Labyrinthine Patterns Research Articles

Combinatorial Study of Surface Pattern Formation in Thin Block Copolymer Films

Surface pattern formation in diblock copolymer films is investigated as a function of film thickness h and molecular mass M. Smooth films are observed for certain h ranges centered about multiples of the lamellar thickness L0, and we attribute this effect to an increase in the surface chain density with h in the outer brushlike copolymer layer. We also observe apparently stable labyrinthine surface patterns for other h ranges, and the average size of these patterns is found to scale as lambda approximately L0(-2.5). Hole and island patterns occur for h ranges between those of the labyrinthine patterns and the smooth regions, and their size similarly decreases with L0 and M.

Front Aggregation and Labyrinthine Pattern in the Drying Process of Two-Dimensional Wet Granular Systems

The aggregation process of granules driven by the motion of water-air fronts in the drying process of two-dimensional wet granular systems is experimentally studied. The fronts and aggregated granules interact with each other during this drying, and a labyrinthine pattern is formed as a result. A qualitative framework of the pattern formation is given, and topological and metrical properties of the pattern are also discussed.

Harmonic patterns in fine granular vibrated beds

We examine the spontaneous formation of numerous previously unreported patterns in deep beds of fine grains vibrated at frequencies ranging from 10 to 100 Hz and accelerations ranging from 1 to 14 times gravity. Similarly to shallow beds, we find stripes, closed cells and labyrinthine patterns. In addition, we observe traveling waves, cannibalizing cells and other behaviors that are unexpected and remain unexplained. All of these patterns vary spatiotemporally, and unlike shallow bed patterns, are harmonic rather than subharmonic.

Self-similar domain growth, localized structures, and labyrinthine patterns in vectorial Kerr resonators

We study domain growth in a nonlinear optical system useful to explore different scenarios that might occur in systems which do not relax to thermodynamic equilibrium. Domains correspond to equivalent states of different circular polarization of light. We describe three dynamical regimes: a coarsening regime in which dynamical scaling holds with a growth law dictated by curvature effects, a regime in which localized structures form, and a regime in which polarization domain walls are modulationally unstable and the system freezes in a labyrinthine pattern.

Splitting Fingers under Strain: Pattern Formation of a Dipolar Fluid in a Polymer Medium

We study the formation and evolution of labyrinthine patterns for dipolar fluids inside a polymer medium under mechanical strain. A new general mechanism is proposed to describe the kinetics of the pattern formation that combines orientation ordering and subsequently breaking characteristics of dipolar fluids with branching characteristics of crack-propagation. This mechanism can lead to many of the patterns found in nature, including spots, stripes, labyrinthine, and starlike formations.

Ovarian Mucinous Cystadenocarcinoma with Yolk- Sac Tumor-in a 71-Year-Old Woman

An ovarian mucinous cystadenocarcinoma admixed with a component of yolk sac tumor was encountered in a 71-year-old woman, the first such tumor to be reported. The preoperative serum level of alpha-fetoprotein was elevated (55.6 ng/mL). A right ovarian cystic tumor, 8 cm in maximal dimension, was removed and found to have a solid component. Microscopically, the cystic lesion was a mucinous cystadenocarcinoma, whereas the solid portion was typical yolk sac tumor with microcystic, reticular, and labyrinthine patterns, Schiller-Duval bodies, scattered giant cells with bizarre nuclei, and intracellular and extracellular hyaline globules. There was no evidence of other germ cell tumor components. Yolk sac tumor showed positive reactions for alpha-fetoprotein, carcinoembryonic antigen, and alpha-1-antitrypsin but was negative for CA125, CA19-9, and human chorionic gonadotrophin. The patient died of recurrent tumor 6 months postoperatively without response to combination chemotherapy.

Numerical Study of the Role of Magnetic Field Ramping Rate on the Structure Formation in Magnetorheological Fluids

A molecular dynamics model is presented to understand the structural formation of MR fluids by including the thermal motion of the particles. The simulation results indicate that the complexity of the lateral pattern as viewed in the direction of the applied field increases with the rate of the application of external magnetic field. We have also found that the maximum range for attractive interaction (escape distance) for two initially straight chains increases with temperature. These results are relevant to understand the mechanisms and conditions for the formation of labyrinthine and columnar patterns found in MR fluids.

Hexagons and interfaces in a vibrated granular layer

The order parameter model based on parametric Ginzburg-Landau equation is used to describe high acceleration patterns in vibrated layer of granular material. At large amplitude of driving both hexagons and interfaces emerge. Transverse instability leading to formation of ``decorated'' interfaces and labyrinthine patterns, is found. Additional sub-harmonic forcing leads to controlled interface motion.

Characterization of Labyrinthine Patterns and Their Evolution

An invariant measure δ is introduced to quantify the disorder in extended locally striped patterns. It is invariant under Euclidean motions of the pattern, and vanishes for a uniform array of stripes. Irregularities such as point defects and domain walls make nonzero contributions to the measure. The evolution of random initial states to labyrinthine patterns is analyzed through the time evolution of δ. This behavior is configuration independent, and exhibits two phases each with a logarithmic decay in δ.

Characterizations of natural patterns

Labyrinthine patterns are observed in systems as diverse as animal coats, slime mold colonies, fish scales, and cloud streets. However, even under well-controlled conditions, repetition of an experiment gives structures with vastly different details. A theoretical analysis of ``universal'' aspects of patterns requires a quantitative description that gives similar values for distinct configurations. We introduce a function to characterize the ``disorder'' of labyrinthine patterns that is the same for structures generated under identical control parameters. Furthermore, patterns with different visual characteristics are described by distinct functions.

Disordered waves in an excitable medium consisting of two layers: Catalytic gel and noncatalytic solution

Disordered waves in the light-sensitive Belousov-Zhabotinsky reaction (``ripples,'' i.e., aperiodic wave front deformations, labyrinthine patterns, and ``frazzle gases,'' i.e., erratically moving wave fragments) were observed experimentally a few years ago, but have so far not been explained and modeled. The system, consisting of a gel with immobilized catalyst covered by a catalyst-free solution, is motionless and horizontally homogeneous. The observed disorder is related to enhancement or breakups of fluctuation valleys. In the model presented here, this destabilization is caused by inhibitor produced in the refractory tail that diffusively surpasses the wave front via the upper solution layer. This is in agreement with experiments both qualitatively, i.e., in the overall appearance of the patterns, as well as quantitatively, regarding the eikonal relationship and autocorrelation analysis.

Self-replicating spots in reaction-diffusion systems

In this article we discuss the self-replicating spot, a particlelike phenomenon that occurs in reaction-diffusion ~RD! systems @1#. The spots consist of localized regions in which the concentrations of the reactants differ from the surrounding concentration fields. They grow, reaching a critical size at which time they divide in two. The two resulting spots again grow and divide. This process, which is visually similar to cell division, continues indefinitely. The long-time behavior depends on the precise values of the external control parameters, but typically consists of a chaotic ‘‘soup’’ in which many spots compete for resources as illustrated in Fig. 1. Those spots that find adequate resources continue to grow and divide. Those that are unable to find adequate resources decay into the background. The spots observed in @1# were found during an attempt to model labyrinthine patterns in the ferrocyanide-iodate-sulfate ~FIS! reaction @2#. Since then, replicating spot patterns have been observed both numerically and experimentally in the FIS reaction @3,4#. There are obvious differences in the Gaspar-Showalter model of the FIS reaction @5# and of the models discussed by us and others. This fact suggests that replication is a generic feature characterizing a broad class of reaction-diffusion systems. In @6#, we presented some arguments in support of this proposition. These arguments included both a heuristic description of the process of replication and demonstrations of analytic features common to several related model RD systems. It turns out that replication is more general than our analysis accounts for. Nevertheless, we think it worthwhile to spell out the details of the theory presented in @6#. We remark here that various aspects of the replication phenomenon have been discussed by other authors @7,8#. Kerner and Osipov have a large body of work on largeamplitude dissipative structures including an analysis of the static division of one-dimensional pulses as the system size is changed. Gurevich and Mints have a body of work on replication of thermal hot spots in composite superconductors. In the article by Petrov, Scott, and Showalter replication

Microscopic model for FitzHugh-Nagumo dynamics

A microscopic reaction model with a FitzHugh-Nagumo mass action law is introduced. A Markov chain that uses a birth-death description of the reaction mechanism and a random walk model for diffusion is constructed and implemented as a lattice-gas automaton. It is shown that the local particle density probability distribution is binomial in the high diffusion limit and the average particle density is governed by the FitzHugh-Nagumo reaction-diffusion equation. The lattice-gas simulations are able to reproduce phenomena such as labyrinthine patterns and Bloch fronts predicted to exist on the basis of the reaction-diffusion equation. The effects of fluctuations on these chemical patterns, the breakdown of the mass-action and reaction-diffusion descriptions, and the existence of phase transitions in the strong reaction limit are discussed.

Ordered structures in Fe3O4 kerosene-based ferrofluids

The patterns of kerosene-based ferrofluid films in applied parallel and perpendicular magnetic fields were studied. The time dependence of quasiperiodic chains in a parallel magnetic field, the perfect hexagonal crystal structure in a weaker perpendicular field, and the labyrinthine pattern in a stronger perpendicular field were observed. The Fe3O4 kerosene-based ferrofluids that we used in this study were synthesized by the coprecipitation method. As the parallel external magnetic field was applied to a ferrofluid film, the ordered quasiperiodic chains were obtained. On the other hand, the initial nonequilibrium disordered quantum columns were formed in an applied perpendicular magnetic field, and an equilibrium hexagonal structure with columns occupying its vortices was formed after two hours. The distance between these columns decreases as the strength of the applied magnetic field increases, and hence, the patterns change gradually from the hexagonal structure to a labyrinthine pattern with the strength of the perpendicular field above a critical value. Thus, a phase transition exists in the ferrofluid film system as the field strength increases.

Scenarios of domain pattern formation in a reaction-diffusion system.

We performed an extensive numerical study of a two-dimensional reaction-diffusion system of the activator-inhibitor type in which domain patterns can form. We showed that both multidomain and labyrinthine patterns may form spontaneously as a result of Turing instability. In the stable homogeneous system with the fast inhibitor one can excite both localized and extended patterns by applying a localized stimulus. Depending on the parameters and the excitation level of the system stripes, spots, wriggled stripes, or labyrinthine patterns form. The labyrinthine patterns may be both connected and disconnected. In the the stable homogeneous system with the slow inhibitor one can excite self-replicating spots, breathing patterns, autowaves and turbulence. The parameter regions in which different types of patterns are realized are explained on the basis of the asymptotic theory of instabilities for patterns with sharp interfaces developed by us in Phys. Rev. E. 53, 3101 (1996). The dynamics of the patterns observed in our simulations is very similar to that of the patterns forming in the ferrocyanide-iodate-sulfite reaction.

Labyrinthine patterns in a self-avoiding growing string

We propose a model of a self-avoiding growing string with excluded volume effects. In this model, the string develops to form a complex shape like a labyrinth. We find simple dynamical scaling relations for several characteristics quantities, including the radius of gyration and radius of curvature. Detailed numerical calculations show that there are two length scales and one time scale characterizing the shape transformations of the growing string.

Theory of self-replication and granulation of spike autosolitons

Recent experimental and numerical studies have shown that self-replication and granulation of the solitary patterns may lead to labyrinthine patterns or turbulence. We propose mechanisms for self-replication and granulation of one of the basic types of solitary patterns — spike autosolitons. We find the conditions for these effects to occur on the basis of an analytical study of the shape and stability of static spike autosolitons in a reaction-diffusion system. We prove that these effects are related to the buildup of the short-wave perturbations of a certain shape.

FIELD-INDUCED LABYRINTHINE PATTERNS IN FERROFLUID EMULSIONS

A ferrofluid emulsion goes through gas — solid phase transition when an external magnetic field is applied. The solid structures are observed as either column, bent-wall, or labyrinthine patterns. The appearance of these different patterns depend upon the rate of the field applied, thickness of the sample cell along the field direction, and the volume fraction of the emulsion droplets used. Using optical microscopy, formed patterns are recorded and analyzed in which a “phase” diagram of the structural transition from column to bent-wall is measured.

Self-replication and splitting of domain patterns in reaction-diffusion systems with the fast inhibitor.

An asymptotic equation of motion for the pattern interface in the domain-forming reaction-diffusion systems is derived. The free boundary problem is reduced to the universal equation of non-local contour dynamics in two dimensions in the parameter region where a pattern is not far from the points of the transverse instabilities of its walls. The contour dynamics is studied numerically for the reaction-diffusion system of the FitzHugh-Nagumo type. It is shown that in the asymptotic limit the transverse instability of the localized domains leads to their splitting and formation of the multidomain pattern rather than fingering and formation of the labyrinthine pattern.

Domain pattern instability of a lipid monolayer induced by adsorption/ desorption of a cyanine dye

Brewster angle microscopy (BAM) is applied for studying the in-plane morphology of a cationic lipid monolayer and its interaction with a novel, water-soluble, J-aggregate forming, negatively charged carbocyanine dye. A methodology of monolayer compression/recompression is followed to influence the dye crystallization upon adsorption from an aqueous subsolution at a charged interface. Crystalline domains of a dendritic type are observed in the lipid monolayer compressed on a pure water subphase. Dye aggregation and crystalline nucleation is established by means of BAM for 2D lipid densities corresponding to the onset of the main phase transition of the composite monolayer. Subphase conditions, which prevent the nucleation of lipid 2D crystalline phase domains, are found to not favor dye crystallization at the interface. Dye adsorption and aggregation is established to be partially irreversible upon monolayer decompression. Dye desorption is related to the observation of stripe-like objects in the lipid monolayer, developing undulation instability, which produces non-equilibrium 2D labyrinthine patterns with time.