Anisotropic Laplacian growths: From diffusion-limited aggregates to dendritic fractals

Saffman-Taylor fingers at intermediate noise

We study Saffman-Taylor flow in the presence of intermediate noise numerically by using both a boundary-integral approach as well as the Kadanoff-Liang modified diffusion-limited aggregation model that incorporates surface tension and reduced noise. For little to no noise, both models reproduce the well-known Saffman-Taylor finger. We compare both models in the region of intermediate noise, where we observe occasional tip-splitting events, focusing on the ensemble-average. We show that as the noise in the system is increased, the mean behavior in both models approaches the cos^{2}(πy/W) transverse density profile far behind the leading front. We also investigate how the noise scales and affects both models.

Fabrication of Multscale Fractal-Like Structures by Controlling Fluid Interface Instability

Nature, in quest for the best designs has shaped its vital systems into fractal geometries. Effectual way of spontaneous fabrication of scalable, ordered fractal-like structures by controlling Saffman-Taylor instability in a lifted Hele-Shaw cell is deployed here. In lifted Hele-Shaw cell uncontrolled penetration of low-viscosity fluid into its high-viscosity counterpart is known to develop irregular, non-repeatable, normally short-lived, branched patterns. We propose and characterize experimentally anisotropies in a form of spatially distributed pits on the cell plates to control initiation and further penetration of non-splitting fingers. The proposed control over shielding mechanism yields recipes for fabrication of families of ordered fractal-like patterns of multiple generations. As an example, we demonstrate and characterize fabrication of a Cayley tree fractal-like pattern. The patterns, in addition, are retained permanently by employing UV/thermally curable fluids. The proposed technique thus establishes solid foundation for bio-mimicking natural structures spanning multiple-scales for scientific and engineering use.

Supercooled water escaping from metastability

The return of supercooled water to a stable equilibrium condition is an irreversible process which, in large enough samples, takes place adiabatically. We investigated this phenomenon in water by fast imaging techniques. As water freezes, large energy and density fluctuations promote the spatial coexistence of solid and liquid phases at different temperatures. Upon synchronously monitoring the time evolution of the local temperature, we observed a sharp dynamic transition between a fast and a slow decay regime at about 266.6 K. We construe the observed phenomenon in terms of the temperature dependence of heat transfers from solid and liquid volumes already at their bulk coexistence temperature towards adjacent still supercooled liquid regions. These findings can be justified by observing that convective motions induced by thermal gradients in a supercooled liquid near coexistence are rapidly suppressed as the nucleated solid fraction overcomes, at low enough temperatures, a characteristic percolation threshold.

Oscillatory electrodeposition of metal films at liquid/liquid interfaces induced by the large surface energy of growing deposits

Electrodeposition of zinc (Zn) at an aqueous ZnSO4/n-butylacetate (BuAc) interface (liquid/liquid (LL) interface) showed a potential oscillation in the region of the current density exceeding the diffusion-limited one, accompanied by formation of two-dimensional Zn film with a concentric pattern at the LL interface. In-situ optical microscopic inspections revealed that the oscillatory growth of the Zn film synchronized with meniscus oscillation of the LL interface. The vigorous growth of the deposits occurs only when the shape of the meniscus becomes hollow on the negative potential side of the potential oscillation. On the other hand, on the positive side, the meniscus becomes almost flat and the deposits formed in the preceding stage are thickened. A mechanism is proposed to explain the oscillatory Zn electrodeposition coupled with the meniscus oscillation, on the basis of the fact that the interfacial tension at the growing metal/aqueous solution interface is extremely large.

Cu2O island shape transition during Cu-Au alloy oxidation

In situ transmission electron microscopy observations of the oxidation of (001) Cu-Au alloys indicate that the Cu2O islands that form undergo a remarkable transformation from an initially compact morphology to a dendritic structure as growth proceeds. Correspondingly, the surface composition becomes nonuniform and the fractal dimension associated with the islands evolves from 2.0 to a stable value of 1.87, indicating a transition in the rate-limiting mechanism of oxidation from oxygen surface diffusion to diffusion of copper through the increasingly gold-rich regions adjacent to the islands.

Anisotropic diffusion-limited aggregation

Using stochastic conformal mappings, we study the effects of anisotropic perturbations on diffusion-limited aggregation (DLA) in two dimensions. The harmonic measure of the growth probability for DLA can be conformally mapped onto a constant measure on a unit circle. Here we map m preferred directions for growth to a distribution on the unit circle, which is a periodic function with m peaks in [-pi,pi) such that the angular width sigma of the peak defines the "strength" of anisotropy kappa= sigma(-1) along any of the m chosen directions. The two parameters (m,kappa) map out a parameter space of perturbations that allows a continuous transition from DLA (for small enough kappa ) to m needlelike fingers as kappa--> infinity. We show that at fixed m the effective fractal dimension of the clusters D(m,kappa) obtained from mass-radius scaling decreases with increasing kappa from D(DLA) approximately 1.71 to a value bounded from below by D(min) = 3 / 2. Scaling arguments suggest a specific form for the dependence of the fractal dimension D(m,kappa) on kappa for large kappa which compares favorably with numerical results.

Patterned and controlled polyelectrolyte fractal growth and aggregations

Two-dimensional patterned and controlled polyelectrolyte aggregations (e.g., tree-like ramified structures) created by microcontact printing have been demonstrated and discussed. Polyelectrolyte-micropatterned aggregations on surfaces were controlled by the micropattern size and shape of PDMS stamps. The formation of aggregates was dependent on the ink and surface conditions, and the aggregates consisted of two distinct layers; strongly adsorbed, primary uniform layers and weakly adsorbed, secondary aggregation layers positioned on top of the primary layers. The adsorption of the primary layers was strong enough not to be washed away, while the aggregated secondary layers were easily removed by washing. The aggregation of secondary layers showed typical tree-like ramified structures of fractal growth and aggregation. Directional and confined stamping led to directing and confining the growth of the fractal polyelectrolyte clusters, respectively. The micropatterned primary uniform layers were not removed by extensive washing, and they were identified by selective nickel plating and charged particle selective adsorption in which the surface formed positive and negative micropatterns. These functional and patterned surfaces have great potentials for advanced devices and sensors.

Nucleation and growth of molecular organic crystals in a liquid film under vapor deposition

We report on the nucleation and growth of tetracene in a thin liquid film which is continuously supersaturated by vapor deposition of molecules onto the film. In a first stage, nucleation and fast anisotropic two-dimensional dendritic growth occurs. In a second stage, the dendrites coarsen into pallet-shaped crystals. These are highly oriented with respect to the plane of the liquid film and reach a lateral size of several 100 microm. The two-dimensional growth mode is explained by the confined growth geometry in the liquid in combination with the anisotropy of the crystal structure.

Interface pattern formation in nonlinear dissipative systems

The problem of interface pattern selection in nonlinear dissipative systems is critical in many fields of science, occurring in physical, chemical and biological systems. One of the simplest pattern formations is the Saffman-Taylor finger pattern that forms when a viscous fluid is displaced by a less viscous fluid. Such finger-shaped patterns have been observed in distinctly different fields of science (hydrodynamics, combustion and crystal growth) and this has led to a search for a unified concept of pattern formation, as first proposed by the classic work of D'arcy Thomson. Two-dimensional finger-shaped patterns, observed in flame fronts and the ensembled average shape of the diffusion-limited aggregation pattern, have been shown to be similar to Saffman-Taylor finger shapes. Here we present experimental studies that establish that the cell shapes formed during directional solidification of alloys can be described by the form of the Saffman-Taylor finger shape equation when a second phase is present in the intercellular region.

Mean-field diffusion-limited aggregation: a "density" model for viscous fingering phenomena

We explore a universal "density" formalism to describe nonequilibrium growth processes, specifically, the immiscible viscous fingering in Hele-Shaw cells (usually referred to as the Saffman-Taylor problem). For that we develop an alternative approach to the viscous fingering phenomena, whose basic concepts have been recently published in a Rapid Communication [Phys. Rev. E 63, 045305(R) (2001)]. This approach uses the diffusion-limited aggregation (DLA) paradigm as a core: we introduce a mean-field DLA generalization in stochastic and deterministic formulations. The stochastic model, a quasicontinuum DLA, simulates Monte Carlo patterns, which demonstrate a striking resemblance to natural Hele-Shaw fingers and, for steady-state growth regimes, follow precisely the Saffman-Taylor analytical solutions in channel and sector configurations. The relevant deterministic theory, a complete set of differential equations for a time development of density fields, is derived from that stochastic model. As a principal conclusion, we prove an asymptotic equivalency of both the stochastic and deterministic mean-field DLA formulations to the classic Saffman-Taylor hydrodynamics in terms of an interface evolution.

Bridge from diffusion-limited aggregation to the Saffman-Taylor problem

We introduce a Monte Carlo mean-field scheme for the diffusion-limited aggregation (DLA) model, in order to simulate processes of viscous fingering. The patterns obtained demonstrate a striking resemblance to natural shapes in Hele-Shaw cells, reproducing the Saffman-Taylor analytical solutions in the stable regime. The corresponding deterministic equations of the mean-field DLA scheme are derived and studied.