Diffusion-limited Cluster Aggregation Research Articles

Light scattering and absorption by fractal aggregates including soot

This paper addresses how well and under what conditions the Rayleigh-Debye-Gans (RDG) approximation describes scattering and absorption of light by fractal aggregates (FA) including soot. The RDGFA theory, which is the prevailing, first order description of this problem, has two assumptions: the monomers, or primary particles, of the aggregate scatter and absorb in the Rayleigh regime, and the aggregate scatters in the diffraction limit weighted by this Rayleigh scattering and absorbs as a system of independent monomer particles. The aggregates studied here are formed via Diffusion Limited Cluster Aggregation (DLCA) and have a fractal dimension D = 1.78 ± 0.04 and prefactor of k0 = 1.35 ± 0.10. The aggregates are a collection of monodisperse spherical monomers with point contacts. Optical calculations were performed with the multiple sphere T-matrix (MSTM) and DDSCAT codes for incident light polarized perpendicular to the scattering plane. The scattering considered is the forward scattering intensity and the angular scattering as parameterized by the scattering wave vector. The total absorption cross section for aggregates is also calculated. This work stresses the systematic study of the effects of the variables of monomers per aggregate, which ranged from one to 502, two monomer size parameters of 0.157 and 0.314, and a wide range of refractive index real and imaginary parts. It also considers soot refractive indices with three representative dispersions. A summary of results for both scattering and absorption includes deviations from RDGFA theory ranging as large as 35% with positive deviations increasing with the real part of the refractive index and negative deviations growing with the imaginary part. These deviation from the RDG limit are shown to be similar to deviations for spheres.

Aggregation kinetics of irreversible patches coupled with reversible isotropic interaction leading to chains, bundles and globules

Abstract In the present study we are performing simulation of simple model of two patch colloidal particles undergoing irreversible diffusion limited cluster aggregation using patchy Brownian cluster dynamics. In addition to the irreversible aggregation of patches, the spheres are coupled with isotropic reversible aggregation through the Kern–Frenkel potential. Due to the presence of anisotropic and isotropic potential we have also defined three different kinds of clusters formed due to anisotropic potential and isotropic potential only as well as both the potentials together. We have investigated the effect of patch size on self-assembly under different solvent qualities for various volume fractions. We will show that at low volume fractions during aggregation process, we end up in a chain conformation for smaller patch size while in a globular conformation for bigger patch size. We also observed a chain to bundle transformation depending on the attractive interaction strength between the chains or in other words depending on the quality of the solvent. We will also show that bundling process is very similar to nucleation and growth phenomena observed in colloidal system with short range attraction. We have also studied the bond angle distribution for this system, where for small patches only two angles are more probable indicating chain formation, while for bundling at very low volume fraction a tail is developed in the distribution. While for the case of higher patch angle this distribution is broad compared to the case of low patch angles showing we have a more globular conformation. We are also proposing a model for the formation of bundles which are similar to amyloid fibers using two patch colloidal particles.

How anisotropy beats fractality in two-dimensional on-lattice diffusion-limited-aggregation growth.

We study the fractal structure of diffusion-limited aggregation (DLA) clusters on a square lattice by extensive numerical simulations (with clusters having up to 10^{8} particles). We observe that DLA clusters undergo strongly anisotropic growth, with the maximal growth rate along the axes. The naive scaling limit of a DLA cluster by its diameter is thus deterministic and one-dimensional. At the same time, on all scales from the particle size to the size of the entire cluster it has a nontrivial box-counting fractal dimension which corresponds to the overall growth rate, which, in turn, is smaller than the growth rate along the axes. This suggests that the fractal nature of the lattice DLA should be understood in terms of fluctuations around the one-dimensional backbone of the cluster.

Effect of surfactant concentration to aggregations of nanogold particles

This research presents a study of aggregation of colloidal gold nanoparticles using 400 nm diameter gold nanoparticles mixed with a surfactant (Plantacare 2000) at various concentrations. When observed under a microscope, we found that the nanoparticles aggregated to form nearly spherical clusters at the beginning of the formation, and then sedimented to the bottom of the container. These clusters moved with Brownian’s motion and collided with each other in the horizontal plane, forming branch-like clusters in 2D. The appearance and size of the clusters were different depending on the concentration of surfactant. The clusters’ size and appearance were rarely changed after mixing with surfactant for 90 minutes, and we found that the cluster’s shapes were nearly spherical at low surfactant concentration (c = 0.25%). At surfactant concentration between 0.50% - 5.00%, the aggregates formed branch-like clusters with skinnier branches and smaller sizes at higher surfactant concentration. Moreover, we also found that, at surfactant concentrations between 2.50% - 5.00%, nanoparticles and aggregates stuck to the bottom of the glass container quickly and rarely moved after 10 minutes. At c = 0.25%, the 2D fractal dimension of the aggregates was measured to be D = 1.88 ± 0.04, since the aggregates were nearly spherical. The fractal dimension decreased to the minimum of D = 1.50 ± 0.12 at c = 1.50%, similar to D ∼ 1.45 found in diffusion-limited cluster aggregation (DLCA). At surfactant concentration above 1.50%, the fractal dimension increased until it reached the value of D ∼ 1.66 at c = 5.00%.

Lattice animals in diffusion limited binary colloidal system.

In a soft matter system, controlling the structure of the amorphous materials has been a key challenge. In this work, we have modeled irreversible diffusion limited cluster aggregation of binary colloids, which serves as a model for chemical gels. Irreversible aggregation of binary colloidal particles leads to the formation of a percolating cluster of one species or both species which are also called bigels. Before the formation of the percolating cluster, the system forms a self-similar structure defined by a fractal dimension. For a one component system when the volume fraction is very small, the clusters are far apart from each other and the system has a fractal dimension of 1.8. Contrary to this, we will show that for the binary system, we observe the presence of lattice animals which has a fractal dimension of 2 irrespective of the volume fraction. When the clusters start inter-penetrating, we observe a fractal dimension of 2.5, which is the same as in the case of the one component system. We were also able to predict the formation of bigels using a simple inequality relation. We have also shown that the growth of clusters follows the kinetic equations introduced by Smoluchowski for diffusion limited cluster aggregation. We will also show that the chemical distance of a cluster in the flocculation regime will follow the same scaling law as predicted for the lattice animals. Further, we will also show that irreversible binary aggregation comes under the universality class of the percolation theory.

Analytical expression for the friction coefficient of DLCA aggregates based on extended Kirkwood–Riseman theory

ABSTRACTWe use a self-consistent field method, which we have previously validated, to calculate the translational friction coefficient of fractal aerosol particles formed by diffusion-limited cluster aggregation (DLCA). Our method involves solving the Bhatnagar–Gross–Krook model for the velocity around a sphere in the transition flow regime. The velocity and drag results are then used in an extension of Kirkwood–Riseman theory to obtain the drag on the aggregate. Our results span a range of primary sphere Knudsen numbers from 0.01 to 100 for clusters with up to N = 2000 primary spheres. Calculated friction coefficients are in good agreement with experimental data and approach the correct continuum and free molecule limits for small and large Knudsen numbers, respectively. Results show that particles exhibit more continuum-like behavior as the number of primary spheres increase, even when the primary particle is in the free molecule regime; as an illustrative example, the friction coefficient for aggregate...

Growth and Aggregation Regulate Clusters Structural Properties and Gel Time.

When particles undergo aggregation, very often they form structures that can be described with fractal geometry concepts. The spatial organization of the particles embedded in these aggregates, quantified by means of their fractal dimension, plays a key role in the clusters' diffusion and aggregation process. Fractal dimensions are typically known for some specific, ideal aggregation scenarios, such as for diluted diffusion limited cluster aggregation (DLCA) or reaction limited cluster aggregation (RLCA). The situation becomes significantly more complicated as soon as the initial particle concentration increases and fractal-dimension changing phenomena, such as particle growth, occur. In this frame, the aim of the present work is twofold: (i) to investigate the clusters' spatial organization in a scenario where growth and aggregation occur simultaneously and (ii) to assess the corresponding aggregation kinetics. To this end, an ad hoc Monte Carlo model has been developed. Both DLCA and RLCA regimes have been explored at several initial primary particle concentrations and for different growth rates. The results were discussed in terms of the characteristic times of growth (τG) and aggregation (τA), as well as the rate at which the structural properties change, vR. It was then possible to propose empirical correlations in the form dm = dm (vR) and tgel = tgel (τA, τG) to relate the evolution of the mass mobility exponent dm and the gel times to the simultaneously occurring processes of growth and aggregation.

Structure and interaction in pathway of charged nanoparticles aggregation in saline water as probed by scattering techniques

The evolution of interaction and structure during the aggregation of anionic silica nanoparticles in saline water has been examined. The net attraction between charged nanoparticles with existing of salt ions is found to be significantly higher than that accounted by van der Waals force of the DLVO theory. The increase in the net attraction with salt leads to systematic transformation of nanoparticles into dimers, tetramers and higher-mers which finally results into fractal aggregate. Such multi-step aggregation is consistent with diffusion limited cluster aggregation (DLCA) fractal morphology but the fractal dimension (∼2.6) obtained is much larger than usually comprehended by DLCA.

A finite-volume fast diffusion-limited aggregation model for predicting the coagulation rate of mixed low-ionized system

Accompanied with the changing of coagulation time, the micro-structure of aerogel can be controlled by adding Polyacrylic acid (PAA) into sol system. We simulate the process of particles aggregation contains attracting molecular chains based on diffusion-limited cluster aggregation (DLCA). Compared with the normal coagulation system, the coagulation rate of the system that contains attracting chains are sped up first and then slowed down. The results of the stimulation point out that the interaction between particles and chains not only accelerates the motion of particles, but also separates the region and constrains the clusters’ motion. These two effects are coexisting but the attracting interaction play a dominant role in the early state while the volume of chains has a dramatic influence on cluster’s motion in late states.

Salt-induced cluster formation of gold nanoparticles followed by stopped-flow SAXS, DLS and extinction spectroscopy.

In this work we investigate the salt-induced cluster formation of monodisperse, spherical gold nanoparticles possessing negative surface charge. Time dependent dynamic light scattering was used to determine the critical coagulation concentration and to study the kinetics of cluster growth. Cryo-TEM as well as small angle X-ray scattering allowed to obtain insights into the cluster morphology at various salt concentrations. The time dependent cluster growth was monitored by stopped-flow small angle X-ray scattering experiments and UV-Visible extinction measurements simultaneously. Our results reveal the transition from reaction limited cluster aggregation (RLCA) at low salt concentrations to diffusion limited cluster aggregation (DLCA) at high salt concentrations. We were able to correlate changes of the plasmon resonance position of the gold nanoparticles to the cluster size. By comparison to theoretical calculations using Mie theory, we found indication that stronger resonance shifts during cluster growth can be related to smaller inter-particle spacings.

Modeling simultaneous deposition and aggregation of colloids

Solid formation and particle deposition in chemical and pharmaceutical processes often leads to heat exchanger or reactor fouling, compromising operability. The aim of this work is to propose a mathematical framework, based on population balance equations, to describe the simultaneous colloid aggregation and deposition. The problem is dealt with by accounting for two distinct cluster mass distributions, the first one describing the free clusters, while the second one the deposited aggregates. Three main events are considered: (i) aggregation of two free clusters, (ii) deposition of free clusters on the reactor wall (i.e. a free cluster turns into a deposited one), and (iii) aggregation of a free cluster with a deposited one. To describe the aggregation of free clusters, three typical aggregation kernels (i.e. diffusion-limited and reaction-limited cluster aggregation as well as simple shear aggregation) have been used. Starting from those kernels, aggregation laws to describe the interaction between a free and a deposited cluster were proposed and a simple cluster-mass dependent deposition law was employed. The simulation results are then discussed in the frame of one dimensionless parameter, function of the relevant characteristic time of the process (i.e. aggregation and deposition).

Relationship between geometric parameters and compositional data: A new approach to karst bauxites exploration

Southern Italy karst bauxites occur in Apulia and Campania regions. In Apulia there are the autochthonous concentrations of Spinazzola (Murge) and San Giovanni Rotondo (Gargano), as well as the allochthonous Salento-type deposits. Typical Campania bauxites are those of the Matese Mts and in the Caserta area. During Cretaceous, both Apulia and Campania experienced sub-aerial carbonate exposures, which promoted bauxite accumulation.Southern Italy bauxite deposits share the same ooidic texture consisting of sub- circular aggregates dispersed in a fine-grained matrix. The ooids are mainly formed by a large core of Al-hematite surrounded by alternating layers of boehmite and Al-hematite reflecting dry and wet climate periods. The ooids of the allochthonous Salento-type bauxite consist of a large boehmite core surrounded by a thinner rim of Al-hematite.Image analysis, performed on these karst bauxites provided geometric data such as ooids circularity and aspect ratio, number of ooids and fractal dimension D. In the Southern Italy bauxite deposits significant correlations exist between the results of image analysis and compositional data. In the allochthonous Salento-type bauxites significant correlations between geometric features (number of ooids and aspect ratio) and some critical elements (Sc, Ni and REE) have been observed. In the autochthonous Apulian bauxites a significant correlation between Ni and circularity has been observed instead.The arrangement and the growth of the ooids can be described with a molecular diffusion pattern, and their growth can be described in terms of fractal geometry. The values of fractal dimension are different between Apulia and Campania bauxite deposits, this suggesting that during long-lasting sub-aerial events, the diffusion-limited cluster aggregation process may prevail over the simpler diffusion-limited one, which is responsible for ooids formation during time-limited exposure events.Image analysis and its integration with other analytical techniques provide interesting information about the genesis of bauxite deposits, and can therefore represent an innovative and useful tool for the exploration of bauxite ores.

Simulation Model of Multiple Cluster Fractals Cultured in Nanocomposite Polymer Electrolyte Film

This work contributes to the melioration of the modeling and simulation of laboratory cultured fractals using poly (vinylidene fluoride-co-hexafluoropropylene)/poly (ethyl methacrylate)-ammonium trifluorome-thanesulfonate nanocomposite polymer electrolyte films as the media of growth. Main focus is given to fractals and fractal growth models particularly DLA (Diffusion Limited Aggregation). The DLA cluster formed through DLA is formed by particles moving in Brownian motion (diffusion) which meet and stick together randomly (aggregation) to form the cluster. The simulation of multiple cluster fractals is done using DLA methods incorporating different parameters such as its sticking coefficient, lattice geometry and number of particles. To compare the simulation with the real patterns obtained, one vital aspect would be the calculation of their fractal dimension values. The computer program developed is able to calculate the fractal dimension value of each of the simulated fractal patterns. Suitable fractal dimension calculation method is employed according to its usefulness and efficiency. Fractal growth modeling and simulation such as done here can contribute to the understanding of other related studies concerning fractal growth found in areas including medical (nervous systems, cancer growth and more).

Kinetically guided colloidal structure formation

The self-organization of colloidal particles is a promising approach to create novel structures and materials, with applications spanning from smart materials to optoelectronics to quantum computation. However, designing and producing mesoscale-sized structures remains a major challenge because at length scales of 10-100 μm equilibration times already become prohibitively long. Here, we extend the principle of rapid diffusion-limited cluster aggregation (DLCA) to a multicomponent system of spherical colloidal particles to enable the rational design and production of finite-sized anisotropic structures on the mesoscale. In stark contrast to equilibrium self-assembly techniques, kinetic traps are not avoided but exploited to control and guide mesoscopic structure formation. To this end the affinities, size, and stoichiometry of up to five different types of DNA-coated microspheres are adjusted to kinetically control a higher-order hierarchical aggregation process in time. We show that the aggregation process can be fully rationalized by considering an extended analytical DLCA model, allowing us to produce mesoscopic structures of up to 26 µm in diameter. This scale-free approach can easily be extended to any multicomponent system that allows for multiple orthogonal interactions, thus yielding a high potential of facilitating novel materials with tailored plasmonic excitation bands, scattering, biochemical, or mechanical behavior.

Grand Canonical Monte Carlo Simulation of Nitrogen Adsorption in a Silica Aerogel Model

In this paper, the Diffusion Limited Cluster Aggregation (DLCA) method is employed to reconstruct the three-dimensional network of silica aerogel. Then, simulation of nitrogen adsorption at 77 K in silica aerogel is conducted by the Grand Canonical Monte Carlo (GCMC) method. To reduce the computational cost and guarantee accuracy, a continuous-discrete hybrid potential model, as well as an adsorbed layer thickness estimation method, is employed. Four different structures are generated to investigate impacts of specific surface area and porosity on adsorptive capacity. Good agreement with experimental results is found over a wide range of relative pressures, which proves the validity of the model. Specific surface area and porosity mainly affect nitrogen uptake under low pressure and high pressure, respectively.

Sensitivity analysis of aggregate morphology on mass-mobility relationship and improved parameterizations

AbstractPast studies have shown that the diffusion-limited cluster aggregation mechanism yields aggregates with a mass fractal dimension (Df) of around 1.8 and power-law prefactor (kf) ranging between 1.2 and 2.5. For a fixed Df, an increasing kf physically manifests as decreasing shape anisotropy or the degree of “stringiness” of an aggregate. In this work, we investigate the effects of changing kf, monomer size (d), and number of monomers (N) of computer-simulated aggregates on their mass-mobility scaling exponent (Dfm) and prefactor (kfm). Our simulation results for a statistically significant number of Df = 1.78 ± 0.10 aggregates yielded Dfm values of 2.20 ± 0.05. These values are in excellent agreement with previous experimental observations. While variations in Dfm were predominantly influenced by Df, kfm showed sensitivity to fluctuations in kf. The validity and accuracy of the empirical power-law exponent 1.08 used for estimating N in three dimensions from two dimensional projection images was als...

From static micrographs to particle aggregation dynamics in three dimensions.

Studies on colloidal aggregation have brought forth theories on stability of colloidal gels and models for aggregation dynamics. Still, a complete link between developed frameworks and obtained laboratory observations has to be found. In this work, aggregates of silica nanoparticles (20 nm) are studied using diffusion limited cluster aggregation (DLCA) and reaction limited cluster aggregation (RLCA) models. These processes are driven by the probability of particles to aggregate upon collision. This probability of aggregation is one in the DLCA and close to zero in the RLCA process. We show how to study the probability of aggregation from static micrographs on the example of a silica nanoparticle gel at 9 wt%. The analysis includes common summary functions from spatial statistics, namely the empty space function and Ripley's K-function, as well as two newly developed summary functions for cluster analysis based on graph theory. One of the new cluster analysis functions is related to the clustering coefficient in communication networks and the other to the size of a cluster. All four topological summary statistics are used to quantitatively compare in plots and in a least-square approach experimental data to cluster aggregation simulations with decreasing probabilities of aggregation. We study scanning transmission electron micrographs and utilize the intensity-mass thickness relation present in such images to create comparable micrographs from three-dimensional simulations. Finally, a characterization of colloidal silica aggregates and simulated structures is obtained, which allows for an evaluation of the cluster aggregation process for different aggregation scenarios. As a result, we find that the RLCA process fits the experimental data better than the DLCA process.

Optical property of nanofluids with particle agglomeration

Nanofluids have been used to improve the efficiency of direct absorption solar thermal collectors (DASC) as working fluids because of their excellent heat transfer and optical absorption properties. However, aggregates could be generated in nanofluids especially in the two-step preparation. A theoretical approach for calculating the extinction coefficients of nanofluids with particle agglomeration is proposed based on diffusion limited cluster aggregation (DLCA) simulation and generalized multi-particle Mie solution (GMM) method, in which the dependent scattering effect between particles is considered. The influences of particle agglomeration, particle diameter and particle volume fraction on the extinction coefficient of nanofluids with particle agglomeration are investigated. Moreover, the harvested solar energy under the real solar irradiation at sea level for the nanofluids with particle agglomeration is also studied. The results show that the particle agglomeration reduces the absorption peak in the short-wavelength as well as enhances the long-wavelength scattering. The nanofluids with particle agglomeration have a higher performance in solar energy harvest.

On the radiative properties of soot aggregates – Part 2: Effects of coating

The effects of weakly absorbing material coating on soot have attracted considerable research attention in recent years due to the significant influence of such coating on soot radiative properties and the large differences predicted by different numerical models. Soot aggregates were first numerically generated using the diffusion limited cluster aggregation algorithm to produce fractal aggregates formed by log-normally distributed polydisperse spherical primary particles in point-touch. These aggregates were then processed by adding a certain amount of primary particle overlapping and necking to simulate the soot morphology observed from transmission electron microscopy images. After this process, a layer of WAM coating of different thicknesses was added to these more realistic soot aggregates. The radiative properties of these coated soot aggregates over the spectral range of 266–1064nm were calculated by the discrete dipole approximation (DDA) using the spectrally dependent refractive index of soot for four aggregates containing Np=1, 20, 51 and 96 primary particles. The considered coating thicknesses range from 0% (no coating) up to 100% coating in terms of the primary particle diameter. Coating enhances both the particle absorption and scattering cross sections, with much stronger enhancement to the scattering one, as well as the asymmetry factor and the single scattering albedo. The absorption enhancement is stronger in the UV than in the visible and the near infrared. The simple corrections to the Rayleigh–Debye–Gans fractal aggregates theory for uncoated soot aggregates are found not working for coated soot aggregates. The core–shell model significantly overestimates the absorption enhancement by coating in the visible and the near infrared compared to the DDA results of the coated soot particle. Treating an externally coated soot aggregate as an aggregate formed by individually coated primary particles significantly underestimates the absorption enhancement by coating in the visible and the near infrared.

Divine Proportion Shape Invariance of Diffusion Limited Cluster–Cluster Aggregates

Aggregation is a non-equilibrium process of fundamental importance for all dispersed particulate systems, and the aggregates so produced are common in nature and technology. The resulting aggregates show fractal morphology with a universal, quantifiable fractal dimension. Current quantitative explanations for the fractal aggregate structure are based on simulation models that successfully describe experimental findings and have the key feature that as aggregates form, they continue to move and eventually connect with other aggregates. If the motion between meetings is diffusive, which is very typical, the process is called diffusion limited cluster aggregation (DLCA). We have shown previously that a three-parameter description, including the fractal dimension, the scaling prefactor, and the shape of these aggregates, is both necessary and sufficient to completely describe their morphology. Here we define a new shape description and find geometric relations between the dimensions of the aggregates. Moreover, we show that with this new shape description the aggregate shape is described by the Divine Proportion and its generalization to any dimension. Finally, we recall a simple, analytical theory for DLCA aggregate structure, the Restricted Hierarchical Model, to find that it accurately predicts shape, fractal dimension, and scaling prefactor to yield a necessary and sufficient three-parameter description of aggregate structure.Copyright 2015 American Association for Aerosol Research