Statistical properties of fractal dendrites and anisotropic diffusion-limited aggregates

Dendritic patterns from shear-enhanced anisotropy in nematic liquid crystals

Controlling the growth morphology of fluid instabilities is challenging because of their self-amplified and nonlinear growth. The viscous fingering instability, which arises when a less viscous fluid displaces a more viscous one, transitions from exhibiting dense-branching growth characterized by repeated tip splitting of the growing fingers to dendritic growth characterized by stable tips in the presence of anisotropy. We controllably induce such a morphology transition by shear-enhancing the anisotropy of nematic liquid crystal solutions. For fast enough flow induced by the finger growth, the intrinsic tumbling behavior of lyotropic chromonic liquid crystals can be suppressed, which results in a flow alignment of the material. This microscopic change in the director field occurs as the viscous torque from the shear flow becomes dominant over the elastic torque from the nematic potential and macroscopically enhances the liquid crystal anisotropy to induce the transition to dendritic growth.

Growth morphology and symmetry selection of interfacial instabilities in anisotropic environments

The displacement of a fluid by another less viscous one in a quasi-two dimensional geometry typically leads to complex fingering patterns. In an isotropic system, dense-branching growth arises, which is characterized by repeated tip-splitting of evolving fingers. When anisotropy is present in the interfacial dynamics, the growth morphology changes to dendritic growth characterized by regular structures. We introduce anisotropy by engraving a six-fold symmetric lattice of channels on a Hele-Shaw cell. We show that the morphology transition in miscible fluids depends not only on the previously reported degree of anisotropy set by the lattice topography, but also on the viscosity ratio between the two fluids, ηin/ηout. Remarkably, ηin/ηout and the degree of anisotropy also govern the global features of the dendritic patterns, inducing a systematic change from six-fold towards twelve-fold symmetric dendrites. Varying either control parameter provides a new method to tune the symmetry of complex patterns, which may also have relevance for analogous phenomena of gradient-driven interfacial dynamics, such as directional solidification or electrodeposition.

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.

Effect of mobility ratio on interaction between the fingers in unstable growth processes

We investigate interactions between thin fingers formed as a result of an instability of an advancing front in growth processes. We show that the fingers can both attract and repel each other, depending on their lengths and the mobility ratio between the invading and displaced phase. To understand the origin of these interactions we introduce a simple resistor model of the fingers. The predictions of the model are then compared to the numerical simulations of two unstable growth processes: dissolution of partially cemented rock fracture and viscous fingering in a regular network of channels.

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.

Growth rate distribution of NH4Cl dendrite and its scaling structure

Scaling structure of the growth rate distribution on the interface of a dendritic pattern is investigated. The distribution is evaluated for an NH4Cl quasi-two-dimensional crystal by numerically solving the Laplace equation with the boundary condition taking account of the surface tension effect. It is found that the distribution has multifractality and the surface tension effect is almost ineffective in the unscreened large growth region. The values of the minimum singular exponent and the fractal dimension are smaller than those for the diffusion-limited aggregation pattern. The Makarov's theorem, the information dimension equals one, and the Turkevich-Scher conjecture between the fractal dimension and the minimum singularity exponent hold.

Fingering and pressure distribution in lifting Hele-Shaw cells with grooves: A computer simulation study

We report a computer simulation study of viscous fingering patterns in a lifting Hele-Shaw cell with grooves. Here circular or square grooves concentric to the plates are etched on the lower plate. Experiments show that the presence of such grooves affect formation of viscous fingers quite strongly. We report a simulated pressure map generated in such grooved cells, when the two plates are separated with a constant force and compare the patterns with experiments. Variation in the simulated patterns for different fluid viscosity and lifting force is also studied.

Fingered growth in channel geometry: a Loewner-equation approach

A simple model of Laplacian growth is considered, in which the growth takes place only at the tips of long, thin fingers. Following Carleson and Makarov [L. Carleson and N. Makarov, J. Anal. Math. 87, 103 (2002)], the evolution of the fingers is studied with use of the deterministic Loewner equation. The method is then extended to study the growth in a linear channel with reflecting sidewalls. One- and two-finger solutions are found and analyzed. It turns out that the presence of the walls has a significant influence on the shapes of the fingers and the dynamics of the screening process, in which longer fingers suppress the growth of the shorter ones.

Scale-invariant competitive growth of side branches in a dendritic crystal

We experimentally investigated statistical properties of side branches of quasi-two-dimensional NH(4)Cl dendritic crystals. The height distributions of the side branches and their number density exhibit scale-invariant power laws. The results are in good agreement with the results of numerical simulations and theories of diffusion-limited needle growth. Our scaling exponents are independent of supersaturation and the statistical properties are universal in dendrites.

Side-branch growth in two-dimensional dendrites. II. Phase-field model

The development of side-branching in solidifying dendrites in a regime of large values of the Peclet number is studied by means of a phase-field model. We have compared our numerical results with experiments of the preceding paper and we obtain good qualitative agreement. The growth rate of each side branch shows a power-law behavior from the early stages of its life. From their birth, branches which finally succeed in the competition process of side-branching development have a greater growth exponent than branches which are stopped. Coarsening of branches is entirely defined by their geometrical position relative to their dominant neighbors. The winner branches escape from the diffusive field of the main dendrite and become independent dendrites.

Side-branch growth in two-dimensional dendrites. I. Experiments

The dynamics of growth of dendrites' side branches is investigated experimentally during the crystallization of solutions of ammonium bromide in a quasi-two-dimensional cell. Two regimes are observed. At small values of the Peclet number a self-affine fractal forms. In this regime it is known that the mean lateral front grows as t(0.5). Here the length of each individual branch is shown to grow (before being screened off) with a power-law behavior t (alpha(n)). The value of the exponent alpha(n) (0.5< or = alpha(n) < or =1) is determined from the start by the strength of the initial disturbance. Coarsening then takes place, when the branches of small alpha(n) are screened off by their neighbors. The corresponding decay of the growth of a weak branch is exponential and defined by its geometrical position relative to its dominant neighbors. These results show that the branch structure results from a deterministic growth of initially random disturbances. At large values of the Peclet number, the faster of the side branches escape and become independent dendrites. The global structure then covers a finite fraction of the two-dimensional space. The crossover between the two regimes and the spacing of these independent branches are characterized.

Numerical study of the shape and integral parameters of a dendrite

We present a numerical study of sidebranching of a solidifying dendrite by means of a phase-field model. Special attention is paid to the regions far from the tip of the dendrite, where linear theories are no longer valid. Two regions have been distinguished outside the linear region: a first one in which sidebranching is in a competition process and a second one further down where branches behave as independent of each other. The shape of the dendrite and integral parameters characterizing the whole dendrite (contour length and area of the dendrite) have been computed and related to the characteristic tip radius for both surface tension and kinetic dominated dendrites. Conclusions about the different behaviors observed and comparison with available experiments and theoretical predictions are presented.

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.

Laplacian growth with separately controlled noise and anisotropy

Conformal mapping models are used to study the competition of noise and anisotropy in Laplacian growth. For this purpose, a family of models is introduced with the noise level and directional anisotropy controlled independently. Fractalization is observed in both anisotropic growth and growth with varying noise. The fractal dimension is determined from the cluster size scaling with cluster area. For isotropic growth d=1.7, at both high and low noise. For anisotropic growth with reduced noise the dimension can be as low as d=1.5 and apparently is not universal. Also, we study the fluctuations of particle areas and observe, in agreement with previous studies, that exceptionally large particles may appear during growth, leading to pathologically irregular clusters. This difficulty is circumvented by using an acceptance window for particle areas.

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.

The Silicate Garden Reaction in Microgravity: A Fluid Interfacial Instability

In the "silicate garden" reaction, crystals of a metal salt are placed in sodium silicate solution. The crystals become coated with a semipermeable membrane of metal silicate reaction product, from which hollow tubes of metal silicate rise convectively upward, against gravity. In the absence of gravity, and free of convective influences, the reaction might be expected to reveal more fundamental organizing principles. Accordingly, we have grown silicate gardens in microgravity, from salts of calcium, cobalt, and magnesium with added nickel. Even in these isotropic conditions, complex structures developed. They included tubes and hollow spheroids, and also novel dendritic "fingers" which grew by continuous plastic deformation. This new mode of growth is favored by the slow, diffusion-limited rate of reaction in microgravity, which greatly reduces the rate of hardening of the reaction products. The magnesium-nickel garden grew almost entirely as a fluid interfacial instability between the metal salt solution inside and the silicate solution outside, by deformation of the semipermeable fluid membrane between them. The resulting shape had similarities to that of a solid front advancing through a supercooled melt. The morphology of such a solid is determined by the diffusion of released latent heat away from it, according to the Laplacian diffusion equation. We suggest that Laplacian-growth morphology arises in a microgravity silicate garden when its development is controlled by the analogous diffusion of dissolved ions away from it. Copyright 1998 Academic Press.