Fractal Formation Research Articles

Formation of Optical Fractals by Chaotic Solitons in Coupled Nonlinear Helmholtz Equations

Formation of Optical Fractals by Chaotic Solitons in Coupled Nonlinear Helmholtz Equations

Periodic and Axial Perturbations of Chaotic Solitons in the Realm of Complex Structured Quintic Swift-Hohenberg Equation

This research work employs a powerful analytical method known as the Riccati Modified Extended Simple Equation Method (RMESEM) to investigate and analyse chaotic soliton solutions of the (1 + 1)-dimensional Complex Quintic Swift–Hohenberg Equation (CQSHE). This model serves to describe complex dissipative systems that produce patterns. We have found that there exist numerous chaotic soliton solutions with periodic and axial perturbations to the intended CQSHE, provided that the coefficients are constrained by certain conditions. Furthermore, by applying a sophisticated transformation, the provided transformative approach RMESEM transforms CQSHE into a set of Nonlinear Ordinary Differential Equations (NODEs). The resulting set of NODEs is then transformed into an algebraic system of equations by incorporating the extended Riccati NODE to assume a series form solution. The soliton solutions to this system of equations can be found as periodic, hyperbolic, exponential, rational-hyperbolic, and rational families of functions. A variety of 3D and contour visuals are also provided to graphically illustrate the axially and periodically perturbed dynamics of these chaotic soliton solutions and the formation of fractals. Our findings are noteworthy because they shed light on the chaotic nature of the framework we are examining, enabling us to better understand the dynamics that underlie it.

Simulation of Polymer Fractal Formation Using a Triangular Network Growth Model.

Fractal formation in spin-coated thin-film polymers is of experimental and theoretical interest. Modeling the determinants and dynamics of this process will deepen our understanding of polymer aggregation and the predictability of thin-film structures. This is especially true if the model used has readily interpretable parameters and has been demonstrated to yield a close match to experimental processes under a variety of conditions. In this work, we adapted and applied a relatively new model of fractal growth comprised of a spreading and contracting triangular network, to model spin-coated, thin-film polymers made of poly(vinyl alcohol) on polydimethylsiloxane substrates. We drew clear connections between model parameters and the process of polymer aggregation and we demonstrated the ability of the model to simulate fractal formation under a wide variety of conditions including varying the degree of hydrolysis of the polymer, changing the spin-coating process, and solvent annealing and reforming of polymer fractals under different drying conditions. We also showed how the model is able to replicate idiosyncratic experimental settings yielding novel fractal patterns.

Insight into Mocha Diffusion Occurring during Ethanol Droplet Spreading on Acrylic Paint.

Mocha diffusion, a significant interfacial phenomenon in pottery painting, remains insufficiently understood in the documented literature regarding its dynamics. This study experimentally investigates Mocha patterns by quantitatively dripping ethanol droplets onto an acrylic paint surface. Results indicate that the spreading radius increases with the ethanol mass fraction, while the fractal period decreases. The fractal dimension of all Mocha patterns approximates 1.4087. Marangoni flow, generated by the volatilization of ethanol, is crucial for the growth and fractal formation of the "dendrites" in this spreading. The scaling analysis is used to interpret the spreading dynamics. This work encourages the interface science community to develop a comprehensive theory for the dynamics of Mocha diffusion and highlights the potential of this intriguing decorative technique.

Magnetoelectric fractals, Magnetoelectric parametric resonance and Hopf bifurcation

In the present work, we study the dynamics of the magnetic nanodisc coupled through a magnetoelectric coupling to a ferroelectric crystal. The model of our interest is nonlinear, and we explore the problem under different limits of weak and strong non-linearity. By applying two electric fields with different frequencies, we control the form of the confinement potential of a ferroelectric subsystem and realize different types of dynamics. We proved that the system is more sensitive to magnetoelectric coupling in the case of double-well potential. In particular, in the case of strong non-linearity, arbitrary small values of magnetoelectric coupling lead to chaotic dynamics. In essence, magnetoelectric coupling plays a role akin to the small perturbations destroying invariant tori according to the KAM theorem. We showed that bifurcations in the system are of Hopf’s type. We observed the formation of magnetoelectric fractals in the system. In the limit of weak non-linearity, we studied a problem of parametric nonlinear resonance and enhancement of magnetic oscillations through magnetoelectric coupling.

Studies of Fractal Microstructure in Nanocarbon Polymer Composites.

The in situ study of fractal microstructure in nanocarbon polymers is an actual task for their application and for the improvement in their functional properties. This article presents a visualization of the bulk structural features of the composites using pulsed acoustic microscopy and synchrotron X-ray microtomography. This article presents details of fractal structure formation using carbon particles of different sizes and shapes-exfoliated graphite, carbon platelets and nanotubes. Individual structural elements of the composite, i.e., conglomerations of the particles in the air capsule as well as their distribution in the composite volume, were observed at the micro- and nanoscale. We have considered the influence of particle architecture on the fractal formation and elastic properties of the composite. Acoustic and X-ray imaging results were compared to validate the carbon agglomeration.

Fullerene-containing modifier of magnetoactive elastomer

Magnetoactive elastomer is a functional material whose properties are controlled by the parameters of the external magnetic field. A modifier that creates ordered structures with controlled nanoscale morphology is capable of intensifying the ability of a material to transfer charge.The modifier was obtained by dissociating C60 fullerene in eugenol at an elevated temperature in a water bath for 6 h. The fullerene content in the samples was 3.5 g/L. The preparation of 3 groups of modifier solutions to study their properties took 60 days. Two groups included solutions obtained through diffuse dissolution, one group - with an additional mechanical action. A study of rheological, optical and electrical conductivity properties was carried out to assess changes in the structure of the solutions. During the studies, thixotropic deposition of the air capsule was noted in some samples. To describe the hydraulic size of deposited objects, a nonlinear dependence is formulated. Spectral analysis of the solutions revealed differences in the optical properties of the samples obtained by various methods. The optical activity of those that have not been subjected to an additional impact is increasing over time. This causes a change in the solution structure and the conformation of the complexes of the solvent molecular structure and C60. Ultimately, this leads to noticeable changes in electrical conductivity properties. The change in the resistivity values of some samples relative to the solvent is associated with the influence of the formed structural aggregation of fullerene molecules, as well as with several types of polarization interactions. Classification of the influence of conformational and electronic characteristics of solvent molecules made it possible to systematize the factors influencing the solvent dissolving ability. The formation of non-centrosymmetric structures in solutions in the form of fractal aggregates of dissociated fullerene was noted. The approach to describing the model for the formation of a cluster structure is based on the principle of increasing the fractional dimension during the dissociation process. Aggregation, limited by diffusion processes, proceeds to limit the reaction rate; at the final stage, spatial limitation dominates.Studying the molecular dynamics of aggregates formation in various solutions allows improved understanding the principles of a fractal structure formation. The results obtained will be used in the development of conductive functional polymers with controlled properties.

ON PARAMETRIC STUDY OF FLUID LEAVES AND FLUID PINEAPPLE

Liquid chain is a commonly known phenomenon, and it is found to be formed under free fall or due to impact of liquid on surface. Usually, these liquid chains have rim and lamina, which are explained by instabilities. In this article, we examined a variant of liquid chain and carried a parametric study experimentally. Since this phenomenon is reason for the shape of leaves in a plant, we call them fluid leaves or liquid leaves. Another fluid phenomenon called fluid pineapple (FP), which is actually the result of an interaction of multiple laminar jets, is the reason for the shape of the pineapple. From the abovementioned fluid phenomena and, it is evident the shape of the living organism is based on fluid physics. Experimental study and the results are provided in detail. All living organisms follows the Fibonacci series and fractal formation. This article shows that the property of fluids to store its molecules in a series is the reason behind the series and fractals' formation in the entire universe. A unified theory is provided to explain the universal presence of leaf shapes and details about the FP in this work.

Indicators of stability of the world economic system

The state of the global economic system is described by a large number of parameters. It is interesting to find out whether it is possible to collapse these parameters into a single symbol (image, logograph) that would describe the state of the global economic system as a whole. A discrete dynamic model (DDM) was used to describe the global economic system. DDM states can be divided into unstable and stable. The degree of instability of the DDM can be used to assess the likelihood that the economic system would emerge from an unstable state. In turn, stable states are divided into stable points, stable cycles and strange attractors. It was discovered that stable states are characterized by areas of convergence that represent fractal formations. The unique nature of fractals encourages us to assess their usefulness and informative value when describing the state of the world economic system. In this work, we are trying to carry out such an assessment and formalize the characteristics of fractals in order to give a rough description of the states of the world economic system. To get input data, we used information on the rate of change in world GDP provided on the World Bank website as of June 29, 2023.

Reverse-degree-based topological indices of two-dimensional coronene fractal structures

Researchers are still drawn to research the physical molecular and chemical structure of benzenoid hydrocarbons, unsaturated, fully conjugated compounds with hexagonal arrangements that exhibit remarkable features in relation to aromaticity. For chemical graphs in many dimensions, structures, or networks, topological indices or numerical descriptors have been employed for decades to link key physicochemical parameters with crucial molecular structural features including melting, boiling point, enthalpy, and cyclicity. For this work, the inverse degrees of the molecular or chemical structure or graphs being studied are used to calculate the reverse-degree-based topological indices. In molecular graph theory, reverse-degree-based topological descriptors are a relatively new method for analyzing chemical networks and structures. In this study, we suggest a reverse-degree-based topological representation. We computed particular types of descriptors of two-dimensional (2-D) coronene fractal formations with a variety of reverse-degree-based topological indices, such as the reverse-degree-based topological index of the first, second, and hyper Zagreb, forgotten, geometric arithmetic, atom-bond-connectivity, and the Randic index.

Fractal microstructure of semiconducting Sb2S3 by surfactant-assisted spin coating and its advantage in optoelectronic properties

Antimony sulfide, Sb2S3 is a mixed band gap semiconductor having a great potential in photovoltaic applications. Its thin films and nanostructures are also a proven anode material in batteries and supercapacitors. This paper presents the synthesis of a unique fern-shaped fractal microstructure of Sb2S3 grown over bare glass substrate using a spin-coating assisted by Triton-X-100 surfactant. The FE-SEM images confirm the fractal formation spanning over hundreds of micrometers. The film exhibits a direct band gap of 1.62 eV. The photoluminescence spectra reveal a strong emission peak at 800 nm (1.55 eV) indicating radiative transitions from sub-band gap states. The surfactant, Triton-X-100, is found to be primarily responsible for forming the ‘Fern’ shaped fractals which have a fractal dimension of 1.84. Without the surfactant, a porous microstructure of Sb2S3 film is observed. The fractality is also associated with lower strain in the film leading to a reduced point defects which would otherwise act as potential recombination centers.

Thermal conductivity and fractal texture formation in β-Si3N4/polyvinylidene fluoride composites

Additive incorporation into polymers can enhance properties, such as the dielectric constant and thermal conductivity. The improved properties of polymer composites have largely been attributed to aggregate and network formation, but the influence of material texture requires clarification. Herein, the relationship between the texture of composite materials and thermal conductivity was investigated based on the fractal nature of composites consisting of polyvinylidene fluoride (PVDF) with β-Si3N4 (SN) as a high-thermal-conductivity filler. The SN particle groups were found to contribute to the formation of thermal conduction paths in the composite material and, thus, improve the thermal conductivity. The characteristics of the particle group clusters involved in SN particle percolation were elucidated by analyzing the interactions between multiple particles as a function of the amount of each fractal dimension. In addition, based on the statistical relationships obtained from the multifractal analysis, the strengths of the interactions between particles during thermal conduction path formation were estimated. Overall, the texture of the SN/PVDF material was characterized by local particle group formation (self-assembly) and subsequent global particle group network construction (self-organization), depending on the amount of SN added. These results suggest that multifractal dimensions can be used as material design indices for the texture of polymer composites.

Network efficiency of spatial systems with fractal morphology: a geometric graphs approach

The functional features of spatial networks depend upon a non-trivial relationship between the topological and physical structure. Here, we explore that relationship for spatial networks with radial symmetry and disordered fractal morphology. Under a geometric graphs approach, we quantify the effectiveness of the exchange of information in the system from center to perimeter and over the entire network structure. We mainly consider two paradigmatic models of disordered fractal formation, the Ballistic Aggregation and Diffusion-Limited Aggregation models, and complementary, the Viscek and Hexaflake fractals, and Kagome and Hexagonal lattices. First, we show that complex tree morphologies provide important advantages over regular configurations, such as an invariant structural cost for different fractal dimensions. Furthermore, although these systems are known to be scale-free in space, they have bounded degree distributions for different values of an euclidean connectivity parameter and, therefore, do not represent ordinary scale-free networks. Finally, compared to regular structures, fractal trees are fragile and overall inefficient as expected, however, we show that this efficiency can become similar to that of a robust hexagonal lattice, at a similar cost, by just considering a very short euclidean connectivity beyond first neighbors.

Penerapan Fraktal dalam Desain Pusat Kerajinan Tenun Ikat Lewokluok di Kecamatan Demon Pagong Kabupaten Flores Timur

The cultural tradition of tie cloth is a typical Indonesian tradition which is the origin of the cultural ecosystem of a region. Ikat cloth has even become an integral part of traditional ceremonial ceremonies, traditional marriage ceremonies, and religious ceremonies. Tie fabrics have different motifs in each region according to the peculiarities of the region itself. Lewokluok Village is a village in East Flores Regency, which has a variety of ikat motifs. The diversity of the ikat motifs is a symbol of unifying the culture of Lewokluok village from the many tribes that inhabit Lewokluok village. The provision of an tie craft center is an effort to overcome the problems faced by tie weaving craftsmen in preserving the tie culture, promoting tie weaving, and marketing the existing tie weaving works. The craft center is a place that is used as a center for production, promotion, tourism, and cultivation activities as well as the cultural development of an area. The design of the Lewokluok tie craft center uses the application of fractals. Fractal is a geometric shape that is irregular in shape, contorted but has similarities with itself. This fractal formation uses the principle of self-similarity from the basic pattern, namely the Lewokluok tie motif which is transformed into the form of building mass. This study uses a descriptive analysis method, starting from data collection consisting of primary data and secondary data. This data is obtained from the results of observations, interviews, and documentation as well as from literature studies and comparative studies of similar objects and themes. This data is then analyzed with various alternatives to produce the final design concept. In the end, it can produce a design center for Lewokluok tie weaving according to its function and purpose and increase the potential for cultural tourism in Lewokluok by applying fractals to the shape of the building that shows the characteristics of Lewokluok tie motifs.

Fractality of Cranial Sutures.

It has long been known that skull suture has a typical fractal structure. Although the fractal dimension has been utilized to assess morphology, the mechanism of the fractal structure formation remains to be elucidated. Recent advances in the mathematical modeling of biological pattern formation provided useful frameworks for understanding this mechanism. This chapter describes how various proposed mechanisms tried to explain the formation of fractal structures in cranial sutures.

Synthesis of Galactoacrylamide and Study of Stimuli‐Responsive Fluorescent Hierarchical Self‐Assembly‐Promoted Specific Interactions with Proteins

AbstractA new sugar based acrylamide, 6‐N‐acrylamido‐D‐galactopyranose (Gal‐acryl) was synthesized from D‐galactose in 6 steps. A solution of Gal‐acryl at basic pH exhibited concentration dependent fractal type assembly formation upon solvent evaporation as revealed by the microscopic images. The fractal dimensions calculated point towards Diffusion Limited Aggregation mode of aggregate formation. The critical aggregation concentration (CAC) of Gal‐acryl in water was found to be 49.3 μM. The increase in fluorescence intensity accompanied by blue shift in emission maxima of coumarin 153 suggests increase in non‐polar environment with Gal‐acryl concentration. The I1/I3ratio of pyrene emission confirmed more non‐polar microenvironment at acidic pH and vice versa at basic pH. Self‐assembly allows Gal‐acryl to exhibit specificity to lectin RCA120 as is demonstrated by fluorescence study further supported by turbidimetric assay. The fluorescence quenching studies with FITC‐RCA120 afforded the Ka value as 6.34×105 M−1, which is much higher than Ka values observed for small galactose derivatives. The present study may help in designing different fractal like glycomaterials with potential applications in drug delivery, bio‐sensing, tissue engineering, and as ligands to asialoglycoprotein receptors that are exclusively expressed by liver cells.

Protein compactness and interaction valency define the architecture of a biomolecular condensate across scales.

Non-membrane-bound biomolecular condensates have been proposed to represent an important mode of subcellular organization in diverse biological settings. However, the fundamental principles governing the spatial organization and dynamics of condensates at the atomistic level remain unclear. The Saccharomyces cerevisiae Lge1 protein is required for histone H2B ubiquitination and its N-terminal intrinsically disordered fragment (Lge11-80) undergoes robust phase separation. This study connects single- and multi-chain all-atom molecular dynamics simulations of Lge11-80 with the in vitro behavior of Lge11-80 condensates. Analysis of modeled protein-protein interactions elucidates the key determinants of Lge11-80 condensate formation and links configurational entropy, valency, and compactness of proteins inside the condensates. A newly derived analytical formalism, related to colloid fractal cluster formation, describes condensate architecture across length scales as a function of protein valency and compactness. In particular, the formalism provides an atomistically resolved model of Lge11-80 condensates on the scale of hundreds of nanometers starting from individual protein conformers captured in simulations. The simulation-derived fractal dimensions of condensates of Lge11-80 and its mutants agree with their in vitro morphologies. The presented framework enables a multiscale description of biomolecular condensates and embeds their study in a wider context of colloid self-organization.

Impact of Minor Alloy Components on the Electrocapillarity and Electrochemistry of Liquid Metal Fractals

AbstractExploring and controlling surface tension‐driven phenomena in liquid metals may lead to unprecedented possibilities for next‐generation microfluidics, electronics, catalysis, and materials synthesis. In pursuit of these goals, the impact of minor constituents within liquid alloys is largely overlooked. Herein, it is showed that the presence of a fraction of solute metals such as tin, bismuth, and zinc in liquid gallium can significantly influence their electrocapillarity and electrochemistry. The instability‐driven fractal formation of liquid alloy droplets is investigated with different solutes and reveals the formation of distinctive non‐branched droplets, unstable fractals, and stable fractal modes under controlled voltage and alkaline solution conditions. In their individually unique fractal morphology diagrams, different liquid alloys demonstrate significantly shifted voltage thresholds in transition between the three fractal modes, depending on the choice of the solute metal. Surface tension measurements, cycle voltammetry and surface compositional characterizations provide strong evidence that the minor alloy components drastically alter the surface tension, surface electrochemical oxidation, and oxide dissolution processes that govern the droplet deformation and instability dynamics. The findings that minor components are able to regulate liquid alloys’ surface tensions, surface element distributions and electrochemical activities offer great promises for harnessing the tunability and functionality of liquid metals.

Mechanism of Stoichiometrically Governed Titanium Oxide Brownian Tree Formation on Stepped Au(111)

Previously observed formation of substoichiometric titanium oxide dendritic structures across terraces of Au(111) is computationally studied and shown to follow the classical fractal formation mechanism of diffusion-limited aggregation (DLA). Globally optimized gas-phase oxide cluster structures are sampled in a variety of landing formations on gold surfaces and shown to favor isomers driving polymerization to Brownian tree fractal structures. Mobility of Ti3O5 monomers is shown to be extremely high, with diffusion barriers of 0.21 eV or less. Through bonding stabilization, polymerization of these monomers is energetically favorable and irreversible on the 111 terrace but geometrically impossible to propagate along the step edge. Simulated scanning tunneling microscopy (STM) images show strong similarity to experiment. By contrast, observation of Ti3O6 aggregating as wires along step edges is explained by the affinity of oxygen to step edges and statistical arguments for aggregation entropy at the step, in addition to low barriers for monomer diffusion and polymerization.

Two states of magnetic frustration in La0.7Sr0.3MnO3 amorphous films with fractal structures: New knowledge about clusters and cluster ensembles

In amorphous La0.7Sr0.3MnO3 films with a fractal structure, two states of magnetic frustration were found, which are the characteristic of cluster spin glass (CSG) and spin glass (SG) states and are associated with the presence of competing, FM and AFM, magnetic interactions, and the geometry of fractal formations. A high density of clusters in them provides effective magnetic interactions of magnetic moments without the participation of free charge carriers. It has been established that the formation of CSG begins with the transition of the central parts of the clusters to the FM state (at T < 98 K) and ends with the formation of the AFM order in the peripheral areas of the clusters, at T < 64 K. Increasing the field to H = 1.2 kOe, which stimulates AFM ordering of peripheral areas, strengthens the state of the cluster glass. A further decrease in temperature (T < 26 ) and an increase in the field (H > 2.5 kOe) causes the transition of peripheral areas from the AFM to the SG state. A phenomenological description of the frustration process is made. The dependences of the order parameter and barrier height on temperature and magnetic field have been studied. The consequences of the CSG ⇒ SG phase transition for the magnetism of the samples and the direct influence of the external field on their magnetic structure are discussed.