Diffusion-limited Aggregation Research Articles

Effects of Moisture Content on the Radiative Properties and Energy-Saving Performance of Silica Aerogel Windows.

The thermal insulation performance of windows is crucial for energy-efficient buildings. Windows are typically the weakest part of the building envelope, regarding thermal insulation. Due to its excellent thermal insulation and high transparency, silica aerogel shows great promise as a window material. However, moisture can impact the effectiveness of the aerogel, leading to poor visibility and reduced thermal insulation. This study simulated a silica aerogel with varying moisture levels using the combination of diffusion-limited cluster aggregation, discrete dipole approximation, and Monte Carlo methods. The effects of the moisture content, thickness, porosity, and particle size on thermal conductivity, solar transmittance, and haze were analyzed. Visual properties of the aerogels were also considered. The energy consumption of a 30 m2 room under different climates was simulated using TRNSYS to assess the energy-saving potential of silica aerogel glass. The findings indicate that a higher moisture content leads to decreased solar transmittance and increased thermal conductivity of aerogels. Silica aerogel glass is more energy efficient than single-layer float glass, with the dry aerogel performing better in cold climates but worse in hot climates. This study provides insights for designing aerogel glass that optimizes solar transmittance and thermal insulation to enhance building comfort and energy efficiency.

Impact of various aggregation kinetics on thermophoretic velocity of asphaltene deposition.

Asphaltene deposition may pose serious challenges to flow assurance of crude oil in well columns. Different aggregation kinetics would partly be responsible for asphaltene particle growth ending in deposition on the surface of well columns. This work primarily investigates the thermophoretic deposition velocity caused by temperature gradients inside well columns for various asphaltene aggregation kinetics, including crossover behaviour, sedimentation, reaction-limited aggregation (RLA), and diffusion-limited aggregation (DLA). To do so, the experimental observations of size distribution for a live crude oil was performed at 80°C and pressure range of 4500-5500 psia. Moreover, various patterns of different size distributions were gathered from the literature for the sake of comparison. Next, a well column in southern Iran was selected to study the kinetic behaviour of thermophoretic velocity of deposition, with a difference between geothermal and static temperatures of around 5 to 50°C. The non-isothermal deposition velocity was shown to decrease from the top to the bottom of the well column, according to the findings of the study. The results also revealed that the thermophoretic velocity decreases as particle size increases and vice versa. This was confirmed by examining a comparably large range of asphaltene particle sizes, ranging from approximately 100 nm to roughly 9 µm. Practical implications of these findings were also discussed which would provide guidance for mitigation of asphaltene deposition in well columns.

Coagulation-flocculation process on a lattice: Monte Carlo simulations

Coagulation-flocculation, the physicochemical process widely used for purification a wastewater, is affected both by chemical details of involved polymers and by the statistics of their conformations on a large scale. The latter aspect is covered in this study by employing a coarse-grained modeling approach based on a combination of two paradigms of statistical mechanics. One is the self-avoiding walk (SAW) which generates a range of conformations for a linear polymer of N SAW monomers. Another one is a non-trivial diffusion limited aggregation (DLA) process of N DLA impurities (referred thereafter as ‘particles’) which describes their coagulation occurring with the probability 0<p⩽1 (p = 1 recovers a standard DLA). DLA of diffusive particles is complemented by their irreversible adsorption on the SAW monomers occurring with the probability equal to one, both processes resulting in formation of the DLA-SAW agglomerates. The dynamics of formation of such agglomerates, as well as their fractal dimensions and internal structure are of practical interest. We consider a range of related characteristics, such as: (i) absolute Na and relative na adsorbing efficiencies of SAW; (ii) effective gyration radius RgDLA-SAW of the DLA-SAW agglomerates; and (iii) the fractal dimension DDLA-SAW of these aggregates. These are studied within a wide range for each parameter from a set {p,NDLA,NSAW} .

Cretaceous Fe-oxidising microbial mats in injection dykes in the Variscan gold-bearing metamorphic basement of the Tisza Mega-unit (SE Hungary)

Large quantities of Cretaceous Fe-oxidising bacteria (FeOB) are documented here for the first time in carbonate injection dykes. These injection dykes occur in fissures of a Palaeozoic metamorphic complex in the Geresd Hills, Hungary, and are made up of microlaminated haematitic carbonate that contain very well preserved helical FeOB stalk fossils similar to recent iron-oxidising Gallionella and Mariprofundus. The presence of filament clusters with semi-parallel textures suggests aerotactic bacterial cell movements. Fractal analysis of the filament clusters showed higher fractal dimension values (Df ∼ 1.9) compared to the typical fractal dimension values of the texturally similar, but inorganic Diffusion-Limited Aggregation (DLA) structures (Df ∼ 1.7). This finding is interpreted to suggest that the observed filamentous texture is of microbial origin. Cretaceous basaltic intrusions that penetrated the gold-bearing pyritic metamorphic basement launched a low-temperature flow of reductive iron-rich fluids. Well‑oxygenated bottom waters percolated through thin pelagic sediments and picked up ferrous iron from the pyritic metamorphic bedrock and made it available for the Fe-oxidising microbes. Reactivation of a major fault zone (Mecsekalja Zone) resulted in the development of extensional faults in the metamorphic basement beneath the seafloor. Pelagic carbonate sediments with FeOB mats were drawn into these fissures, forming a network of injection dykes. This study shows that fractal dimension analysis and directional analysis may be a helpful method to differentiate FeOB textures from inorganic textures in ancient rocks. The presented results are expected to provide useful analogues to aid the reconstruction of similar palaeoenvironments worldwide.

Experimental, theoretical and numerical simulation-based investigations on the fabricated Cu2ZnSn thin-film-based Schottky diodes with enhanced electron transport for solar cell

Copper-zinc-tin Cu2ZnSn (CZT) thin films are promising materials for solar cell applications. This thin film was deposited on a fluorine-doped tin oxide (FTO) using an electrochemical deposition hierarchy. X-ray diffraction of thin-film studies confirms the variation in the structural orientation of CZT on the FTO surface. As the pH of the solution is increased, the nature of the CZT thin-film aggregate changes from a fern-like leaf CZT dendrite crystal to a disk pattern. The FE-SEM surface micrograph shows the dendrite fern leaf and sharp edge disks. The 2-D diffusion limitation aggregation under slippery conditions for ternary thin films was performed for the first time. The simulation showed that by changing the diffusing species, the sticking probability was responsible for the pH-dependent morphological change. Convincingly, diffusion-limited aggregation (DLA) simulations confirm that the initial structure of copper is responsible for the final structure of the CZT thin films. An experimental simulation with pH as a controlled parameter revealed phase transition in CZT thin films. The top and back contact of Ag-CZT thin films based on Schottky behavior give a better electronic mechanism in superstrate and substrate solar cells.

New Insights into the Formation of Aggregates of Bidisperse Nano- and Microplastics in Water Based on the Analysis of In Situ Microscopy and Molecular Simulation.

Microplastics (MPs) and nanoplastics (NPs) in water pose a global threat to human health and the environment. To develop efficient removal strategies, it is crucial to understand how these particles behave as they aggregate. However, our knowledge of the process of aggregate formation from primary particles of different sizes is limited. In this study, we analyzed the growth kinetics and structures of aggregates formed by polystyrene MPs in mono- and bidisperse systems using in situ microscopy and image analysis. Our findings show that the scaling behavior of aggregate growth remains unaffected by the primary particle size distribution, but it does delay the onset of rapid aggregation. We also performed a structural analysis that reveals the power law dependence of aggregate fractal dimension (df) in both mono- and bidisperse systems, with mean df consistent with diffusion-limited cluster aggregation (DLCA) aggregates. Our results also suggest that the df of aggregates is insensitive to the shape anisotropy. We simulated molecular forces driving aggregation of polystyrene NPs of different sizes under high ionic strength conditions. These conditions represent salt concentration in ocean water and wastewater, where the DLVO theory does not apply. Our simulation results show that the aggregation tendency of the NPs increases with the ionic strength. The increase in the aggregation is caused by the depletion of clusters of ions from the NPs surface.

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

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.