Complex motion of precipitation bands

Diffusion-controlled annihilation A+B→0: Coalescence, fragmentation, and collapse of nonidentical A-particle islands submerged in the B-particle sea

We present a systematic analysis of diffusion-controlled interaction and collapse of two nonidentical spatially separated d-dimensional A-particle islands in the B-particle sea at propagation of the sharp reaction front A+B→0 at equal species diffusivities. We show that at a sufficiently large initial distance between the centers of islands 2ℓ and a relatively large initial ratio of island-to-sea concentrations, the evolution dynamics of the island-sea-island system demonstrates remarkable universality and, depending on the system dimension, is determined unambiguously by two dimensionless parameters Λ=N_{0}^{+}/N_{Ω} and q=N_{0}^{-}/N_{0}^{+}, where N_{0}^{+} and N_{0}^{-} are the initial particle numbers in the larger and smaller of the islands, respectively, and N_{Ω} is the initial number of sea particles in the volume Ω=(2ℓ)^{d}. We find that at each fixed 0<q≤1, there are threshold values Λ_{★}(q) and Λ_{s}(q)≥Λ_{★}(q) that depend on the dimension and separate the domains of individual death of each of the islands Λ<Λ_{★}(q), coalescence and subsequent fragmentation (division) of a two-centered island Λ_{★}(q)<Λ<Λ_{s}(q), and collapse of a single-centered island formed by coalescence Λ>Λ_{s}(q). We demonstrate that regardless of d, the trajectories of the island centers are determined unambiguously by the parameter q, and we reveal a detailed picture of the evolution of islands and front trajectories with an increase in Λ, focusing on the scaling laws of evolution at the final collapse stage and in the vicinity of starting coalescence and fragmentation points.

Diffusion-controlled coalescence, fragmentation, and collapse of d-dimensional A-particle islands in the B-particle sea

We present a systematic analysis of diffusion-controlled interaction and collapse of two identical spatially separated d-dimensional A-particle islands in the B-particle sea at propagation of the sharp reaction front A+B→0 at equal species diffusivities. We show that at a sufficiently large initial distance between the centers of islands 2ℓ compared to their characteristic initial size and a relatively large initial ratio of island to sea concentrations, the evolution dynamics of the island-sea-island system is determined unambiguously by the dimensionless parameter Λ=N_{0}/N_{Ω}, where N_{0} is the initial particle number in the island and N_{Ω} is the initial number of sea particles in the volume Ω=(2ℓ)^{d}. It is established that (a) there is a d-dependent critical value Λ_{★} above which island coalescence occurs; (b) regardless of d the centers of each of the islands move toward each other along a universal trajectory merging in a united center at the d-dependent critical value Λ_{s}≥Λ_{★}; (c) in one-dimensional systems Λ_{★}=Λ_{s}, therefore, at Λ<Λ_{★} each of the islands dies individually, whereas at Λ>Λ_{★} coalescence is completed by collapse of a single-centered island in the system center; (d) in two- and three-dimensional systems in the range Λ_{★}<Λ<Λ_{s} coalescence is accompanied by subsequent fragmentation of a two-centered island and is completed by individual collapse of each of the islands. We discuss a detailed picture of coalescence, fragmentation, and collapse of islands focusing on evolution of their shape and on behavior of the relative width of the reaction front at the final collapse stage and in the vicinity of starting coalescence and fragmentation points. We demonstrate that in a wide range of parameters, the front remains sharp up to a narrow vicinity of the coalescence, fragmentation, and collapse points.

Diffusion-controlled formation and collapse of a d-dimensional A-particle island in the B-particle sea

We consider diffusion-controlled evolution of a d-dimensional A-particle island in the B-particle sea at propagation of the sharp reaction front A+B→0 at equal species diffusivities. The A-particle island is formed by a localized (point) A-source with a strength λ that acts for a finite time T. We reveal the conditions under which the island collapse time t_{c} becomes much longer than the injection period T (long-living island) and demonstrate that regardless of d the evolution of the long-living island radius r_{f}(t) is described by the universal law ζ_{f}=r_{f}/r_{f}^{M}=sqrt[eτ|lnτ|], where τ=t/t_{c} and r_{f}^{M} is the maximal island expansion radius at the front turning point t_{M}=t_{c}/e. We find that in the long-living island regime the ratio t_{c}/T changes with the increase of the injection period T by the law ∝(λ^{2}T^{2-d})^{1/d}, i.e., increases with the increase of T in the one-dimensional (1D) case, does not change with the increase of T in the 2D case and decreases with the increase of T in the 3D case. We derive the scaling laws for particles death in the long-living island and determine the limits of their applicability. We demonstrate also that these laws describe asymptotically the evolution of the d-dimensional spherical island with a uniform initial particle distribution generalizing the results obtained earlier for the quasi-one-dimensional geometry. As striking results, we present a systematic analysis of the front relative width evolution for fluctuation, logarithmically modified, and mean-field regimes, and we demonstrate that in a wide range of parameters the front remains sharp up to a narrow vicinity of the collapse point.

Self-organization in precipitation reactions far from the equilibrium

Far from the thermodynamic equilibrium, many precipitation reactions create complex product structures with fascinating features caused by their unusual origins. Unlike the dissipative patterns in other self-organizing reactions, these features can be permanent, suggesting potential applications in materials science and engineering. We review four distinct classes of precipitation reactions, describe similarities and differences, and discuss related challenges for theoretical studies. These classes are hollow micro- and macrotubes in chemical gardens, polycrystalline silica carbonate aggregates (biomorphs), Liesegang bands, and propagating precipitation-dissolution fronts. In many cases, these systems show intricate structural hierarchies that span from the nanometer scale into the macroscopic world. We summarize recent experimental progress that often involves growth under tightly regulated conditions by means of wet stamping, holographic heating, and controlled electric, magnetic, or pH perturbations. In this research field, progress requires mechanistic insights that cannot be derived from experiments alone. We discuss how mesoscopic aspects of the product structures can be modeled by reaction-transport equations and suggest important targets for future studies that should also include materials features at the nanoscale.

Propagation Properties of the Precipitation Band in an AlCl₃/NaOH System

When inherently immobile solid particles collectively form precipitates in a reaction-diffusion system involving a redissolution reaction, a propagation phenomenon may occur in which a dynamic pattern of precipitation bands forms. This propagating precipitation phenomenon has been studied by many researchers. However, two completely different processes-i.e., the reaction-diffusion of reactants and the crystal growth of products-progress simultaneously in the system, thereby rendering the phenomenon complex. There are no well-established experimental laws for this propagating precipitation phenomenon, such as the spacing, time, and width laws associated with the well-known Liesegang phenomenon, which is static in the sense that precipitation bands form and remain at the same position. In fact, it has not been clarified which of the processes controls the propagation phenomenon. Accordingly, we have investigated the apparent diffusion coefficient associated with the dynamics of propagating precipitation band in an AlCl3/NaOH system for the case in which a large excess of outer electrolytes (i.e., OH(-)) diffuses into gel in which inner electrolytes (i.e.,Al(3+)) are homogeneously distributed. An isolated precipitation band of Al(OH)3 was formed horizontally in a test tube and propagated vertically in proportion to the square root of time. In our experimental results, we found that the apparent diffusion coefficient, D(p), possesses an exponential dependence on the initial concentrations of the outer electrolyte, and the inner electrolyte; the measured relation was D(p) = D[Al(3+)](-0.6)[OH(-)](0.6), where D = (0.63 ± 0.04) × 10(5) cm(2)/s. From our model equations based on the prenucleation theory, which take into account a redissolution reaction, we found that the dynamics of the reaction front of the outer and the inner electrolytes was an important factor in controlling the propagation of the precipitation band. In our simulation results, we obtained a similar dependence of the apparent diffusion coefficient on the electrolyte concentrations.

Protrusion and rising of a gel-based precipitation layer

Two types of unstable growth of a precipitation layer in gel are discussed. A cation and an anion that are reactive diffuse from opposite ends of the gel to its center. A white turbid zone forms due to their reactions. When the concentration ratios for both the ions are far from stoichiometry, the turbid zone expands toward the lower-concentration side. However, when the ratio is nearly stoichiometric, unstable growth occurs. In a glass tube, a protrusion of the precipitation region from the turbid zone grows, which forms a long needle-like shape. When a free surface is present on the gel, the precipitation region protrudes from the gel surface to form a rising structure. Mapping the growing structure on a concentration diagram and using scanning electron microscopy to examine contained particles suggest that the reaction is restricted to a narrow region and the reaction product migrates through a path formed in the protrusive structure to form a bulk solid at the edge.

Coarsening of precipitation patterns in a moving reaction-diffusion front

Precipitation patterns emerging in a two-dimensional moving front are investigated on the example of NaOH diffusing into a gel containing AlCl3 . The time evolution of the precipitate Al(OH)_{3} can be observed since the precipitate redissolves in the excess outer electrolyte NaOH and thus it exists only in a narrow optically accessible region of the reaction front. The patterns display self-similar coarsening with a characteristic length xi increasing with time as xi(t) approximately sqrt[t] . A theory based on the Cahn-Hilliard phase-separation dynamics, including redissolution, is shown to yield agreement with the experiments.

Moving blue band caused by cooperation of diffusion and phase separation

Diffusions of Cu(2+) and Fe(3+) in gelatin generate a moving blue band. It is formed by a diffusion of Cu(2+) and a phase separation of gelatin with diffusing Fe(3+). The diffusing Fe(3+) forms Fe(OH)(3) colloids and gathers gelatin molecules from the surroundings. The diffusion of gelatin molecules generates the concentration gradient, resulting in a gel/sol transition in the dilute phase. In the region where the concentration of Fe(3+) is high enough, the gel remains hard, while a sol phase appears under the hard gel. The absorption spectrum of Cu(2+) depends on the concentration ratio of Cu(2+) to gelatin. As a consequence, we can see a blue band in the restricted region between the diffusing front of Cu(2+) and the phase separation front. The movement of the blue band is caused by a coupling of a simple diffusion and the phase separation.