Fractal-like Structures Research Articles

Self-organized fractal-like structures formation on the silicon wafer surface using the femtosecond laser pulses

In this paper interaction of a different number of femtosecond laser pulses and various fluences with silicon wafer has been studied. Effects of the pulse number and the laser fluence on self-organized structures created during femtosecond laser pulses processing are investigated. The scanning electron microscopy images of the sample show that by increasing the number of laser pulses, spike-like structures form on the surface, which gradually transform to hills and holes structure. The increment of the laser fluence causes the microstructures to grow in diameter. However, the increase in diameter is limited by surface stability. When the micro-structures are at their maximum size and a stable geometrical shape, further fluence increment can cause new self-organized nanostructures to be created over the microstructures’ surface. The hierarchical micro and nanostructures created after femtosecond laser pulse interaction have a similar shape. Thus fractal-like structures formed in the femtosecond pulse laser processing can be useful in the surface nano-structuring applications.

Pattern Analysis of Approximant Characteristics in Fractal-Like Multilayered Systems with Metalayers

The effect structural changes in fractal-like multilayered structures with metamaterial inserts have on the stability of the shape of fractal patterns in their optical spectra is investigated. Calculations using the model of approximants allows scaling parameters to be obtained that establish the relationship between structural features and their optical spectra.

FoGDbED: Fractional-order Gaussian derivatives-based edge-relevant structure detection using Caputo-Fabrizio definition

Edge-relevant structure features (ERSFs), e.g., object edges, boundaries and contours, junctions, etc. play an important role in low and middle level image processing tasks, such as image segmentation, as well as in higher-level computer vision tasks, such as scene analysis and content understanding. Commonly-used ERSF detection methods employ the integer-order differential-based methods, which are noise-sensitive and have less selectivity of ERSFs. Hence, they are difficult to effectively extract object edges or boundaries, especially in natural images with rich fractal-like structures. This paper presents a highly selective and noise-robust ERSF detection approach based on the fractional-order Gaussian derivatives (FoGDs) by using the definition of Caputo-Fabrizio derivative, termed FoGDbED. FoGDbED is constructed based on the concept of robust edge feature selection and inflexion point localization, whose detection mask can be designed with a close-form expression of FoGDs. Theoretical analysis and experimental results show that the proposed FoGDbED operator is capable of extracting complex ERSFs in natural images especially detecting object edges and junctions in natural images with serious noises to achieve a better visual effect.

Chameleon Metals: Autonomous Nano-Texturing and Composition Inversion on Liquid Metals Surfaces.

Studies on passivating oxides on liquid metals are challenging, in part, due to plasticity, entropic, and technological limitations. In alloys, compositional complexity in the passivating oxide(s) and underlying metal interface exacerbates these challenges. This nanoscale complexity, however, offers an opportunity to engineer the surface of the liquid metal under felicitous choice of processing conditions. We inferred that difference in reactivity, coupled with inherent interface ordering, presages exploitable order and selectivity to autonomously present compositionally biased oxides on the surface of these metals. Besides compositional differences, sequential release of biased (enriched) components, via fractal-like paths, allows for patterned layered surface structures. We, therefore, present a simple thermal-oxidative compositional inversion (TOCI) method to introduce fractal-like structures on the surface of these metals in a controlled (tier, composition, and structure) manner by exploiting underlying stochastic fracturing process. Using a ternary alloy, a three-tiered (in structure and composition) surface structure is demonstrated.

Polarized radiative transfer, rotation measure fluctuations, and large-scale magnetic fields

ABSTRACT Faraday rotation measure (RM) at radio wavelengths is commonly used to diagnose large-scale magnetic fields. It is argued that the length-scales on which magnetic fields vary in large-scale diffuse astrophysical media can be inferred from correlations in the observed RM. RM is a variable which can be derived from the polarized radiative transfer equations in restrictive conditions. This paper assesses the usage of rotation measure fluctuation (RMF) analyses for magnetic field diagnostics in the framework of polarized radiative transfer. We use models of various magnetic field configurations and electron density distributions to show how density fluctuations could affect the correlation length of the magnetic fields inferred from the conventional RMF analyses. We caution against interpretations of RMF analyses when a characteristic density is ill defined, e.g. in cases of lognormal-distributed and fractal-like density structures. As the spatial correlations are generally not the same in the line-of-sight longitudinal direction and the sky plane direction, one also needs to clarify the context of RMF when inferring from observational data. In complex situations, a covariant polarized radiative transfer calculation is essential to capture all aspects of radiative and transport processes, which would otherwise ambiguate the interpretations of magnetism in galaxy clusters and larger scale cosmological structures.

Mechanical response of Ti–6Al–4V hierarchical architected metamaterials

A snap-fit manufacturing technique has been designed for making hierarchically architected lattice structures from Ti–6Al–4V alloy sheets. By patterning the unit cell of a small-scale lattice along the struts of a self-similar unit cell of a large-scale lattice, fractal-like geometry with a hierarchy order of 2 was obtained. A subsequent vacuum brazing operation was performed to create fractal-like structures with an octahedron-of-octahedra half-cell geometry. Hierarchical architected lattices with a fractal number 4–15 and a relative density ranging from 0.7% to 12% have been manufactured by allowing a fixed small-scale cell size of approximately 10 mm and a varying large-scale cell size from 40 to 190 mm, and their mechanical responses under compression been experimentally examined. The measured compressive elastic moduli and strengths of the hierarchical lattices agreed well with micromechanical and finite element predictions. The Ti–6Al–4V hierarchical lattice materials exhibit very competitive mechanical properties when compared to other cellular materials, and provide a new solution for lightweight engineering materials for use at maximum service temperature up to 450 °C.

Structure Formation during Phase Separation of Poly(D,L-Lactide)–Tetraglycol–Antisolvent Ternary System

The diffusion and structure formation processes during phase separation of poly(D,L)-lactide solutions in tetraglycol and their antisolvent deposition in aqueous medium are experimentally studied. The effect of the molecular weight of polymers, the initial concentrations of their solutions, and the composition of antisolvent medium on the type of structures formed is studied. We find that either fractal-like or spongy polymer structures are formed as a result of phase separation processes. We show that the structure type depends on the intensity of diffusion processes in the polylactide–tetraglycol–antisolvent ternary system.

Finite-Difference Time-Domain Simulation of Fractal-Like Microlens Arrays for High Outcoupling Efficiency of Organic Light-Emitting Diodes.

The outcoupling efficiencies (OCEs) of organic light-emitting diodes (OLEDs) were studied for a fractal-like two-dimensional structure consisting of three layers of semicircular microlens on a glass substrate using a finite-difference time-domain (FDTD) method. The OCE with only one semicircular microlens layer was 29.5%, 1.75 times larger than that of the basic OLED. Additional layers with smaller diameters on the first layer did not improve the OCE. The OCE remained constant or slightly decreased with the increase of the number of layers. Two possible origins of this result were suggested; first, the possibility that the escaped light enters the nearby microlens becomes higher with the introduction of an additional protruded layer; second, the Mie scattering effect becomes important with the decrease of the diameter of the semicircular microlens from 20 μm to 0.8 μm. An additional FDTD simulation was performed for the OLED with only one microlens array as a function of the diameter. The OCE decreased approximately monotonously with the decrease of the diameter from 20 μm to 0.2 μm. In particular, the OCE became lower than that of the basic OLED when the diameter decreased from 0.5 μm to 0.2 μm. This is consistent with the observation that smaller fractal-like structures on the large microlens array did not further enhance the OCE.

Porous supraparticle assembly through self-lubricating evaporating colloidal ouzo drops

The assembly of colloidal particles from evaporating suspension drops is seen as a versatile route for the fabrication of supraparticles for various applications. However, drop contact line pining leads to uncontrolled shapes of the emerging supraparticles, hindering this technique. Here we report how the pinning problem can be overcome by self-lubrication. The colloidal particles are dispersed in ternary drops (water, ethanol, and anise-oil). As the ethanol evaporates, oil microdroplets form (‘ouzo effect’). The oil microdroplets coalesce and form an oil ring at the contact line, levitating the evaporating colloidal drop (‘self-lubrication’). Then the water evaporates, leaving behind a porous supraparticle, which easily detaches from the surface. The dispersed oil microdroplets act as templates, leading to multi-scale, fractal-like structures inside the supraparticle. Employing this method, we could produce a large number of supraparticles with tunable shapes and high porosity on hydrophobic surfaces.

Scaling properties of fractal-like structures

This work is aimed at solving one of the fundamental problems of modern fractal optics. The goal is to establish general laws that relate structural features of fractal-like systems and their associated spectral characteristics. Establishing such laws allows finding new diagnostic approaches for exploration of fractal-like systems and their formation processes. Nanocluster systems and multilayer structures, also with inclusion of metamaterial inserts, were chosen as the objects of research. A wide variety of applications of varying physical nature is enabled as fractal optics successfully penetrate into interdisciplinary technologies and theoretical concepts. In this work, particular interest is paid to scaling properties of structures of the studied systems and their optical characteristics. Based on theoretically obtained amplitude and phase distributions of light waves that diffract or interfere in the analyzed systems, the features of scaling in light fields are determined. When analyzing the obtained modeling results, a comparison is made between the structural features of the studied systems revealed by the scaling parameters in diffraction patterns and transmission spectra.

THE BOX-COUNTING DIMENSION OF PASCAL’S TRIANGLE r mod p

We consider Pascal’s Triangle [Formula: see text] to be the entries of Pascal’s Triangle that are congruent to [Formula: see text]. Such a representation of Pascal’s Triangle exhibits fractal-like structures. When the Triangle is mapped to a subset of the unit square, we show that such a set is nonempty and exists as a limit of a sequence of coarse approximations. We then show that for any given prime [Formula: see text], any such sequence converges to the same set, regardless of the residue(s) considered. As an obvious consequence, this allows us to conclude that the fractal (box-counting) dimension of this nonempty, compact representation of Pascal’s Triangle [Formula: see text] is independent of [Formula: see text].

Structural response of black locust (Robinia pseudoacacia L.) and small-leaved lime (Tilia cordata Mill.) to varying urban environments analyzed by terrestrial laser scanning: Implications for ecological functions and services

Quantitative measurements of structure and morphology of urban trees are hardly exhausted so far, especially in regard to variations caused by altering urban environments. However, structure and functions of trees are heavily interwoven. In fact, knowledge about structural attributes is essential for a better understanding of urban ecosystem functions and services. In order to scrutinize spatially explicit and detailed structural attributes under varying urban environments, we acquired terrestrial laser scans and applied the according methodological approaches to the common urban tree species black locust (Robinia pseudoacacia L.) and small-leaved lime (Tilia cordata Mill.). We analyzed 52 small-leaved limes and 41 black locust trees within the city of Munich (Germany). Species as well as growing location had a significant effect on the height-diameter relation. We also found greater crown volumes for small-leaved lime. Black locust however, displayed more crown projection area and likely more shade efficient crown shapes at similar volumes. Stem inclination of black locust was found to be higher in parks than in street canyons with town squares lying in between. Furthermore, black locust displayed strong crown asymmetry in park areas, likely caused by competition with neighbors. The angles of main branches did not differ significantly between both species nor between the growing location. Branch angles, branch bending, the length of the branches as well as species and growing location had a significant effect on vertical crown center position, i.e. general crown shape. Surface complexity of lime is lower than of black locust, with its lowest manifestation in parks. Fractal-like crown surface structures, increasing surface roughness and complexity, were found to be more pronounced for black locust than for small-leaved lime. Thereby, black locust featured the highest crown surface complexity in parks, the lowest in street canyons. The results suggest that studies on spatially explicit tree structures may contribute to more target oriented tree plantings and thus, more effective exploitation of ecosystem services and benefits.

Flame morphology and self-acceleration of syngas spherically expanding flames

The self-acceleration of spherically expanding flames were investigated using a constant volume combustion chamber for CO/H2/O2/N2 mixtures over a wide range of initial pressure from 0.2 to 0.6 MPa, CO/H2 ratio from 50/50 to 10/90 and equivalence ratio from 0.4 to 1.5. The adiabatic flame temperature was kept constant by adjusting O2/N2 ratio at different equivalence ratios. Schlieren images were recorded to investigate the flame front evolution of spherically expanding flames. Local acceleration exponents were extracted using a proper equation to study the process of flame self-acceleration. Results show that the flame cells develop on the smooth flame fronts and finally reach fractal-like structures due to the hydrodynamic and diffusional-thermal instabilities, resulting in flame self-accelerative propagation. The critical Peclet number corresponding to the onset of self-acceleration, Pecr increases nonlinearly with the Markstein length, Ma. The observation further reveals that the onset of self-acceleration is mainly controlled by the diffusional-thermal effect. There exists two distinct flame propagation regimes in the self-acceleration, namely quick transition accelerative and quasi self-similar accelerative regimes. The quick transition regime is controlled by the destabilization effect of hydrodynamic perturbation and stabilization effect of flame stretch. While the quasi self-similar regime is primarily affected by the cascading process of flame front cells controlled by hydrodynamic instability. The self-similar acceleration exponent, αs varies with the initial pressure and Lewis number, Le. The values of αs are measured to be 1.1–1.25 (smaller than 1.5), indicating the flame dose not attain self-turbulization.

Soot template TiO2 fractals as a photoactive gas sensor for acetone detection

Using candle soot fractal-like structures as a template to introduce porosity, TiO2 surfaces with visible-blind photoactivity for acetone gas sensing applications were developed, which would otherwise be impossible with thin films of TiO2 formed by chemical vapor deposition (CVD). The photoactivity performance was obtained at low operating potentials, with a high UV response repeatability (∼99%) even towards low UV intensities of μW·cm−2, thus making it suitable for light-assisted gas sensing applications. The acetone gas sensing functionality can be used for non-invasive diabetic diagnoses as the sensing mechanism takes advantage of the developed material’s photoactivity, thus enhancing both sensitivity and selectivity towards acetone gas. This enables the sensor to discriminate between acetone concentrations associated with both healthy and diabetic concentrations in human breath while exhibiting a repeatability and detection limit of >97% and 10 ppb, respectively. The excellent performance is attributed to the high film porosity and fractal like network structures, thus enhancing the number of gas-solid interactions as well as having titania crystalline structure with good optical excitement in the UV range.

Topological Network Properties of Fractal-Like Metallic Nanoparticle Patterns and Their Effects on Optical Resonances

Fractal-like nanoparticle two-dimensional patterns forming in diffusion-limited aggregation show variant spatial patterns. However, they have invariant statistical properties in their network topologies, even though their formation is completely in self-assembled processes. One of the outputs from these topological properties is optical resonances at invariant frequencies, which is a required feature of a metamaterial alternative. Fractal-like metallic patterns studied here in both experiments and theoretical models exhibit similar resonance frequencies in the infrared-ray range, and they depend on the unit length of nanoparticles composing arbitrary fractal-like structures. The scheme of analysis applied here using complex network theory does not only reveal the topological properties of the nanoparticle network, but points out their optical and possibly other physical potentials arising from their geometrical properties.

Selective Molecular Transport in Thermoresponsive Polymer Membranes: Role of Nanoscale Hydration and Fluctuations

For a wide range of modern soft functional materials the selective transport of sub-nanometer-sized molecules (`penetrants') through a stimuli-responsive polymeric membrane is key to the desired function. In this study, we investigate the diffusion properties of penetrants ranging from non-polar to polar molecules and ions in a matrix of collapsed Poly(N-isopropylacrylamide) (PNIPAM) polymers in water by means of extensive molecular dynamics simulations. We find that the water distributes heterogeneously in fractal-like cluster structures embedded in the nanometer-sized voids of the polymer matrix. The nano-clustered water acts as an important player in the penetrant diffusion, which proceeds via a hopping mechanism through `wet' transition states: the penetrants hop from one void to another via transient water channels opened by rare but decisive polymer fluctuations. The diffusivities of the studied penetrants extend over almost five orders of magnitude and thus enable a formulation of an analytical scaling relation with a clear non-Stokesian, exponential dependence of the diffusion coefficient on the penetrant's radius for the uncharged penetrants. Charged penetrants (ions) behave differently as they get captured in large isolated water clusters. Finally, we find large energetic activation barriers for hopping, which significantly depend on the hydration state and thereby challenge available transport theories.

Effect of the Alkanoyl Group Position at the Glycerol Backbone on the Monolayer Characteristics Demonstrated by 2-Monopalmitoyl-rac-glycerol.

The influence of the position of the aliphatic chain at the glycerol backbone has been basically unknown. Solely the results of 2-monopalmitoyl-rac-glycerol obtained at ≥13 °C indicated an essential influence of the position of the palmitoyl group at the glycerol backbone, substantiated by a disordered packing of the alkyl chains. Therefore, the present study extends the comprehensive characterization of 2-monopalmitoyl-rac-glycerol monolayers to the low-temperature range for highlighting the effect of the position of the aliphatic chain at the glycerol backbone of monoalkanoylglycerolester monolayers. Systematic studies of the thermodynamic behavior, the morphological features, and the 2D lattice structures of 2-monopalmitoyl-rac-glycerol monolayers at ≤10 °C allow useful conclusions. Large differences between the π-A isotherms of 1- and 2-monopalmitoyl-rac-glycerol monolayers and their thermodynamic analysis indicate that the change of the substitution from position 1 to position 2 of glycerol backbone is consistent with a shortening of the alkyl chain by roughly two CH2 groups. Quantum chemical calculations of the molecular structure and packing calculations are in reasonable agreement with the thermodynamic results. Considerable diversity in the mesoscopic domain topography exists between the monoalkanoylglycerol esters with the aliphatic chain positioned at the end of the glycerol backbone (1-position) and those with the aliphatic chain in the middle of the glycerol backbone (2-position). The new faceted shape of the 2-monopalmitoyl-rac-glycerol domains, before they develop branched fractal-like structures at the edges, is the essential difference to the round or cardioid-like 1-monoalkanoylglycerol domains. In the low-temperature range, well-defined orthorhombic lattice structures exist at all surface pressures. Comparing all GIXD data from the three racemic compounds (1-monostearoyl-rac-glycerol, 1-monopalmitoyl-rac-glycerol, and 2-monopalmitoyl-rac-glycerol) shows that 2-monopalmitoyl-rac-glycerol behaves as 1-monomyristoyl-rac-glycerol, i.e., the shift from position 1 to position 2 of the glycerol backbone is equivalent to a shortening of the alkyl chain.

Fractal structures in freezing brine

The process of initial ice formation in brine is a highly complex problem. In this paper, we propose a mathematical model that captures the dynamics of nucleation and development of ice inclusions in brine. The primary emphasis is on the interaction between ice growth and salt diffusion, subject to external forcing provided by temperature. Within this setting two freezing regimes are identified, depending on the rate of change of the temperature: a slow freezing regime where a continuous ice domain is formed; and a fast freezing regime where recurrent nucleation appears within the fluid domain. The second regime is of primary interest, as it leads to fractal-like ice structures. We analyse the critical threshold between the slow and fast regimes by identifying the explicit rates of external temperature control that lead to self-similar salt-concentration profiles in the fluid domains. Subsequent heuristic analysis provides estimates of the characteristic length scales of the fluid domains depending on the time-variation of the temperature. The analysis is confirmed by numerical simulations.

Self-similar resistive circuits as fractal-like structures

Fractals play a central role in several areas of modern physics and mathematics. In the present work we explore resistive circuits where the individual resistors are arranged in fractal-like patterns. These circuits have some of the characteristics typically found in geometric fractals, namely self-similarity and scale invariance. Considering resistive circuits as graphs, we propose a definition of self-similar circuits which mimics a self-similar fractal. General properties of the resistive circuits generated by this approach are investigated, and interesting examples are commented in detail. Specifically, we consider self-similar resistive series, tree-like resistive networks and Sierpinski’s configurations with resistors.

Successive range expansion promotes diversity and accelerates evolution in spatially structured microbial populations.

Successive range expansions occur within all domains of life, where one population expands first (primary expansion) and one or more secondary populations then follow (secondary expansion). In general, genetic drift reduces diversity during range expansion. However, it is not clear whether the same effect applies during successive range expansion, mainly because the secondary population must expand into space occupied by the primary population. Here we used an experimental microbial model system to show that, in contrast to primary range expansion, successive range expansion promotes local population diversity. Because of mechanical constraints imposed by the presence of the primary population, the secondary population forms fractal-like dendritic structures. This divides the advancing secondary population into many small sub-populations and promotes intermixing between the primary and secondary populations. We further developed a mathematical model to simulate the formation of dendritic structures in the secondary population during succession. By introducing mutations in the primary or dendritic secondary populations, we found that mutations are more likely to accumulate in the dendritic secondary populations. Our results thus show that successive range expansion can promote intermixing over the short term and increase genetic diversity over the long term. Our results therefore have potentially important implications for predicting the ecological processes and evolutionary trajectories of microbial communities.