Diffusion-limited Aggregation Research Articles

Fractal growth of copper powder on point and plate electrodes based on diffusion-limited aggregation model

Fractal growth of copper powder on point and plate electrodes based on diffusion-limited aggregation model

The Pathologically Evolving Aggregation-State of Cells in Cancerous Tissues as Interpreted by Fractal and Multi-Fractal Dispersion Theory in Saturated Porous Formations

A recent author’s fractal fluid-dynamic dispersion theory in porous media has focused on the derivation of the associated nonergodic (or effective) macrodispersion coefficients by a 3-D stochastic Lagrangian approach. As shown by the present study, the Fickian (i.e., the asymptotic constant) component of a properly normalized version of these coefficients exhibits a clearly detectable minimum in correspondence with the same fractal dimension (d ≅ 1.7) that seems to characterize the diffusion-limited aggregation state of cells in advanced stages of cancerous lesion progression. That circumstance suggests that such a critical fractal dimension, which is also reminiscent of the colloidal state of solutions (and may therefore identify the microscale architecture of both living and non-living two-phase systems in state transition conditions) may actually represent a sort of universal nature imprint. Additionally, it suggests that the closed-form analytical solution that was provided for the effective macrodispersion coefficients in fractal porous media may be a reliable candidate as a physically-based descriptor of blood perfusion dynamics in healthy as well as cancerous tissues. In order to evaluate the biological meaningfulness of this specific fluid-dynamic parameter, a preliminary validation is performed by comparison with the results of imaging-based clinical surveys. Moreover, a multifractal extension of the theory is proposed and discussed in view of a perspective interpretative diagnostic utilization.

Modeling Colloidal Particle Aggregation Using Cluster Aggregation with Multiple Particle Interactions.

In this study, we investigate the aggregation dynamics of colloidal silica by generating simulated structures and comparing them to experimental data gathered through scanning transmission electron microscopy (STEM). More specifically, diffusion-limited cluster aggregation and reaction-limited cluster aggregation models with different functions for the probability of particles sticking upon contact were used. Aside from using a constant sticking probability, the sticking probability was allowed to depend on the masses of the colliding clusters and on the number of particles close to the collision between clusters. The different models of the sticking probability were evaluated based on the goodness-of-fit of spatial summary statistics. Furthermore, the models were compared to the experimental data by calculating the structures' fractal dimension and mass transport properties from simulations of flow and diffusion. The sticking probability, depending on the interaction with multiple particles close to the collision site, led to structures most similar to the STEM data.

Crossover effects and dynamic scaling properties from Eden growth to diffusion-limited aggregation

The diffusion-limited aggregation (DLA) model is a classical fractal aggregation model that describes numerous fractal phenomena, and the Eden model is a typical discrete growth model for investigating cell clusters, such as bacteria or tissue cultures. There is a significant difference between these two discrete models in their growth rules and dynamic scaling properties. In this paper, we introduce a generalized growth theory to describe effectively these two distinct growth classes based on the competitive growth rules. By controlling the adhesion coefficient, a continuous transition exists from dense polymerization to finger growth, i.e., the evolution from Eden growth to DLA in both two-dimensional and three-dimensional cases. To simulate effectively the generalized growth system in the most physically relevant case of three-dimensions, we employ a proper technique named the Marsaglia method for correctly obtaining distributed points to avoid the inappropriate simulation schemes commonly used. Furthermore, the crossover of these competitive growth exhibits non-trivial scaling behavior, and the quantitative relation between the corresponding critical exponents and the adhesion parameter is also investigated. Concurrently, the porosities of the generalized growth system are calculated, which exhibit obviously nontrivial dynamic properties depending on the growth dimensions.

Fractal analysis on Ag2O thin film using a data-driven approach

The synthesis of fractal Ag oxide (Ag2O) on the surface of Ag thin film has been achieved at room temperature by using Synchrotron X-ray irradiation. We have performed an automated quantitative analysis of a batch of 1879 fractal Ag2O patterns in a scanning electron microscopy (SEM) image within a radius of 2000 mm outward from the center of the X-ray beam. The morphology is similar to that of the diffusion-limited cluster aggregation (DLCA) model. The fractal dimension (D) of Ag2O is between 1.7 and 1.5 from the center to the edge. The area distribution density of fractal Ag2O follows a quadratic function with radius R. It is found that the branches’ number of fractal Ag2O is a key factor affecting the fractal dimension. The more branches the fractal has, the greater the D is. This is the first time that Ag fractal has been investigated by combining automated data analysis methods with batch experimental data. This data-driven approach provides a new research perspective for rationally regulating materials’ fractal morphology and performance.

On the Radial Growth of Ballistic Aggregation and Other Aggregation Models

For a class of aggregation models on the integer lattice Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {Z}}}^d$$\\end{document}, d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 2$$\\end{document}, in which clusters are formed by particles arriving one after the other and sticking irreversibly where they first hit the cluster, including the classical model of diffusion-limited aggregation (DLA), we study the growth of the clusters. We observe that a method of Kesten used to obtain an almost sure upper bound on the radial growth in the DLA model generalizes to a large class of such models. We use it in particular to prove such a bound for the so-called ballistic model, in which the arriving particles travel along straight lines. Our bound implies that the fractal dimension of ballistic aggregation clusters in Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {Z}}}^2$$\\end{document} is 2, which proves a long standing conjecture in the physics literature.

Disorder-induced non-linear growth of fingers in immiscible two-phase flow in porous media

Immiscible two-phase flow in porous media produces different types of patterns depending on the capillary number Ca and viscosity ratio M. At high Ca, viscous instability of the fluid–fluid interface occurs when the displaced fluid is the more viscous, and leads to viscous fingering, which is believed to exhibit the same growth behavior as the viscously-unstable fingers observed in Hele–Shaw cells by Saffman and Taylor [“The penetration of a fluid into a porous medium or Hele–Shaw cell containing a more viscous liquid,” Proc. R. Soc. London 245, 312 (1958)], or as diffusion-limited aggregates (DLA). In such Laplacian growth processes, the interface velocity depends linearly on the local gradient of the physical field that drives the growth process (for two-phase flow, the pressure field). However, a non-linear power-law dependence between the flow rate and the global pressure drop, reminiscent of what has also been observed for steady-state two-phase flow in porous media, was evidenced experimentally for the growth of viscously-unstable drainage fingers in two-dimensional porous media, 20 years ago. Here, we revisit this flow regime using dynamic pore-network modeling and explore the non-linearity in the growth properties. We characterize the previously unstudied dependencies of the statistical finger width and non-linear growth law's exponent on Ca, and discuss quantitatively, based on theoretical arguments, how disorder in the capillary barriers controls the growth process' non-linearity, and why the flow regime crosses over to Laplacian growth at sufficiently high Ca. In addition, the statistical properties of the fingering patterns are compared to those of Saffman–Taylor fingers, DLA growth patterns, and the results from the aforementioned previous experimental study.

A triangular model of fractal growth with application to adsorptive spin-coating of polymers.

Over the last 40 years, applied mathematicians and physicists have proposed a number of mathematical models that produce structures exhibiting a fractal dimension. This work has coincided with the discovery that objects with fractal dimension are relatively common in the natural and human-produced worlds. One particularly successful model of fractal growth is the diffusion limited aggregation (DLA) model, a model as notable for its simplicity as for its complex and varied behavior. It has been modified and used to simulate fractal growth processes in numerous experimental and empirical contexts. In this work, we present an alternative fractal growth model that is based on a growing mass that bonds to particles in a surrounding medium and then exerts a force on them in an iterative process of growth and contraction. The resulting structure is a spreading triangular network rather than an aggregate of spheres, and the model is conceptually straightforward. To the best of our knowledge, this model is unique and differs in its dynamics and behavior from the DLA model and related particle aggregation models. We explore the behavior of the model, demonstrate the range of model output, and show that model output can have a variable fractal dimension between 1.5 and 1.83 that depends on model parameters. We also apply the model to simulating the development of polymer thin films prepared using spin-coating which also exhibit variable fractal dimensions. We demonstrate how the model can be adjusted to different dewetting conditions as well as how it can be used to simulate the modification of the polymer morphology under solvent annealing.

Marangoni spreading on liquid substrates in new media art.

With the advent of new media art, artists have harnessed fluid dynamics to create captivating visual narratives. A striking technique known as dendritic painting employs mixtures of ink and isopropanol atop paint, yielding intricate tree-like patterns. To unravel the intricacies of that technique, we examine the spread of ink/alcohol droplets over liquid substrates with diverse rheological properties. On Newtonian substrates, the droplet size evolution exhibits two power laws, suggesting an underlying interplay between viscous and Marangoni forces. The leading edge of the droplet spreads as a precursor film with an exponent of 3/8, while its main body spreads with an exponent of 1/4. For a weakly shear-thinning acrylic resin substrate, the same power laws persist, but dendritic structures emerge, and the texture of the precursor film roughens. The observed roughness and growth exponents (3/4 and 3/5) suggest a connection to the quenched Kardar-Parisi-Zhang universality class, hinting at the existence of quenched disorder in the liquid substrate. Mixing the resin with acrylic paint renders it more viscous and shear-thinning, refining the dendrite edges and further roughening the precursor film. At larger paint concentrations, the substrate becomes a power-law fluid. The roughness and growth exponents then approach 1/2 and 3/4, respectively, deviating from known universality classes. The ensuing structures have a fractal dimension of 1.68, characteristic of diffusion-limited aggregation. These findings underscore how the nonlinear rheological properties of the liquid substrate, coupled with the Laplacian nature of Marangoni spreading, can overshadow the local kinetic roughening of the droplet interface.

Prediction of temperature-dependent nucleation and growth rates from crystallization-related heat release.

We propose a method for determining the time and, therefore, temperature-dependent relative nucleation and growth rates during crystallization. We do so by linking the partial differential equationgoverning the time dynamics of the crystal size distribution to kinetic (Avrami) parameters describing heat release. This approach is tested in silico by nucleating and growing diffusion limited aggregates with time-varying morphology and growth rates unhindered by impingement. The associated heat release is analyzed, showing that nucleation and growth rates could be extracted with high fidelity.

The motion of three-dimensional fractal aggregates in homogeneous shear flow

A model for the motion of aggregates in low Reynolds number flow has been established by utilizing the Stokesian dynamics and applying the quaternion as global attitude parameter. The model has been validated by the theorical solution of sphere doublet in shear flow, the simple helical aggregates, and the experiment on the settling of some specific objects in still water. The motion of fractal aggregates has been further studied, aiming to get better understanding of the dynamic behavior of fine-grained sediment flocs in shear flow. The fractal aggregates have been generated using the diffusion-limited aggregation model, which has similar fractal dimension as natural flocs. The results illustrate that fractal aggregates undergo a complex rotation in shear flow, which exhibit a bi-periodic characteristic. The motion of a particle within the fractal aggregate shows three-dimensional trajectory in a simple shear flow, affected by its initial orientation. The major rotation period is approximately 4π/γ̇, which corresponds to the rotation period of a sphere with shear strength γ̇. The deviation decreases with the increase in the size of the fractal aggregate.

In situ surface evolution dynamics of external-electric-field-triggered structural oscillation on Au(111)

Surface nanostructures serve as an essential role in determining intrinsic physical features and device performance in solid materials. Here, this work systematically investigates the surface dynamical evolution at the nanoscale on Au(111) induced by a “tip-to-surface” external electric field through a scanning tunneling microscope (STM). The Au(111) steps exhibit a “compact to fractal” reconstruction under a relatively high electric field, in which the transition is strengthened with increasing the applied electric field. Inversely, a “fractal to compact” morphological transition enables to be self-triggered at Au(111) surface steps upon a normal STM imaging electric field (very low). These two reversible structural changes are attributed to the diffusion-limited aggregation mechanism where the diffusion barriers were quantified as 0.64–0.75 eV varied with the regularity in step edges. In addition, we further simulate a “terrace-edge-kink” model to trace the effect of atomic coordination on structural transition, determining the surface step edge stability. This study presents insight into morphological and structural transformation at surface steps induced by variant external electric fields, establishing a deep understanding in the nature of surface evolution dynamics.

Fractal simulation of hydrophobic flocculation fragmentation and reconstruction of coal particles

Based on a diffusion limited cluster aggregation model, this study analyzed the fragmentation and reconstruction of flocs during the hydrophobic flocculation process of coal particles. The flocs gyration radius and gyration fractal dimension calculated by the program characterize the geometric size and spatial structure of the coal particle flocs, respectively. With advancing step length, the distributions of gyration radius first increased and then decreased and finally basically stabilized at approximately 1.70 ± 0.2, indicating that the reconstruction of flocs is reversible in geometric size. However, the gyration fractal dimensions first decreased and then increased and finally stabilized at 1.49, 1.63, 1.60 and 1.51 with increasing step length, showing variability. This indicated that the spatial structure of the flocs is irreversible. The above conclusions of the simulation analysis were confirmed by the floc particle size distribution curves and the image analysis of surface fractal dimensions. The study clarifies the morphological changes and fragmentation-reconstruction changes of flocs and improves the hydrophobic flocculation mechanism of coal particles.

Two methods for thermal conduction analysis of bi-porous sintered structures

Loop heat pipes (LHPs) have consistently garnered attention for effectively managing thermal conditions in critical electronic components of spacecraft and satellites. Particularly intriguing are LHPs equipped with bi-porous capillary wicks, demonstrating commendable flow and thermal dissipation capabilities. Despite its significance for evaluating the phase-transition heat transfer in porous structures, the effective thermal conductivity (ETC) has not been understood mechanistically. This study aims to establish an efficient procedure for predicting ETC in bi-porous wicks through a comparative analysis of analytical and numerical methods. The analytical ETC expression was formulated using thermal resistance network and neck formation theory. Meanwhile, the numerical model utilized the Finite volume method (FVM) combined with a 3D porous structure created by Diffusion-Limited Aggregation (DLA) algorithm. The investigation revealed a congruence between two methods and the measured ETCs of interstitial-pore samples containing fine nickel powders. However, undesirable results were shown for coarse samples due to the irregular particle shapes. In samples with formation pores, the numerical model excels in characterizing the spatial distribution and scale of pores, albeit at the expense of time consumption and model complexity compared to the analytical one. Both methods prove valuable for ETC modeling of sintered bi-porous structures according to desired applications.

Diffusion limited aggregation, resetting and large deviations of Brownian motion

Models of fractal growth commonly consider particles diffusing in a medium and that stick irreversibly to the forming aggregate when making contact for the first time. As shown by the well-known diffusion limited aggregation (DLA) model and its generalisations, the fractal dimension is sensitive to the nature of the stochastic motion of the particles. Here, we study the structures formed by finite-lived Brownian particles, i.e. particles constrained to find the aggregate within a prescribed time, and which are removed otherwise. This motion can be modelled by diffusion with stochastic resetting, a class of processes which has been widely studied in recent years. In the short lifetime limit, a very small fraction of the particles manage to reach the aggregate. Hence, growth is controlled by atypical Brownian trajectories, that move nearly in straight line according to a large deviation principle. In d dimensions, the resulting fractal dimension of the aggregate decreases from the DLA value and tends to 1, instead of increasing to d as expected from ballistic aggregation. In the zero lifetime limit one recovers the non-trivial model of ‘aggregation by the tips’ proposed long ago by R. Jullien [J. Phys. A: Math. Gen. 19, 2129 (1986)].

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.

Nucleation and growth mechanisms in the electrochemical synthesis of Ag/polyaniline nanocomposites

Nucleation and growth mechanisms in the synthesis of polyaniline nanowires (PANI NWs) and of silver nanoparticles (AgNPs) which are assembly to PANI NWs, are proposed. Aniline polymerization proceeds over a graphite electrode at potentials that go from 0.9 V to 1.15 V (vs Ag/AgCl), using: 0.25, 0.15, or 0.05 M of aniline in HNO3 solution. In the polyaniline nucleation and growth is postulated a diffusional 3D model that involves the dendrites formation and the unidirectional growing step. At lowest aniline concentration, the transients displayed the typical peaked shape, characteristic of a 3D nucleation and diffusion-controlled growth. Silver nanoparticles are synthesized on the polyaniline/graphite substrate. Current-time transients at −0.1 V and −0.2 V (vs Ag/AgCl), displayed a decreasing in the current which is fit to Cotrell modified diffusion-limited aggregation model. Nanowires and nanoparticles morphology observed by SEM corroborated the proposed mechanisms.

Effect of temperature on the appearance of flocculated asphaltene in an unperturbed unstable American shale oil: Insights from a colloidal modeling

This work aimed to study the onset of flocculation of asphaltenes using optical microscopy and to evaluate the effect of temperature on the destabilization of an unstable undisturbed American shale oil. After magnetic stirring for 24 h, the flocculated asphaltene was redispersed. Increasing the temperature to 40 °C led to asphaltenes flocculating. After heating, the sample was cooled back to room temperature. We calculated the precipitate volume from microscopy data and found that after cooling the system had a larger precipitate volume than in the previously studied scenarios. When comparing the volumes of precipitates before and after cooling, it is possible to observe that the volume of smaller particles does not vary significantly in each class, but the volume of larger particles increases with cooling, indicating an irreversible phenomenon. Statistical analysis of the optical microscopy results allowed a detailed analysis of the phenomenon. To study the colloidal effect, a theoretical analysis was performed based on Hamaker theory and De Gennes scaling theory, presenting compatible results when the parameters of the repulsive term were adjusted. When the parameters of both the attractive term and the repulsive term of the model were kept constant, greater stability was verified with the increase of temperature. Conversely, it was observed that the reduction of the parameter related to the length of the alkyl chains can explain the experimental macroscopic behavior of the shale oil here studied and also some different behaviors reported in the literature, i. e., increasing of flocculated volume with increasing temperature. This size reduction of the length of the alkyl chains can be attributed to a desolvation effect. Hence, the appropriate parameterization of the model showed great potential to mimic the behavior of increasing and also decreasing stability with the increase of temperature. The sets of parameters were related to reaction-limited aggregation (RLA) and diffusion-limited aggregation (DLA), which allowed to interconnect the discussion on experimental stability of asphaltenes, molecular simulation, and the theoretical discussion. Furthermore, the proper use of this theory seems to be a promising way to represent microscopic interactions in asphaltene systems, including previous results from the literature. This work presents (i) an experimental case that shows reversibility of asphaltene flocculation only with magnetic stirring; (ii) the effect of temperature on the stability of asphaltenes, and also (iii) brings a contribution with the volume of precipitate, making it possible to conclude on the irreversibility in the thermal cycle applied. In addition, the proper use of this modeling can predict the stability changes as a function of temperature, being a promising way to represent microscopic interactions in asphaltene systems, including previous results from the literature.

Monitoring effects of hydrodynamic cavitation pretreatment of sodium oleate on the aggregation of fine diaspore particles through small-angle laser scattering

Hydrodynamic cavitation (HC) enhanced fine particle aggregation could be largely due to the generation of tiny bubbles and their role in bridging particles. However, the lack of adequate characterizations of aggregates severally limits our further understanding of the associated aggregation behaviors. In this study, the aggregation of fine diaspore particles was comparatively investigated in sodium oleate (NaOl) solutions with and without HC pretreatment through the small-angle laser scattering (SALS) technique in a shear-induced aggregation (SIA) system. Results showed that HC pretreatment caused the formation of bulk nanobubbles (BNBs), which significantly modified the particle interactions and thereby modified the size and mass fractal dimension (Df) of aggregates under different SIA conditions. Although HC pretreatment did not noticeably alter the gradual change trend of aggregate size and structure characteristics under specific variables, BNBs bridging facilitated the aggregation process towards the diffusion-limited cluster aggregation model, resulting in the formation of larger but looser aggregates. This effect was more pronounced under relatively high NaOl concentrations. Apart from BNBs, the aggregation was also affected by cavitation bubbles formed during shear cavitation, which was more significant under high stirring intensity conditions (i.e., 1800 rpm) than the low stirring intensity conditions (i.e., 600 rpm).

Modeling Processes of Emergence and Growth of Fractal Structures and Chamber Defects of Thermal Power Units. Part 2

The paper describes further development of model studies with the purpose to draw analogies of the occurrence of possible inhomogeneities and defects of the fractal type on the surface of various objects under the action of laser radiation for different conditions and schemes of a laser simulation experiment in comparison with real processes in the chambers of thermal machines due to the contacts of the working substances with their surface. The subject of this consideration is related to the fact that in laser schemes one can relatively easily and in a controlled manner to study their topology in various modifications (in the range of 0D–3D structures formed on surfaces of various geometric shapes of the fractal type). This provides great opportunities for studying various frequently occurring undesirable random defects and inhomogeneities of complex configuration with the development of unexpected processes in the dynamic modes of operation of various types of power units in their actual operation. This is determined by geometric analysis within the framework of the correspondding non-linear models with different non-linear fractal images. In this aspect, we have used two models to consider the development of emerging fractal fractured structures: first, diffusion propagation/growth of cracks of different geometry with a set of local segments and fragments in different directions; secondly, with diffusely limited aggregation of their distribution (Diffusion-Limited Aggregation – DLA) using the formalism of cellular automata in percolation approximation for different algorithms for the formation of fractal structures. We present only the final results of calculations without details of their procedure, which is sufficient for a visual and qualitative determination of the process of growth of fracturing and defects. In this case, the final goal of consideration is to study the emergence of possible and controllable local areas / collectors, starting from surface structures, and their association with real objects on the surface of chambers with a working substance in power systems. The possible practical significance of such modeling and analysis is discussed to forecast their working conditions.