Diffusion-limited Aggregation Model Research Articles

Study on the morphological characteristics of coal particle flocs under fractal theory

Based on fractal theory, the geometric size, surface morphology, and spatial structure of flocs in the hydrophobic flocculation process of coal particles were examined via particle size distribution detection, image analysis technology, and numerical simulation. The floc size distribution curve was classified and discussed via the dual-domain fractal rule, and the variation law of the composition and distribution of the coarse- and fine-grained coal particle flocs with the flocculation process was clarified. The various levels and characteristics of the floc growth geometry were explained by the analysis of the fractal dimension of the individual floc surface and the multiple spectra of the floc groups in the metallographic microscopy image. The hydrophobic flocculation process of coal particles was simulated via the diffusion-limited cluster aggregation model. The gyration fractal dimension was used to examine floc fragmentation and reconstruction and clarify the growth law of the floc spatial structure during the hydrophobic flocculation process.

Simulation of Dendrite Growth with a Diffusion-Limited Aggregation Model Validated by MRI of a Lithium Symmetric Cell during Charging

Lithium metal batteries offer high energy density but are challenged by dendrite growth, which can lead to short circuits and battery failure. Multiple models with varying degrees of accuracy and computational cost have been developed to understand and predict dendrite growth. This study presents a simple model to simulate macroscale dendrite growth on lithium metal electrodes. The model uses a 3D single-particle Diffusion-Limited Aggregation (DLA) algorithm with an electric field bias to simulate dendrite growth. The electric field bias was introduced into the model with an important parameter, namely the biasing factor c, which determines the balance between diffusion and electric field effects. Before performing the simulation with the proposed model, the dendrite growth in a lithium symmetric cell during charging was measured by sequential 3D magnetic resonance imaging (MRI). These data were then used to validate the simulation, as the dendrite structure in each measured MRI time frame was used a starting point for a new simulation, the results of which were then validated with the measured dendrite structure of the next time frame. The best agreement between the simulated and measured dendrite structures using the overlap and displacement of deposition sites metrics was obtained at the biasing factor c = 0.7. This agreement was also good in terms with the fractal dimension of the dendrite structures. The proposed method offers a simple, accurate, and scalable framework for predicting dendrite growth over long deposition periods, making it a valuable tool for studying dendrite suppression under real-world battery charging conditions.

On the diffusive model of C<sub>60</sub> fullerene clusterization in liquids

Purpose of research. The purpose of the study is to explain the mechanisms of clustering of fullerene molecules in liquid media observed using various structural nuclear -physical methods, as well as the interpretation of experimental data within the framework of a microscopic diffusion model. Methods. The article gives a brief overview of the results of positron annihilation spectroscopy, small-angle neutron scattering, translucent electron microscopy, with which the geometric parameters of fullerene clusters in solutions were established. The theoretical method of research is the microscopic diffusion model “Limited diffusion aggregation”, which describes the kinetic processes of clusterization. Results. The diffusion limited aggregation model displays adequately the mechanism of formation of C60 molecules in the form of fractal units in the volume of fluid, observed in experiments on positron annihilation spectroscopy and small-angle neutron scattering. The structural indicators of aggregate fullerene particles in carbon disulfide, o-Xylene, toluene, water and other solvents are analyzed. The properties of various diffusion models (the reaction limited aggregation model and the diffusion limited aggregation model) are considered and their combination in relation to the assessmentsof the kinetics of fullerene aggregation in solutions. A quantitative comparison of the results of the discussed models was carried out using the example of carbon disulfide. Conclusion. The diffusion microscopic models adequately describe the phenomenon of fullerene aggregation in polar and non-polar solvents, which are recorded by various nuclear methods (positron annihilating spectroscopy and small-angle neutron scattering), the most reliable is the diffusion limited aggregation model, more than that, it is the basis of a numerical definition of structural structures parameters of units. Compared to neutron scattering, with the annihilation of positrons, in the aggregation of fullerene the [Ps–C60] molecular complexes participate in the clusterization, but this does not affect the change in the size of the cluster and the reliability of the results.

Formation dynamics of branching structure in the slippery DLCA model.

We numerically investigated the aggregation dynamics and resulting network structures of colloidal gels using the slippery diffusion-limited cluster aggregation (DLCA) model. In this model, bonds are irreversibly formed upon the particle contacts, but the angles among them are not fixed, unlike the conventional DLCA. This allows clusters to be deformed in the process of aggregation. By characterizing the aggregation dynamics and using a reduced network scheme, our simulation revealed two distinct branching structure formation routes depending on the particle volume fraction ϕ. In lower volume fraction systems (ϕ ≤ 8%), the deformations of small-size clusters proceed prior to the percolation. When the Maxwell criterion is satisfied and the clusters become mechanically stable, the formation of the branching structure is nearly completed. After forming the branching structures, they aggregate and form a larger percolating network. Then, the aggregation proceeds through the elongation and straightening of the chain parts of the network. In higher volume fraction systems (ϕ > 8%), on the other hand, the clusters percolate, and a fine and homogeneous branching structure is formed at the early stage of the aggregation. In the aging stage, it collapses into a denser and more heterogeneous structure and becomes more stable. Our quantitative analyses of the branching structure will shed light on a new strategy for describing the network formation and elasticity of colloidal gels.

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

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.

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.

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.

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.

Simulation of noble metal nanocluster systems formation during deposition from a colloid solution

A model was proposed for the convective flow of the liquid phase of a colloidal solution of glycerol and noble metal nanoparticles (Ag, Au, Ag/Au) near the substrate. The diffusion approximation is used to describe the formation of nanocluster systems on a substrate. The diffusion-limited aggregation model was implemented by applying a cellular automaton in the Neumann neighborhood. A diverse structure of model systems of nanoclusters, which adequately describes the structural features of the experimental samples, was obtained by varying the aggregation probability parameter. The proposed models can be useful for calibrating the parameters of the experimental production of systems of noble metal nanoclusters, as well as describing in the first approximation the processes that have a decisive effect on nanocluster structures.

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.

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).

Fractal Growth of Quasi Two-Dimensional Copper Dendrites by Template-free Electrodeposition

Fractal dendrites are extensively observed in industry, especially in the electrochemical deposition process. The fractal dendrite electrodeposition behavior of quasi-two-dimensional Cu (Q2D-Cu) metal based on the wire is examined via direct electrodeposition using a thin layer reactor. Here, to explain the fractal growth mechanism, the directional migration and random walking of ions are introduced in the traditional diffusion-limited aggregation model, and fractal patterns consistent with the experimental results are successfully simulated. In addition, the Cu fractal dendrite structure is finely adjusted by varying electrodeposition conditions, demonstrating its great potential for further optimization. The CuO/Q2D-Cu fractal dendrite photothermal device fabricated through in situ assembly of CuO nanowires on Cu fractal dendrite has good photothermal conversion ability. Therefore, metal fractal dendrites, which are considered harmful in the electroplating industry, have application prospects in the photothermal field.

Toward a Realistic Model of Diffusion-Limited Aggregation: Rotation, Size-Dependent Diffusivities, and Settling.

In this Brownian dynamics simulation study on the formation of aggregates made of spherical particles, we build on the well-established diffusion-limited cluster aggregation (DLCA) model. We include rotational effects, allow diffusivities to be size-dependent as is physically relevant, and incorporate settling under gravity. We numerically characterize the growth dynamics of aggregates and find that their radius of gyration, R g, grows approximately as R g ∼ t 1.02 for classical DLCA but slows to an approximate growth rate of R g ∼ t 0.71 when diffusivity is size-dependent. We also analyze the fractal structure of the resulting aggregates and find that their fractal dimension, d, decreases from d ≈ 1.8 for classical DLCA to d ≈ 1.7 when size-dependent rotational diffusion is included. The addition of settling effects further reduces the fractal dimension observed to d ≈ 1.6 and appears to result in aggregates with a vertical extent marginally smaller than their horizontal extent.

A Computational Template for Three-Dimensional Modeling of the Vascular Scaffold of the Human Thyroid Gland.

We recently designed an innovative scaffold-bioreactor unit for the bioengineering of a three-dimensional (3D) bioartificial human thyroid gland or its miniaturized replica as a part of a microfluidic chip test system. This device is based on the evidence that the 3D geometry of the intraglandular stromal/vascular scaffold (SVS; i.e., the fibrous and vascular matrix) of mammalian viscera plays a key role in guiding growth and differentiation of in vitro seeded cells. Therefore, we initiated a research program focused on computer-aided reconstruction of the 2nd to 4th order intralobar arterial network (IAN) of the human thyroid gland as a reliable surrogate for its 3D SVS, to be used as an input for rapid prototyping of a biomaterial replica. To this end, we developed a computational template that works within the Mathematica environment, giving rise to a quasi-fractal growth of the IAN distribution, constrained within an approximation of the thyroid lobe shape as a closed surface. Starting from edge detection of planar images of real human thyroid lobes acquired by in vivo real-time ultrasonography, we performed data approximation of the lobar profiles based on splines and Bezier curves, providing 3D lobar shapes as geometric boundaries for vessel growth by a diffusion-limited aggregation model. Our numerical procedures allowed for a robust connection between development of lobar arterial trees and thyroid lobe shape, led to a vascular self-similarity consistent with that of a cadaveric lobar arterial cast, and reproduced arterial vessels in a proportion not statistically different from that described for the real human thyroid gland. We conclude that our algorithmic template offers a reliable reproduction of the extremely complex IAN of the adult human thyroid lobe, potentially useful as a computational guidance for bioprinting of thyroid lobe matrix replicas. In addition, due to the simplicity and limited number of morphometrical parameters required by our system, we predict its application to the design of a number of patient-tailored human bioartificial organs and organs-on-chip, including parenchymal viscera and bones. Impact statement The study introduces the computer simulation of the three-dimensional (3D) intrinsic vascular matrix of the human thyroid gland, offering a general concept applicable to a number of other human viscera. Indeed, it provides a flexible software tool for reproduction of a 3D surrogate of the organ's 3D stromal matrix, suitable for eventual 3D bioprinting with biomaterials, and recellularization with organ-specific stem cells/progenitors. The final expectation is the design of patient-tailored 3D organ's matrices upon clinical request.

Sustainable synthesis and theoretical studies of polyhedral gold nanoparticles displaying high SERS activity, NIR absorption, and cellular uptake

The preparation of gold nanoparticles with multimodal properties such as near-infrared absorption, high surface-enhanced Raman scattering, cell internalization and low cytotoxicity is a challenging task. In this study, we developed a sustainable protocol to develop gold nanoparticles, using trisodium citrate as reducing, stabilizing, and shape-modulating agent at ambient conditions (25 °C). Reduction of gold salt at room temperature at a peculiar ratio R(Ccitrate/CHAuCl4) and concentration of reactants resulted in the formation of non-spherical, homogenous mixture of gold nanoparticles with polyhedral shapes. The protocol is extremely simple and does not even require stirring or mechanical shaking. The as-synthesized gold nanoparticles, even though multi-shaped, displayed a single monomodal peak in dynamic light scattering with polydispersity index of 0.098, representative of a fairly good monodisperse system. We investigated the optical properties of the nanoparticles both experimentally and by two-dimensional Finite-Difference-Time-Domain modeling. Due to the presence of shapes such as nanorods, nanotriangles, and prismatic, these nanoparticles exhibited a high surface-enhanced Raman scattering activity and a wide absorption range extending up to the near-infrared region, which makes them useful candidate for photothermal therapy too. We characterized the nanoparticles by electron microscopy, UV–vis–NIR spectroscopy, Raman spectroscopy, and X-ray photoelectron spectroscopy. We have also developed an enhanced numerical diffusion limited aggregation model to simulate the growth of the particles, including the particle interfacial energy as a parameter of the system. Numerical results matched with the experimental data, and the model revealed being effective in reproducing size, shape, and morphological characteristics of non-spherical nanoparticles obtained under real experimental conditions. Finally, in vitro studies and nanoparticles cellular uptake were performed on a model cell line of mouse brain endothelium (bEnd.3) to assess biocompatibility. The main advantage of the proposed method lies in its simplicity with no other requirement like refluxing at elevated temperature, mechanical stirring, electromagnetic radiations, ultrasound, toxic chemicals, or seed mediation.

Computational Complexity of Biased Diffusion-Limited Aggregation

Diffusion-Limited Aggregation (DLA) is a cluster growth model that consists of a set of particles that are sequentially aggregated over a two-dimensional grid. In this paper, we introduce a biased version of the DLA model, in which particles are limited to move in a subset of possible directions. We denote $k$-DLA the model where the particles move only in $k$ possible directions. We study the biased DLA model from the perspective of Computational Complexity, defining two decision problems The first problem is Prediction, whose input is a site of the grid $c$ and a sequence $S$ of walks, representing the trajectories of a set of particles. The question is whether a particle stops at site $c$ when sequence $S$ is realized. The second problem is Realization, where the input is a set of positions of the grid, $P$. The question is whether there exists a sequence $S$ that realizes $P$, i.e. all particles of $S$ exactly occupy the positions in $P$. Our aim is to classify the Prediciton and Realization problems for the different versions of DLA. We first show that Prediction is P-Complete for 2-DLA (thus for 3-DLA). Later, we show that Prediction can be solved much more efficiently for 1DLA. In fact, we show that in that case, the problem is NL-Complete. With respect to Realization, we show that restricted to 2DLA the problem is in P, while for 1DLA is in L.