Formation Of Periodic Patterns Research Articles

Gap junctions in Turing-type periodic feather pattern formation.

Periodic patterning requires coordinated cell-cell interactions at the tissue level. Turing showed, using mathematical modeling, how spatial patterns could arise from the reactions of a diffusive activator-inhibitor pair in an initially homogenous 2D field. Most activators and inhibitors studied in biological systems are proteins, and the roles of cell-cell interaction, ions, bioelectricity, etc. are only now being identified. Gap junctions (GJs) mediate direct exchanges of ions or small molecules between cells, enabling rapid long-distance communications in a cell collective. They are therefore good candidates for propagating nonprotein-based patterning signals that may act according to the Turing principles. Here, we explore the possible roles of GJs in Turing-type patterning using feather pattern formation as a model. We found 7 of the 12 investigated GJ isoforms are highly dynamically expressed in the developing chicken skin. In ovo functional perturbations of the GJ isoform, connexin 30, by siRNA and the dominant-negative mutant applied before placode development led to disrupted primary feather bud formation. Interestingly, inhibition of gap junctional intercellular communication (GJIC) in the ex vivo skin explant culture allowed the sequential emergence of new feather buds at specific spatial locations relative to the existing primary buds. The results suggest that GJIC may facilitate the propagation of long-distance inhibitory signals. Thus, inhibition of GJs may stimulate Turing-type periodic feather pattern formation during chick skin development, and the removal of GJ activity would enable the emergence of new feather buds if the local environment were competent and the threshold to form buds was reached. We further propose Turing-based computational simulations that can predict the sequential appearance of these ectopic buds. Our models demonstrate how a Turing activator-inhibitor system can continue to generate patterns in the competent morphogenetic field when the level of intercellular communication at the tissue scale is modulated.

Periodic Alignment of Binary Droplets via a Microphase Separation of a Tripolymer Solution under Tubular Confinement.

We report the spontaneous formation of a characteristic periodic pattern through the phase separation of a tripolymer solution comprising polyethylene-glycol (PEG)/dextran (DEX)/gelatin. When this tripolymer solution is introduced into a glass capillary with a PEG-coated inner surface, we observe the time-dependent growth of microphase separation. Remarkably, a self-organized, periodic alignment of DEX- and gelatin-rich microdroplets ensues, surrounded by a PEG-rich phase. This pattern demonstrates considerable stability, enduring for at least 8 h. The fundamental characteristics of the experimentally observed periodic alignment are successfully replicated via numerical simulations using a Cahn-Hilliard model underpinned by a set of simple, theoretically derived equations. We propose that this type of kinetically stabilized periodic patterning can be produced across a broad range of phase-separation systems by selecting appropriate boundary conditions such as at the surface within a narrow channel.

Hopf bifurcation and patterns in a modified SIR model

Infectious diseases have constantly threatened human safety because the diffusion of the susceptible and infected may make more individuals infected and even die. In this paper, a modified SIR model with both external stimulus and diffusion is considered to illustrate the dynamical mechanism of the periodic outbreak and pattern formation. Firstly, we propose a modified SIR model based on the propagation behaviour of infectious diseases to show the effects of the different parameters and diffusion on the outbreak. The Hopf bifurcation and multiscale methods are performed to analyze the stability of this model, which explains the dynamical mechanism of the periodic outbreak. Then, the pattern formation and Turing instability are discussed through comparison principles to reveal the role of periodic disturbances and diffusion in selecting pattern formation. Also, we find rich patterns that may occur when the frequency modulation is close to the intrinsic frequency. Finally, our theoretical results are verified by numerical simulation.

Chemotaxis driven formation of time periodic pattern and steady state pattern

This paper is concerned with a chemotaxis-fluid model with realistic boundary conditions matching the experiments of Kessler et al. Different from the initial and boundary value problem, the existence of time periodic solution is proved under no any smallness assumptions on chemotactic coefficient. Subsequently, we investigate the reason of pattern formation. It is generally known that there is no pattern formation for the homogeneous boundary value problem. Therefore, the pattern formation is partly caused by non-homogeneous boundary conditions, but chemotaxis is also an indispensable condition. In fact, it is proved that the density of bacteria will tend to be evenly distributed when the chemotactic intensity tends to 0. At last, the idea of finding time periodic pattern is also applied to prove the steady-state pattern formation of chemotaxis-Navier-Stokes model.

Structural Color Generation on Transparent and Flexible Substrates by Nanosecond Laser Induced Periodic Surface Structures

AbstractLaser patterning to fabricate micro/nanoscale structures attracts enormous attention as they find many functional applications owing to their controllable structure and unique properties. The present work proposes a unique solution‐based method to fabricate laser‐induced periodic surface structure (LIPSS) patterns on a thin film of Titanium (IV) butoxide and Ag organometallic (OM) solution‐coated glass substrates with near infra‐red (NIR) nanosecond (ns) laser. The formation of periodic patterns has been confirmed, the thin film solution concentration affects the LIPSS pattern thickness and a continuous pattern is achieved with the diluted solution. Parametric studies demonstrate that the LIPSS patterns could be achieved with optimal solution dilution, laser fluence, scan speed, and pulse duration. A high power and faster scan speed result in discontinued, and defective patterns. The photothermal process such as material reorganization followed by electromagnetic absorption that takes place during laser patterning is mainly responsible for the fabrication of controlled structures. LIPSS patterned surfaces exhibit iridescent surface coloration while illuminated with white light, and the color spectrum from red to blue is distinguished. LIPSS patterned substrates serve as mold and structures are transferrable to the poly dimethyl siloxane (PDMS) film and exhibit iridescent structural coloration like LIPSS patterned substrates.

Crystalline phases of laser-driven dipolar Bose-Einstein condensates

Although crystallization is a ubiquitous phenomenon in nature, crystal formation and melting still remain fascinating processes with several open questions yet to be addressed. In this work, we study the emergent crystallization of a laser-driven dipolar Bose-Einstein condensate due to the interplay between long-range magnetic and effectively infinite-range light-induced interactions. The competition between these two interactions results in a collective excitation spectrum with two roton minima that introduce two different length scales at which crystalline order can emerge. In addition to the formation of regular crystals with simple periodic patterns due to the softening of one of the rotons, we find that both rotons can also soften simultaneously, resulting in the formation of exotic, complex periodic or aperiodic density patterns. We also demonstrate dynamic state-preparation schemes for achieving all the found crystalline ground states for experimentally relevant and feasible parameter regimes

Cellular automata approach to modeling self-organized periodic patterns in nanoparticle-doped liquid crystals.

Elementary cellular automata provide one of the simplest ways to generally describe the phenomena of pattern formation. However, they are considered too simple to be able to describe in detail the more complex phenomena occurring in real experimental systems. In this article, we demonstrate the an application of these methods to optical systems, providing an understanding of the mechanisms behind the formation of periodic patterns in nanoparticle-doped liquid crystals. Our extremely simplified model also explains the observed linear relationship between periodicity and system size.

Adhesion and wear of diamond: formation of nanocracks and adhesive craters of torn out material.

Mechanical transformation of rough diamonds into brilliant ones is usually achieved by polishing using microsized abrasive diamond particles. It is shown that in addition to formation of periodic pattern of 'partial' Hertzian cone cracks on the diamond surface, nano-sized domains (50-150 nm in diameter) of crumbled material are observed. Because these domains are located in the centres of the regions (250-500 nm in diameter) partially surrounded by the Hertzian cone cracks, where the stresses are close to the stress field of hydrostatic compression, the material removal cannot be explained by creation of tensile or shear cracks. It is argued that the creation of these domains of crumbled material is due to adhesive interactions between sliding diamond particles and the diamond surface. Employing a two-term law of friction, the scheme of ultimate equilibrium between the particle and the surface is presented. The distributions of contact stresses are calculated for two approaches: (i) the extended Johnson-Kendall-Roberts model and (ii) the 'soft' model of adhesive contact. Thus, adhesion between the sliding diamond particle and the surface leads to creation of periodic pattern of the crumbling domains with the steps 500-1000 nm and adhesive tearing out of the material from the domains. This article is part of the theme issue 'Nanocracks in nature and industry'.

A review of continuous modeling of periodic pattern formation with modified phase-field crystal models

A review of continuous modeling of periodic pattern formation with modified phase-field crystal models

Nonlocal Competition and Spatial Multi-peak Periodic Pattern Formation in Diffusive Holling-Tanner Predator-prey Model

Nonlocal Competition and Spatial Multi-peak Periodic Pattern Formation in Diffusive Holling-Tanner Predator-prey Model

Chemical Tracking of Temperature by Concurrent Periodic Precipitation Pattern Formation in Polyacrylamide Gels.

In nature, nonequilibrium systems reflect environmental changes, and these changes are often “recorded” in their solid body as they develop. Periodic precipitation patterns, aka Liesegang patterns (LPs), are visual sums of complex events in nonequilibrium reaction–diffusion processes. Here we aim to achieve an artificial system that “records” the temperature changes in the environment with the concurrent LP formation. We first illustrate the differences in 1-D LPs developing at different temperatures in terms of band spacings, which can demonstrate the time, ramp steepness, and extent of a temperature change. These results are discussed and augmented by a mathematical model. Using scanning electron microscopy, we show that the average size of the CuCrO4 precipitate also reflects the temperature changes. Finally, we show that these changes can also be “recorded” in the 2-D and 3-D LPs, which can have applications in long-term temperature tracking and complex soft material design.

Global dynamics and pattern formation for predator-prey system with density-dependent motion.

In this paper, we concern with the predator-prey system with generalist predator and density-dependent prey-taxis in two-dimensional bounded domains. We derive the existence of classical solutions with uniform-in-time bound and global stability for steady states under suitable conditions through the Lyapunov functionals. In addition, by linear instability analysis and numerical simulations, we conclude that the prey density-dependent motility function can trigger the periodic pattern formation when it is monotone increasing.

Impact of Pre-Patterned Structures on Features of Laser-Induced Periodic Surface Structures.

The efficiency of light coupling to surface plasmon polariton (SPP) represents a very important issue in plasmonics and laser fabrication of topographies in various solids. To illustrate the role of pre-patterned surfaces and impact of laser polarisation in the excitation of electromagnetic modes and periodic pattern formation, Nickel surfaces are irradiated with femtosecond laser pulses of polarisation perpendicular or parallel to the orientation of the pre-pattern ridges. Experimental results indicate that for polarisation parallel to the ridges, laser induced periodic surface structures (LIPSS) are formed perpendicularly to the pre-pattern with a frequency that is independent of the distance between the ridges and periodicities close to the wavelength of the excited SPP. By contrast, for polarisation perpendicular to the pre-pattern, the periodicities of the LIPSS are closely correlated to the distance between the ridges for pre-pattern distance larger than the laser wavelength. The experimental observations are interpreted through a multi-scale physical model in which the impact of the interference of the electromagnetic modes is revealed.

Superhydrophobic Polymer Composite Surfaces Developed via Photopolymerization.

Fabrication of superhydrophobic materials using incumbent techniques involves several processing steps and is therefore either quite complex, not scalable, or often both. Here, the development of superhydrophobic surface-patterned polymer–TiO2 composite materials using a simple, single-step photopolymerization-based approach is reported. The synergistic combination of concurrent, periodic bump-like pattern formation created using irradiation through a photomask and photopolymerization-induced nanoparticle (NP) phase separation enables the development of surface textures with dual-scale roughness (micrometer-sized bumps and NPs) that demonstrate high water contact angles, low roll-off angles, and desirable postprocessability such as flexibility, peel-and-stick capability, and self-cleaning capability. The effect of nanoparticle concentration on surface porosity and consequently nonwetting properties is discussed. Large-area fabrication over an area of 20 cm2, which is important for practical applications, is also demonstrated. This work demonstrates the capability of polymerizable systems to aid in the organization of functional polymer–nanoparticle surface structures.

Evolution of Micron-Spaced Patterns within Precipitating Patterns of In Situ Synthesized Silver Nanoparticles in a Nanodot-Embedded PVA/PVP Film.

Periodic pattern formation beyond conventional precipitation reactions of electrolytes is of greater importance for gaining insights into the driving forces behind spontaneous spatiotemporal pattern formation in living matter. The Liesegang phenomenon is considered to be one of the important models for understanding well-defined periodic patterns. In this study, we have used biomolecule-derived photoluminescent carbon nanodots as reducing agents that were embedded in thin polymer films. The poor water content of polyvinyl alcohol/polyvinyl pyrrolidone films has been found to dictate the temporal scale of reaction-diffusion kinetics. Moreover, the precursors for the synthesis of nanodots have been varied to decipher the role of thiol groups present in glutathione in micron-spaced pattern formation of silver nanoparticles. A method to develop periodic patterns of plasmonic silver nanoparticles is of significant interest from technological aspects. Moreover, the formation of a micron-spaced pattern has been rationed by invoking a lowered nucleation threshold in terms of slow reaction-controlled aggregation. We expect that such an understanding of the chemical reaction-based pattern formation will help in resolving the formation of artistic spatiotemporal patterns in nature.

Pattern selection in reaction diffusion systems.

Turing's theory of pattern formation has been used to describe the formation of self-organized periodic patterns in many biological, chemical, and physical systems. However, the use of such models is hindered by our inability to predict, in general, which pattern is obtained from a given set of model parameters. While much is known near the onset of the spatial instability, the mechanisms underlying pattern selection and dynamics away from onset are much less understood. Here, we provide physical insight into the dynamics of these systems. We find that peaks in a Turing pattern behave as point sinks, the dynamics of which is determined by the diffusive fluxes into them. As a result, peaks move toward a periodic steady-state configuration that minimizes the mass of the diffusive species. We also show that the preferred number of peaks at the final steady state is such that this mass is minimized. Our work presents mass minimization as a potential general principle for understanding pattern formation in reaction diffusion systems far from onset.

Production of V2C MXene using a repetitive pattern of V2AlC MAX phase through microwave heating of Al-V2O5-C system

The present work focuses on the preparation of V2C MXene from Al-V2O5-C mixture heated in a microwave and etched under HF-based process. In the first stage of process, microwave heating caused the formation of an uncommon periodic pattern of V2AlC-Al2O3 MAX phase. In the second stage of process the aggressive leaching (temperature of 80 °C for 96 h) was performed in HF to remove the interlayer aluminum and Al2O3 particles as impurities. The leaching was not completed at room temperature even after 24 h. XRD and FESEM analyses confirmed the successful formation of V2C MXene and its distorted layer, respectively. HRTEM, XPS and Raman analyses also confirmed a perfect structure of V2C MXene. It seems that the sintering of compacted Al-V2O5-C mixture in microwave and subsequently aggressive leaching of these samples encourage the formation of periodic pattern of MAX phase together with few-layer V2C MXene. Finally, this work led to the successful preparation of MXene using cheap starting materials and the influence of microwave heating on the reduction of V2O5 by Al which could be a driving force for the formation of periodic pattern in V2AlC.

A perspective on nanoscale pattern formation at surfaces by ion-beam irradiation

The formation of periodic patterns on the surfaces of many solid materials undergoing ion-beam irradiation has long been known. The advent of high resolution characterization techniques elucidated the nanoscopic traits of this self-organization process, enabling a wide range of applications for the nanostructures thus produced, from optoelectronic to biomedical. Meanwhile, full theoretical understanding of the technique has been challenged by its multiscale nature, whereby the external perturbation implemented by the ion beam acts at a much slower rate (typically, one ion arrives per square-nm every second) than the microscopic processes, like collision cascades and material transport, which try to relax such external perturbations (collision cascades or surface diffusion attempts usually relax after a few picoseconds). Here, we present a Perspective on the main developments that have led to the current understanding of nanoscale pattern formation at surfaces by ion-beam irradiation, from the points of view of experiments, applications, and theory, and offer an outlook on future steps that may eventually facilitate full harnessing of such a versatile avenue to materials nanostructuring.

Periodic subcellular structures undergo long-range synchronized reorganization during C. elegans epidermal development.

Periodic pattern formation on the cellular and tissue scale is an important process and has been extensively studied. However, periodic pattern formation at the subcellular level still remains poorly understood. The C. elegans epidermis displays a highly ordered parallel stripe pattern as part of its subcellular structure, making it an ideal model to study the formation and reorganization of periodic patterns within cells. Here, we show that the initial formation of periodic striped patterns in the C. elegans epidermis is dependent on actin and spectrin, and requires the apical membrane attachment structures for maintenance. The periodic subcellular structures do not accommodate cell growth by continuously making new stripes. Instead, they increase the number of stripes by going through one round of uniform duplication, which is independent of the increasing epidermal length or the developmental cycles. This long-range synchronized reorganization of subcellular structures is achieved by physical links established by extracellular collagens together with extension forces generated from epidermal cell growth. Our studies uncover a novel strategy employed by evenly spaced and interlinked subcellular structures to maintain their integrity and equidistribution during cell growth and tissue development.

Bistability, wave pinning and localisation in natural reaction–diffusion systems

A synthesis is presented of recent work by the authors and others on the formation of localised patterns, isolated spots, or sharp fronts in models of natural processes governed by reaction–diffusion equations. Contrasting with the well-known Turing mechanism of periodic pattern formation, a general picture is presented in one spatial dimension for models on long domains that exhibit sub-critical Turing instabilities. Localised patterns naturally emerge in generalised Schnakenberg models within the pinning region formed by bistability between the patterned state and the background. A further long-wavelength transition creates parameter regimes of isolated spots which can be described by semi-strong asymptotic analysis. In the species-conservation limit, another form of wave pinning leads to sharp fronts. Such fronts can also arise given only one active species and a weak spatial parameter gradient. Several important applications of this theory within natural systems are presented, at different lengthscales: cellular polarity formation in developmental biology, including root-hair formation, leaf pavement cells, keratocyte locomotion; and the transitions between vegetation states on continental scales. Philosophical remarks are offered on the connections between different pattern formation mechanisms and on the benefit of subcritical instabilities in the natural world.