Diffusion-limited Aggregation Research Articles

Optimal number of linkers per monomer in linker-mediated aggregation.

We study the dynamics of diffusion-limited irreversible aggregation of monomers, where bonds are mediated by linkers. We combine kinetic Monte Carlo simulations of a lattice model with a mean-field theory to study the dynamics when the diffusion of aggregates is negligible and only monomers diffuse. We find two values of the number of linkers per monomer which maximize the size of the largest aggregate. We explain the existence of the two maxima based on the distribution of linkers per monomer. This observation is well described by a simple mean-field model. We also show that a relevant parameter is the ratio of the diffusion coefficients of monomers and linkers. In particular, when this ratio is close to ten, the two maxima merge at a single maximum.

New insights on concentration inequalities for self-normalized martingales

We propose new concentration inequalities for self-normalized martingales. The main idea is to introduce a suitable weighted sum of the predictable quadratic variation and the total quadratic variation of the martingale. It offers much more flexibility and allows us to improve previous concentration inequalities. Statistical applications on autoregressive process, internal diffusion-limited aggregation process, and online statistical learning are also provided.

Diffusion-Limited Cluster Aggregation: Impact of Rotational Diffusion

Diffusion-limited cluster aggregation (DLCA) is a well established model for the formation of highly porous low-density non-equilibrium structures. One of the main conclusions of the previous studies considering this model is that the rotational diffusion of aggregating clusters does not change their structure characterized by a universal fractal dimension of $d_f=1.7-1.8$. In contradiction to this assumption, we demonstrate that the rotation movement of clusters significantly changes the structure of forming aggregates. The fractal dimension of rotating clusters is lower than the one found in the standard DLCA model and decreases with the increasing ratio of rotational and translational diffusion constants $D_r/D_t$, which offers a possibility to tune the structures of the aggregates below the conventional DLCA fractal dimension limit.

Diffusion limited aggregation on the Boolean lattice

In the Diffusion Limited Aggregation (DLA) process on $\mathbb{Z}^{2}$, or more generally $\mathbb{Z}^{d}$, particles aggregate to an initially occupied origin by arrivals on a random walk. The scaling limit of the result, empirically, is a fractal with dimension strictly less than $d$. Very little has been shown rigorously about the process, however. We study an analogous process on the Boolean lattice $\{0,1\}^{n}$, in which particles take random decreasing walks from $(1,\dots,1)$, and stick at the last vertex before they encounter an occupied site for the first time; the vertex $(0,\dots,0)$ is initially occupied. In this model, we can rigorously prove that lower levels of the lattice become full, and that the process ends by producing an isolated path of unbounded length reaching $(1,\dots,1)$.

Fractal simulation of thin film nucleation growth process using a diffusion-limited aggregation model

In order to study the initial nucleation and growth process of the films, a two-dimensional (2D) DLA model was established, and fractal dimension was calculated by sandbox method. Compared with the experimental results, the model can well characterize the morphology of nucleation and growth in the initial stage of film growth. In addition to the number of particles and deposition probability involved in other studies, the size of substrate and the location of central particles are also considered in this work. The growth morphology, fractal dimension and the number of simulated steps are presented in this paper.

Numerical modeling of the gas-contributed thermal conductivity of aerogels

Due to the complex microstructure in aerogels and the intricate heat transfer mechanism of solid-gas coupling heat conduction, modeling of the gas-contributed thermal conductivity of this type of material is quite difficult. The present work introduces a novel numerical methodology for computing the gas-contributed thermal conductivity of aerogels by analyzing their microstructural characteristics and heat transfer mechanism of the thermal coupling between the gas phase and the solid backbone of the system. Specifically, structures of aerogels are reconstructed by an improved three-dimensional diffusion-limited cluster-cluster aggregation (DLCA) method, and the contribution of the solid-gas coupling heat transfer to the gas-contributed thermal conductivity of aerogels is quantified. The present numerical model is fully validated by the available experimental data for different aerogels with porosity ranging from 78% to 97.7%. The proposed numerical method is flexible and versatile because it is capable to account for both the geometrical and topological details of the aerogel structure.

Computer-generated mesoporous materials and associated structural characterization

This study aims to computationally generate and fully characterize realistic three-dimensional mesoporous materials. Notably, a new algorithm reproducing gas adsorption porosimetry was developed to calculate the specific surface area and pore size distribution of computer-generated structures. The diffusion-limited cluster-cluster aggregation (DLCCA) method was used to generate point-contact or surface-contact mesoporous structures made of monodisperse or polydisperse spherical particles. The generated structures were characterized in terms of particle overlapping distance, porosity, specific surface area, interfacial area concentration, pore size distribution, and average pore diameter. The different structures generated featured particle radius ranging from 2.5 to 40 nm, porosity between 35 and 95%, specific surface area varying from 35 to 550 m2/g, and average pore diameter between 3.5 and 125 nm. The specific surface area and pore size distribution of computer-generated mesoporous materials were in good agreement with experimental data reported for silica aerogels. Finally, widening the particle size distribution and increasing the particle overlapping were shown to strongly decrease the specific surface area and increase the average pore size of the mesoporous structures. The developed computational tools and methods can accelerate the discovery and optimization of mesoporous materials for a wide range of applications.

A fractal dimension minimum in electrodeposited copper dendritic patterns

Dendritic growth processes occur in a plethora of natural systems and play a relevant role in materials science and engineering. Models based on diffusion-limited aggregation (DLA) have been extensively used to study the mechanisms underlying this class of particle aggregation phenomena. One of the main features of DLA clusters is their fractal character, evinced by self-similarity at different scales. The physical and chemical properties of systems in which DLA aggregates emerge are determined by their fractality. In this work, two-dimensional copper dendritic patterns are grown by electrodeposition in a circular cell at applied voltages in the 5 – 10 V range using a 1 M copper sulfate aqueous solution. The box counting method confirms that the electrodeposited dendritic structures manifestly exhibit fractal character. Although the growth rate increases exponentially with the applied voltage, it is reported for the first time that the fractal dimension has a minimum value for patterns grown at an applied voltage between 7 and 8 V. The results reflect the active role of applied voltage in the physical mechanisms driving dendritic growth in the system, mainly controlled by the copper ion concentration gradient. However, it also suggests the existence of two growth regimes: one corresponding to patterns grown under voltages lower than circa 7–8 V; the other to patterns grown at higher voltages. This evidence is supported by the determined mass values of the copper electrodeposits.

Hybrid L-systems–Diffusion Limited Aggregation schemes

Diffusion Limited Aggregation (DLA) and Lindenmayer Systems (L-Systems) are two important fractal structures generators, whose characteristics have been often discussed in the literature. On their own, both DLA and L-Systems have been applied to model very different phenomena related to non-equilibrium growth models and self-similarity patterns generation. This paper introduces hybrid L-Systems–Diffusion Limited Aggregation schemes, as a novel combinative approach to model novel growth structures. The most intuitive possibilities for hybridizing these systems are discussed: using L-Systems to guide DLA simulations, and embedding DLA schemes into L-Systems variables. Different computational simulations illustrate the performance of some of these hybrid L-Systems–DLA schemes. Possibilities for extending the application of this hybrid proposal to alternative research fields are also discussed in the paper.

Self-assembly of ellipsoidal particles at fluid-fluid interfaces with an empirical pair potential

Colloidal particles adsorbed at fluid-fluid interfaces interact via mechanisms that can be specific to the presence of interfaces, for instance, lateral capillary interactions induced by nonspherical particles. Capillary interactions are highly relevant for self-assembly and the formation of surface microstructures, however, these are very challenging to model due to the multibody nature of capillary interactions. This work pursues a direct comparison between our computational modelling approach and experimental results on surface microstructures formed by ellipsoidal particles. We begin by investigating the accuracy of using pairwise interactions to describe the multibody capillary interaction by contrasting exact two- and three-particle interaction energies and we find that the pairwise approximation appears reasonable for the experimentally relevant configurations studied. We then develop an empirical pair potential and use it in Monte-Carlo type simulations to efficiently model the structure formation process for relevant particle properties such as aspect ratio, contact angle and surface coverage, and succeed in reproducing our experimental observations where we spread sterically-stabilised ellipsoidal particles onto an oil-air interface at high surface coverage. At lower surface coverages, we find that the self-assembly process falls into the diffusion-limited colloid aggregation universality class.

Simulated and experimental analysis of electrodeposited β' Brass composition with DLA morphology

Electrodeposition of two ions like Copper and Zinc in the alloy of β' Brass is done in thin gap 2D geometry. The growth follows a Diffusion Limited Aggregation (DLA) like morphology. It is observed that the ratio of the two ions deposited remains almost constant with the distances from cathode. This ratio also depends on the composition of the cathodes. This has been shown experimentally and using a unique two-ion simulation model developed by us. The compositional analysis by EDX of the electrodeposited layers at different separations of 0.5 cm, 1.0 cm and 1.5 cm from the negative electrode shows a uniform ratio of 2.3087 for the two ions of copper and zinc. A two-ion simulation model made by us with different masses for the ions and for the same composition of cathode shows a uniform ratio of 2.3133 for the two ions systems and therefore, support the experimental observation to a high degree of accuracy. The results obtained have been explained theoretically.

A completely green approach to the synthesis of dendritic silver nanostructures starting from white grape pomace as a potential nanofactory

A simple, eco-friendly, cost-effective and rapid microwave-assisted method has been developed to synthetize dendritic silver nanostructures, composed of silver nanoparticles (AgNPs), using white grape pomace aqueous extract (WGPE) as both reducing and capping agent. With this aim, WGPE and AgNO3 (1 mM) were mixed at different ratio, and microwave irradiated at 700 W, for 40 s. To understand the role of bioactive compounds involved in the green synthesis of AgNPs, preliminary chemical characterization, FT-IR analysis and 1H NMR metabolite profiling of WGPE were carried out. The effects of bioactive extract concentration and stability over time on AgNPs formation were also evaluated. WGPE-mediated silver nanostructures were then characterized by UV–vis, FTIR analyses, and scanning electron microscopy. Interestingly, the formation of dendritic nanostructures, originated from the self-assembly of Ag rounded nanoparticles (average diameter of 33 ± 6 nm), was observed and ascribed to the use of microwave power and the presence of organic components within the used WGPE, inducing an anisotropic crystal growth and promoting a diffusion-limited aggregation mechanism. The bio-dendritic synthetized nanostructures were also evaluated for potential applications in bio-sensing and agricultural fields. Cyclic voltammetry measurements in 0.5 M phosphate + 0.1 M KCl buffer, pH 7.4 showed that green AgNPs possess the electroactive properties typical of AgNPs produced using chemical protocol. The biological activity of synthetized AgNPs was evaluated by in-vitro antifungal activity against F. graminearum. Additionally, a phytotoxicity evaluation of synthetized green nanostructures was carried out on wheat seed germination. Results highlighted the potential of WGPE as green agent for bio-inspired nanomaterial synthesis, and of green Ag nanostructures, which can be used as antifungal agent and in biosensing applications.

Topological Network Properties of Fractal-Like Metallic Nanoparticle Patterns and Their Effects on Optical Resonances

Fractal-like nanoparticle two-dimensional patterns forming in diffusion-limited aggregation show variant spatial patterns. However, they have invariant statistical properties in their network topologies, even though their formation is completely in self-assembled processes. One of the outputs from these topological properties is optical resonances at invariant frequencies, which is a required feature of a metamaterial alternative. Fractal-like metallic patterns studied here in both experiments and theoretical models exhibit similar resonance frequencies in the infrared-ray range, and they depend on the unit length of nanoparticles composing arbitrary fractal-like structures. The scheme of analysis applied here using complex network theory does not only reveal the topological properties of the nanoparticle network, but points out their optical and possibly other physical potentials arising from their geometrical properties.

Application of the Stochastic Lattice Model for Describing Aggregation Processes in the Disperse Systems Flowing through a Tubular Reactor

The paper deals with results of the numerical experiment conducted to study the kinetics of the aggregation process occurring in a dispersed system flowing through a tubular reactor. The model is based on the discrete-event simulation paradigm (DES). Previously, a lattice model was used to simulate the process of aggregation of a dispersed system in a periodic reactor. The main result of the given work is the development of a lattice aggregation model applicable to a flow reactor. For creating the model the mathematical apparatus of a random walk on mathematical lattices has been used. As shown in the work, this approach allows removing the problem linked to accounting the effect of many-particles collisions, and as a result it gives the way for describing both a diffusion-limited aggregation and the competition of different kinetic mechanisms taking place on different characteristic times. In this paper, the algorithm that allows taking into account the impact of fresh flow with a dispersed phase entering the flow reactor in the working volume of which aggregation takes place on the kinetic parameters of the process has been submitted.

Versatility of electrochemically grown dendrites in the undergraduate laboratory

We describe an experiment to fabricate atomic-scale contacts using electrochemically grown silver wires. The formation of a single-wire junction is directly observed and captured by an optical microscope, while electrical conductance of the wire, simultaneously recorded, is shown to be quantized. Further, a diffusion-limited aggregate (DLA) simulation is performed to compare the observed fractal formed by the silver dendrites. Our experiment directly exposes undergraduate students to exciting contemporary physics ranging from atomic-scale switches to fractal formation, all on a single experimental platform.

Silica-Resorcinol-Formaldehyde Aerogels Nanostructure Modelling

The work is devoted to the study and modelling of the internal nanostructure of silica-resorcinol-formaldehyde aerogels. These aerogels are highly porous (up to 98 %) functional materials with a high specific surface area, low thermal conductivity, and good sound insulation. This material is of interest for use in thermal insulation, gas sorption, and also as a soundproof material. In addition, silica-resorcinol-formaldehyde aerogels are a material for further pyrolysis to obtain silica-carbon aerogels. Several aerogel samples experimental data were analysed. The experimental data includes amount of solvent, silica to resorcinol-formaldehyde ratio, pore size distribution, density, and specific surface area. The size and volume of different pores and specific surface area vary depending on the gelation conditions of the samples. This fact indicates the gelation conditions have a significant effect on the final aerogel structure. However, the input parameters of the model are the pore size distribution and sample density. Analysis and comparison of two methods for the generation of porous structures of silica-resorcinol-formaldehyde aerogels including Diffusion-Limited Cluster Aggregation (DLCA) and Reaction-Limited Cluster Aggregation (RLCA) are carried out and the advantages of these two methods are shown in this paper.

Mechanisms of precipitate formation during the purification of an Fc-fusion protein.

Protein precipitates that arise during bioprocessing can cause manufacturing challenges, but they can also aid in clearance of host-cell protein (HCP) and DNA impurities. Such precipitates differ from many protein precipitates that have been studied previously in their heterogeneous composition, particularly in the presence of high concentrations of the product protein. Here, we characterize the precipitates that form after neutralization of protein A purified and viral-inactivated material of an Fc-fusion protein produced in Chinese hamster ovary cells. The physical growth of precipitate particles was observed by optical microscopy, transmission electron microscopy, dynamic light scattering, and small-angle and ultra-small-angle X-ray scattering to characterize the precipitate microstructure and growth mechanism. The precipitate microstructure is well-described as a mass fractal with fractal dimension approximately 2. The growth is governed by a diffusion-limited aggregation mechanism as indicated by a power-law dependence on time of the size of the principal precipitate particles. Optical microscopy shows that these primary particles can further aggregate into larger particles in a manner that appears to be promoted by mixing. Absorbance experiments at varying pH and salt concentrations reveal that the growth is largely driven by attractive electrostatic interactions, as growth is hindered by an increase in ionic strength. The solution conditions that resulted in the most significant particle growth are also correlated with the greatest removal of soluble impurities (DNA and HCPs). Proteomic analysis of the precipitates allows identification of unique HCP impurities, depending on the buffer species (acetate or citrate) used for the viral inactivation. Most of these proteins have pI values near the precipitation pH, supporting the likely importance of electrostatic interactions in driving precipitate formation.

(Invited) Drying Patterns from Nanofluid Droplet Evaporation

Droplet evaporation has received more attention for the applications including concentrating chemicals, coating, printing, spraying, drying, painting, biomedical sampling, etc. Deposition pattern form the evaporative colloidal droplet has many applications in the industrial applications. The process to form the configuration in the drying droplet has been investigated with simulation and experiments. Experimentally, the suspensions of colloidal droplet with the particle size at nano- and micro-scale nanoparticles in the deionized water were prepared. The suspensions are carried out by well stirring first and then putting in an ultrasonication bath to ensure the even initial dispersion of the particles in the water. The clean silicon wafer was used as a substrate for the colloidal sessile droplet drying. Right after the suspensions were prepared, we placed a tiny sessile droplet to dry in the open conditions. The drying patterns were directly recorded with the optical microscope. The formation of branched particle aggregation structures was observed during the droplet evaporation. The experiments demonstrated the dried patterns such as finger, coffee ring, uniform coverage and the combined structures. We have applied the kinetic Monte Carlo model and the diffusion-limited aggregation simulation to explain the self-assemble processes. A lattice-gas Kinetic Monte Carlo (KMC) is developed to explain the of the finger structure while the diffusion-limited aggregation simulation is development to explain the transition from the coffee ring to uniform coverage. The mathematical models control the variables which indicate the presence of liquid, particle or substrate in cells of the cubical lattice. The simulation domain mimics the shape of a sessile nanofluid droplet. The particle, liquid, vapor interaction, and the other factors are considered. The experiments can be reflected by the simulations.

Tuning the morphology of electrosprayed BiVO4 from nanopillars to nanoferns via pH control for solar water splitting

Electrosprayed BiVO4 adopts a nanopillar structure formed by diffusion-limited aggregation, which maximizes the surface area of the nanopillars. However, increasing the interfacial area between an electrode and the electrolyte through nanostructuring enhances the overall interfacial activity for solar water splitting. For this purpose, the pH of the precursor solution used for electrospraying BiVO4 was altered by adding ammonium hydroxide, thereby inducing a drastic change in the morphology of BiVO4. The previously demonstrated nanopillar morphology of electrosprayed BiVO4 was transformed into a nanofern structure that increased the photocurrent density of BiVO4 from 0.82 to 1.23 mA·cm−2 at 1.2 V vs. Ag/AgCl. The produced films were characterized by scanning electron microscopy, X-ray photoelectron spectroscopy, Raman spectroscopy, and electrochemical impedance spectroscopy.

Orthogonal fractal growth of CsI domains forming a ladder-like structure

A unique structure of a thin layer consisting of cesium iodine, manifested by a “ladder-like” fractal structure formed by spin-coating is reported herein. The ladder-like structure is made of mm-size domains, each comprising of a highly correlated, perpendicularly interconnected, network of CsI lines. Each line served as the growth origin of 2–3 levels of short, perpendicularly-oriented CsI crystals, yielding a fractal dimension of 1.53.The observed structure differs from common Diffusion Limited Aggregation (DLA) shapes by the absence of any morphological indicators that may point on the origin of growth. Furthermore, the perfect orthogonal alignment of all junctions in the CsI structure is very rare in DLA type of growth.A formation mechanism is presented, based on studying the evolution of this structure at different spinning rates and on a variety of substrates. It is proposed that this unique structure originates from a rare combination of conditions: strong anisotropy in surface energy between different facets arising from the primitive ionic crystal of CsI, the strong water-breaking property of cesium ions and an unusual effect of mesoporous substrates in preventing premature nucleation.