experiments to verify this theoretical finding. Oscillations were indeed observed in the density and width of the precipitate ring, suggesting that the obtained pattern is that of a breathing pulse. The properties of the pulse studied in this Letter, capture the characteristics of a pulse of mineral deposit conjectured earlier (Earth-Sci. Rev. 29 (1990) 163) using a model of geochemical pattern formation.

On dynamic self-organization: examples from magmatic and other geochemical systems

Standard Liesegang banding is the display of parallel bands of precipitate formed periodically when co-precipitate ions interdiffuse in a gel medium. The most striking resemblance with Liesegang patterns in Nature lies in the diverse scenery of banded textural features commonly observed in some geological materials, such as geodes, agates, malachites, as well as stratigraphic units of certain rock formations. Here, we explore the possible relationship between the Liesegang banding scenario and magmatic-type pattern formation, such as zonations in km-scale circular zoned plutons and anorogenic ring complexes, cyclic layering in large mafic - ultramafic layered intrusions and orbicular granites, as well as in mm-scale crystal zonations. We also investigate magmatic processes such as fractional crystallization, and the ranges of T and p that are compatible with operating conditions for Liesegang banding. For geochemical self-organization to operate via a Liesegang-type mechanism, a necessary condition is that the system be transiently out of equilibrium, and be described by complex nonlinear kinetic laws. We examine the viability of the development of geochemical patterns, in relation with the various requirements for the growth of Liesegang structures.

Propagating Fronts in Thin Tubes: Concentration, Electric, and pH Effects in a Two-Dimensional Precipitation Pulse System

In this paper, we studied the dynamics of a CaCO3 precipitate deposition pulse in a thin, long tube connecting two reservoir sinks of coprecipitates. The pulse profile, as well as the time t(c) and distance x(c) of the first appearance of precipitate, is studied as a function of the initial concentration of CO(3)(2-) in the right reservoir, [CO(3)(2-)](0), and later as a function of an applied external electric field at different voltages. The time variations of the pulse location and the pH at the center of the tube are determined. The distance from the calcium chloride sink (x) at any fixed time decreases as [CO(3)(2-)](0) increases. The time evolution of the front location exhibits a crossover between an early time regime and a late time regime. The pH-time curve shows a marked resemblance with a sigmoid shape. At any time, the pH consistently increases with [CO(3)(2-)](0). In the presence of a constant electric field applied across the tube (fixed voltage), t(c) decreased with the field strength, whereas x(c) exhibited a correlated increase. Irregularities in the variation of distance with the applied voltage (at a fixed time) were noted. The pH experiences a slight increase with the applied voltage. The pulse width exhibits a nonlinear time dependence, of the form w = a + bt(1/6). The shape of the deposition pulse deviates from a Gaussian distribution. This study is of special interest in the experimental simulation and modeling of precipitate deposition and potential clogging in microcapillary channels.