Fractal Growth Model Research Articles

Directional region control of the thermal fractal diffusion of a space body**Project supported by the National Natural Science Foundation of China (Grant Nos. 61273088, 61473173, and 61473174).

We present a directional region control (DRC) model of thermal diffusion fractal growth with active heat diffusion in three-dimensional space. This model can be applied to predict the space body heat fractal growth and study its directional region control. When the nonlinear interference term and the inner heat source term are generalized functions, the relationship between the particle aggregation probability and the interference terms can be obtained using the norm theory. We can then predict the aggregation form of particles in different regions. When the nonlinear interference terms in the model are expressed as a trigonometric function and its composite function, our simulations show that the DRC method of thermal fractal diffusion is effective and has reference value for the directional control of actual fractal growth systems.

Механизм формирования пленок SnS химическим осаждением из водных растворов

The results of investigation of the growth mechanism of SnS thin films prepared by hydrochemical deposition by means of a scanning probe microscopy are presented. According to the results of a comparative analysis of layers morphology at different growth stages and the use of fractal formalism growth model of SnS films by clustercluster aggregation with elements of self-organization is proposed. It is shown that there is a solid layer, which again are deposited particles of the solution. The process has expressed periodic self-organizing nature. The study showed that the proposed model of film growth SnS when hydrochemical deposition agrees well with ex-periments and proves difficult coagulation mechanism of the formation of hard phase of the saturated aqueous solution.

Simulation and assessment of realistic breast lesions using fractal growth models

A new method of generating realistic three dimensional simulated breast lesions known as diffusion limited aggregation (DLA) is presented, and compared with the random walk (RW) method. Both methods of lesion simulation utilize a physics-based method for inserting these simulated lesions into 2D clinical mammogram images that takes into account the polychromatic x-ray spectrum, local glandularity and scatter. DLA and RW masses were assessed for realism via a receiver operating characteristic (ROC) study with nine observers. The study comprised 150 images of which 50 were real pathology proven mammograms, 50 were normal mammograms with RW inserted masses and 50 were normal mammograms with DLA inserted masses. The average area under the ROC curve for the DLA method was 0.55 (95% confidence interval 0.51–0.59) compared to 0.60 (95% confidence interval 0.56–0.63) for the RW method. The observer study results suggest that the DLA method produced more realistic masses with more variability in shape compared to the RW method. DLA generated lesions can overcome the lack of complexity in structure and shape in many current methods of mass simulation.

Nonlinear evolution of surface morphology under shadowing

Fluorocarbon thin-film deposition is studied, which shows an anomalous high dynamic growth exponent and therefore does not fit in any universal class of fractal surface growth models. A detailed analysis of the nonlinear behavior of the surface morphology evolution is carried out, quantifying several features of the shadowing instability. A synergy effect with the Kardar-Parisi-Zhang nonlinearity, which couple the large scales induced by shadowing with intermediate scales, may explain the anomalous high growth exponent.

Characteristic analysis on morphological evolution of suspended particles in water during dynamic flocculation process

The evolution of floc morphology during constant-speed and variable-speed flocculation was examined to understand floc growth mechanisms. Flocculation-test results were reported in terms of floc average size and fractal dimension, derived from in situ optical sampling and image analysis. The morphological evolution was also described using a fractal growth model, which defined flocculation as the combined processes of aggregation and restructuring. During constant-speed flocculation, aggregation rate increased with increasing shear, but breakage became significant. Also, a decrease was observed after reaching the peak of size, possibly due to floc settling in low shear and the irreversibility of breakage in high shear. The development of floc morphology in variable-speed flocculation indicated that the surface nature of initial flocs was critical to form larger final flocs, because aggregates with irregular shape had more connection spots than those with smooth surface, thus producing a higher aggregation...

Characteristic analysis on temporal evolution of floc size and structure in low-shear flow

A series of flocculation tests were performed to investigate the effect of low-shear rates ( G = 3–16 s −1) on flocculation of kaolin suspension by polyaluminum chloride (PACl), with the goal of understanding floc growth mechanisms. Results were reported in terms of floc average size ( d p ) and boundary fractal dimension ( D pf ), derived from a non-intrusive optical sampling and digital image analysis technique. As expected, the rate of floc aggregation increased with increasing G, resulting in faster changes in aggregate size and structure in the initial stage of flocculation. Nevertheless, steady state was attained faster for D pf than for d p at the same shear rates, possibly due to the self-similarity of fractal aggregates. An interesting finding was that at G = 3 s −1, an obvious plateau was observed for the average-size evolution at steady state; for shear rates of 6 and 7 s −1, the flocs exhibited some decrease after reaching the peak of size, mainly as a result of floc settling at steady state; and for G = 11–16 s −1, a decrease in floc size was possibly attributed to the irreversibility of PACl-floc breakage. The process of floc growth was described using a fractal growth model, which defined flocculation as the result of the combined processes of aggregation and restructuring. The conceptual model could effectively characterize temporal changes in floc size and structure, and found that fragmentation followed by reformation seemed to be more effective in forming larger and more compact aggregates than the restructuring process due to erosion and reformation, which may provide useful insights for the design of flocculation reactors.

Using optical methods to reveal structural features in porous nanocomposite films of the tin dioxide-silicon dioxide system

Nanocomposite films of the tin dioxide-silicon dioxide system have been obtained by the sol-gel method using various solvents (ethanol, butanol) and their optical properties have been studied in the 300–700 nm wavelength range. The obtained data were used to calculate the Uhrbach energy parameters, slope parameters, and band-gap energies of the films. The results are interpreted within the framework of the model of fractal nanocluster growth and evolution during the sol-gel process. The employed optical method can be effectively used to study the optical properties of nanocomposites for optimizing their production technology.

Multifractal pattern and process during a recent period of forest expansion in a temperate mountainous region of China

Understanding the mechanisms that drive spatial pattern change over space and time is a central goal of landscape ecology, but these patterns are often complex. Multifractals, an extension of the concept of fractal geometry, is especially suited to the description of complex pattern. In our paper, we examine the scaling properties of forest spatial pattern over 32 years of forest expansion in a warm temperate deciduous forest near Beijing city using multifractal analysis. An idealized fractal growth model was used to simulate spatial patterns and to link patterns to processes. We show that the despite substantial increases in forest cover, the scaling properties of the spatial pattern of the forest landscape remains similar over time. The variance of Rényi spectra among different years implied that the changes in the forest landscape over time may be a fractal growth process. The fact that these scaling behaviors persist across a heterogeneous landscape suggests that simple rules could exist for some processes of forest expansion, facilitating prediction and further modeling efforts.

Fractal Growth of Minerals: a Modified DLA Model

A modified fractal growth model of minerals was considered in the square lattice 300 × 300. One seed is placed at the center of the square lattice as a growth focus and the attraction domain is set to diamond. Three main parameters are considered in this model which probability of the surface adhesion is Pg with the range of 0 to 99% to eliminate the noise impact and adsorption distance d varies from 0 to 9 units on behalf of the gravitational field strength between particles and fill gaps s varied from 0.00 to 5.00 units distance to perform the influence of environmental temperature on the minerals growth system. Three model sets are carried out with two fixed parameters and the other changed parameter. The results show the fractal dimension of DLA group increase with the decrease of Pg and the increases of d and s, as well as the structure of minerals from disseminated, star or radial gradual transition to agglomerate. The model is a two-dimensional simulation for three-dimensional fractal growth and can not fully reflect the complex growth dynamics of mineral aggregate in three-dimensional confined space.

Dendritic zircon formation by deterministic volume-filling fractal growth: Implications for the mechanisms of branch formation in dendrites

In this paper, I demonstrate that the complex structure observed in natural dendritic zircon, which crystallized from a mafic magma, can be approximated by a simple model of deterministic volume-filling fractal growth. The model comprises a novel dendritic growth process that reveals a previously unrecognized underlying structure to complex dendritic crystals, and provides an ideal way of maximizing surface area at the crystal-melt interface during dendritic crystal growth. This addresses the important kinetic effect of creating interfacial area. In turn, this relates to dissipation of the latent heat of fusion from the moving crystal-melt interface, which, in addition to chemical diffusion, is known to be a fundamental rate-controlling process in dendritic crystal growth. A new mechanism of tip-splitting branch formation in dendritic crystals is proposed. Although both types of branch formation seem to occur simultaneously, the newly proposed mechanism operates at larger length scales and independently of classical side-branching linked with periodic perturbations caused by thermal noise. It is shown that this type of modeling potentially has applications in metallurgical engineering; and in predicting the relative lengths, orientations, and spacings of branches in mathematical modeling studies of natural and experimentally grown dendritic crystals.

Stimuli-Responsive Controlled Growth of Mono- and Bidimensional Particles from Basic Zirconium Sulfate Hydrosols

A thermostimulated sol-gel transition in a system prepared by mixing a ZrOCl(2) acidified solution to a hot H(2)SO(4) aqueous solution was studied by dynamic rheological measurements and quasi-elastic light scattering. The effect of temperature and of molar ratio R(S) = [Zr]/[SO(4)] on the gelation kinetics was analyzed using the mass fractal aggregate growth model. This study shows that the linear growth of aggregates occurs at the early period of transformation, while bidimensional growth occurs at the advanced stage. The bidimensional growth can be shifted toward monodimensional growth by decreasing the aggregation rate by controlling the temperature and/or molar ratio R(S). EXAFS and Raman results gave evidence that the linear chain growth is supported by covalent sulfate bonding between primary building blocks. At the advanced stage of aggregation, the assembly of linear chains through hydrogen bonding gave rise to the growth of bidimensional particles.

Interconnected Hydroxyapatite Nanofibers in the Biomimetic Coating of a Bioinert Ceramic Substrate

A bioinert ceramic substrate, α‐Al2O3, has been coated with hydroxyapatite (HAp) by the biomimetic route using simulated body fluid (SBF) solution at room temperature. The substrate was incubated at 37°C in SBF for 6 days with a periodic replacement with freshly prepared SBF at 48‐h intervals. After 6 days, continuous nanofiber‐like structures of HAp (5–35 μm in length, 0.05 μm diameter) were obtained, connecting the intra‐ and interglobular clusters, within the coated mineral layer on the substrate surface. This is a unique and new observation, and this phenomenon has been demonstrated by a simple fractal growth model.

Kinetic Simulation of Fractal Aggregation on Liquid Surfaces

A modified fractal growth model based on the deposition,diffusion, and aggregation (DDA) with cluster rotationis presented to simulate two-dimensional fractal aggregationon liquid surfaces. The mobility (including diffusion and rotation)of clusters is related to its mass, which is given by Dm = D0s−γD and θm = θ0s−γθ, respectively. We concentrate on revealing the details of theinfluence of deposition flux F, cluster diffusion factorγD and cluster rotation factorγθ on the dynamics of fractal aggregationon liquid surfaces. It is shown that the morphologies ofclusters and values of cluster density and fractal dimensiondepend dramatically on the deposition flux and migration factors of clusters.

307 フラクタル次元成長モデルによる火花点火エンジンの燃焼シミュレーション(OS-1 着火・燃焼解析,噴流解析,研究発表講演)

A combustion process in an S.I. engine has been simulated using the Fractal Dimension Growth Model. The computed pressure and the rate of heat release agree well with measured ones for three different excess air ratios at the same engine speed. However, the model needs improvements for the cases of different engine speeds.

Fractal Aggregation Under Rotation**The project supported by the Fund of Natural Science Foundation for Young Scientists of Zhejiang Province under Grant No. RC02069 and Natural Science Foundation of Zhejiang Province of China under Grant No. 101032

By means of the Monte Carlo simulation, a fractal growth model is introduced to describe diffusion-limited aggregation (DLA) under rotation. Patterns which are different from the classical DLA model are observed and the fractal dimension of such clusters is calculated. It is found that the pattern of the clusters and their fractal dimension depend strongly on the rotation velocity of the diffusing particle. Our results indicate the transition from fractal to non-fractal behavior of growing cluster with increasing rotation velocity, i.e. for small enough angular velocity ω the fractal dimension decreases with increasing ω, but then, with increasing rotation velocity, the fractal dimension increases and the cluster becomes compact and tends to non-fractal.

Fractal growth of a conventional calamitic liquid crystal

We report observation of liquid crystal phase ordering via fractal growth aggregates for a calamitic, non-bent-core mesogen. Fractal growth of a conventional smectic- C (Sm-C) phase from the isotropic melt after a temperature quench was experimentally investigated with respect to time, cell gap, quench depth, and quench rate. The determined fractal dimensions relating to the area as well as the perimeter of the growing aggregates suggest a phase formation process via a percolation mechanism. Computer simulations of the phase ordering process give further evidence for percolation growth, qualitatively reproducing the observed textures and quantitatively leading to the same fractal dimensions. We propose a general model of fractal smectic liquid crystal growth, which accounts for all of the different systems observed so far, bent-core or "banana" phases as well as the observation of fractal phase ordering of a conventional Sm-C phase. The model is based on the "breaking" of the commonly observed growth anisotropy by strong in-layer molecular interactions. These are provided by hydrogen bonding in the Sm-C case discussed here and by steric interactions in the case of the bent-core phases discussed in previous publications.

Fractal approach for modeling the morphology evolution of olefin polymerization with heterogeneous catalysts

AbstractFractal theory and methodology were used to investigate the morphology of titanium–magnesium‐supported polyethylene catalysts and their relevant polymer particles. Through an analysis of the submicrostructures using scanning electron microscopy images, the surface fractal dimensions of the related particles were estimated with the box‐counting method. With consideration given to the fact that the growth process of a polymer is an evolving fractal process, which is controlled on the one hand by the initial conditions, including the initial fractal dimensions of the catalysts and the initial reaction conditions, and on the other hand by the previous morphology characteristics of the system, a novel polymerization fractal growth model was constructed. The simulation results showed good agreement with the experiment data. Moreover, the morphology evolution with the prepolymerization technique was predicted, and it was suggested that the duration of polymerization was 10–30 min. It was proven that the use of the surface fractal dimension as an important parameter describing the surface morphologies of the particles, either of catalysts or polymers, was real and effective. © 2003 Wiley Periodicals, Inc. J Appl Polym Sci 90: 1463–1470, 2003

Fractal Pattern Growth in Ti-Implanted Steel with High Ion Flux

We report on the formation of metal nanometre phase and fractal patterns insteel using metal vapour vacuum arc source ion implantation with high ionflux. The dense nanometre phases are cylindrical and welldispersed in the Ti-implanted layer with an ion flux up to 50 µA/cm2. The collision fractal pattern is formed in Ti-implanted steelwith an ion flux of 25 µA/cm2 and the disconnected fractal pattern isobserved with an ion flux of 50 µA/cm2. The average density ofnanometre phases decreases from 1.2×1011/cm2 to6.5×1010/cm2 as the ion flux increases from 25 µA/cm2 to50 µA/cm2. Fractal pattern growth is in remarkable agreement withSander's diffusion-limited aggregation model. The alloy clusters havediffused and aggregated in chains forming branches to grow a beautiful treeduring Ti implantation with an ion flux ranging from 75 µA/cm2 to85 µA/cm2. We discuss the model of fractal pattern growth during ionimplantation with high ion flux.

Simulation of the fractal growth in the restricted geometry

A modified fractal growth model based on the diffusion limited aggregation (DLA) with spatial restriction of a random walk is considered. This model is of interest as new insight towards the quantum confinement phenomenon for the small clusters containing tens and hundreds of constituents. The main effect of this restriction occurs upon kinetic parameters of the growing fractals. The final properties of them little depend on the spatial restriction, and several tens of particles can provide usual DLA aggregates.

Geometric fractal growth model for scale-free networks.

We introduce a deterministic model for scale-free networks, whose degree distribution follows a power law with the exponent gamma. At each time step, each vertex generates its offspring, whose number is proportional to the degree of that vertex with proportionality constant m-1 (m>1). We consider the two cases: First, each offspring is connected to its parent vertex only, forming a tree structure. Second, it is connected to both its parent and grandparent vertices, forming a loop structure. We find that both models exhibit power-law behaviors in their degree distributions with the exponent gamma = 1+ln(2m-1)/ln m. Thus, by tuning m, the degree exponent can be adjusted in the range, 2 < gamma < 3. We also solve analytically a mean shortest-path distance d between two vertices for the tree structure, showing the small-world behavior, that is, d approximately ln N/ln K macro, where N is system size, and k macro is the mean degree. Finally, we consider the case that the number of offspring is the same for all vertices, and find that the degree distribution exhibits an exponential-decay behavior.