Cluster-cluster correlation during irreversible diffusion-limited aggregation

Summary In this paper, we discuss the evolution of the scattered intensityI(q) during irreversible diffusion-limited cluster-cluster aggregation. We analyse twodimensional simulations and interpret the results within the framework of a recently proposed theoretical approach. The theory describes the correlation among different clusters which develops during the irreversible aggregation process. The model is based on two coupled differential equations, controlling the growth of the average cluster mass and the time dependence of the probability of finding pairs of clusters as a function of their distance. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Numerical accuracy of “equivalent” spherical approximations for computing ensemble-averaged scattering properties of fractal soot aggregates

Numerical accuracy is quantitatively assessed in conjunction with the application of four “equivalent” spherical approximations in the computation of the optical properties of small aggregate soot particles. The approximations are based on equal volume, equal surface area, the radius of gyration, and a collection of independent spheres with the same volume and the same volume-to-projected area ratio as the original nonspherical particle. A diffusion-limited cluster-cluster aggregation algorithm is used to specify the geometries of soot particles. Furthermore, the Generalized Multi-particle Mie (GMM) method is utilized to compute the single-scattering properties of individual soot aggregate particles assumed to be randomly oriented in space. The ensemble-averaged single-scattering properties of the particles are obtained by accounting for the probability distribution functions (PDF) of the number of monomers per aggregate at two wavelengths, 0.628 and 1.1 μm. It is shown that all of the aforementioned equivalent-spherical approximations lead to large errors in the computation of the phase function.     

Self-organization in aggregating colloids

Summary The interceluster organization of a two-dimensional colloidal system undergoing diffusion-limited cluster-cluster aggregation is considered. During this process the system evolves into a stationary, scaling state. In this state the organization arises from effective intercluster repulsion due to the mutually exclusive depletion zones surrounding each cluster. It is shown that in the scaling regime the relevant size of the depletion zones is determined by the separation between clusters, rather than the cluster size. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Analysis of structural features of tin dioxide-based fractal nanocomposites by atomic-force microscopy and x-ray diffraction

Tin dioxide-based gas-sensitive layers with a fractal structure were prepared by the sol-gel method. The properties of nanocomposite layers were studied by atomic-force microscopy and x-ray diffraction. The main stages of the fractal system’s evolution, i.e., diffusion-limited aggregation, cluster-cluster aggregation, percolation transition, and network nanostructure formation were studied.     

Colloidal aggregation with sedimentation: concentration effects

The results of computer models for colloidal aggregation, that consider both Brownian motion and gravitational drift experienced by the colloidal particles and clusters, are extended to include concentrations spanning three orders of magnitude. In previous publications and for a high colloidal concentration, it was obtained that the aggregation crosses over from diffusion-limited colloidal aggregation (DLCA) to another regime with a higher cluster fractal dimension and a speeding up followed by a slowing down of the aggregation rate. In the present work we show, as the concentration is decreased, that we can still cross over to a similar regime during the course of the aggregation, as long as the height of the sample is increased accordingly. Among the differences between the mentioned new regimes for a high and a low colloidal concentration, the cluster fractal dimension is higher for the high concentration case and lowers its value as the concentration is decreased, presumably reaching for low enough concentrations a fixed value above the DLCA value. It is also obtained the fractal dimension of the sediments, arising from the settling clusters that reach the bottom and continue a 2D-like diffusive motion and aggregation, on the floor of the container. For these clusters we now see two and sometimes three regimes, depending on concentration and sedimentation strength, with their corresponding fractal dimensions. The first two coming from the crossover already mentioned, that took place in the bulk of the sample before the cluster deposition, while the third arises from the two-dimensional aggregation on the floor of the container. For these bottom clusters we also obtain their dynamical behavior and aggregation rate.Received: 7 January 2004, Published online: 25 March 2004PACS: 61.43.Hv Fractals; macroscopic aggregates (including diffusion-limited aggregates) - 82.70.Dd Colloids - 05.10.Ln Monte Carlo methods     

Complexation in dense gases: concentrations of argon monomers,dimers and trimers

A classical statistical mechanical cluster formalism is presented for computing the thermodynamic properties and equilibrium state of aggregation of a fluid composed of structureless monomeric units which are capable of forming weakly bound van der Waals complexes. The general procedure is not restricted to any particular definition of the clusters, provided that the definition used satisfies certain reasonable criteria. The formalism is illustrated by applying it to two different cluster definitions. Calculations based on these definitions have been performed for the concentrations of argon monomers, dimers and trimers. It is found that gas imperfections due to cluster-cluster interactions can significantly affect the calculated values of these concentrations.     

Off lattice simulations of cluster-cluster aggregation in dimensions 2–6

Cluster-cluster aggregation has been simulated in dimensions two to six using both linear and brownian cluster trajectories. Relatively efficient off lattice algorithms have allowed large clusters to be generated and values for the fractal dimensionalities of the aggregates have been obtained without finite concentration effects. The values for the fractal dimensionality are in good aggreement with lattice model simulations for euclidean dimensionalities 2–4. The effective dimensionality (D_β) obtained from the dependence of the radius of gyration on cluster size increases with increasing cluster size for all of our models (particularly for d ≥ 4). For clusters in the accessible size range (up to 10³-10⁴) D_β is slightly larger for cluster-cluster aggregation via linear trajectories than for brownian trajectories. For cluster-cluster aggregation via brownian trajectories, the limiting (large cluster size) fractal dimensionality is estimated to be 1.46 ± 0.04 for d=2,1.82 ± 0.10 for d = 3, 2.10 ± 0.15 for d = 4, 2.35 ± 0.15 for d = 4, 2.65 ± 0.25 for d = 6. For cluster- cluster aggregation via linear trajectories, the limiting fractal dimensionality is estimated to be 1.55 ± 0.04 for d = 2, 1.91 ± 0.10 for d = 3≥ 2.5 ± 0.06 for d = 5 and ≥2.64 ± 0.05 for d = 6.     

Dense cluster formation during aggregation and gelation of attractive slippery nanoemulsion droplets

Using time-resolved small angle neutron scattering, we have measured the wave-number-dependent structure factor S(q) of monodisperse nanoemulsions that aggregate and gel after we suddenly turn on a strong, short-range, slippery attraction between the droplets. At high q, peaks in S(q) appear as dense clusters of droplets form, and S(q) increases strongly toward low q, as these dense clusters become locked into a rigid gel network, despite the fluidity of the films between the droplets. The long-time high-q structure of nanoemulsion gels formed by slippery diffusion-limited cluster aggregation is universal in shape and remarkably independent of the droplet volume fraction, phi.     

Effects of Monomer Size Distribution on the fractal dimensionality of diffusion-limited aggregates

Computer simulations of diffusion-limited aggregation (DLA) for monomers to investigate the effects of size and of lognormal distribution on the fractal dimensionality of the aggregates were conducted on a two-dimensional lattice. The results show the DLA clusters posses multifractal characteristics. For clusters consisting of monodisperse monomers, the bifurcation point on the graph of the pair correlation function (PCF) for each cluster is located right at the monomers size under investigation The textural dimension (D_f1) has a stable value of about 1.65, whereas the structural dimension (D_f2) decreased with increase in monomer size. For the cases with monomers in log-normal distributions, the textural dimension is around 1.67; however, the structural dimension decreases with increasing polydispersity of monomer size.     

Effect of monomer geometry on the fractal structure of colloidal rod aggregates

The fractal structure of clusters formed by diffusion-limited aggregation of rodlike particles is characterized over three decades of the scattering vector q, and displays an unexpected dependence on the aspect ratio of the constituent monomers. Monte Carlo simulations of aggregating Brownian rods corroborate the experimental finding that the measured fractal dimension is an increasing function of the monomer aspect ratio. Moreover, increasing the rod aspect ratio eliminates the structural distinction between diffusion- and reaction-limited cluster aggregation that is observed for spheres.     

On anomalous diffusion of fractal clusters under certain realistic physical conditions

Summary Diffusion of a fractal cluster of dimensiond _f in a three-dimensional space is investigated. The diffusion process is assumed to be modelled by a standard parabolic diffusion equation, although a more general case represented by the Fokker-Planck-Kolmogoroff equation is also introduced. The mean-square displacement of the cluster mass centre is analysed and its anomalous behaviour is presented and critically discussed. The results obtained can be applied to describe some effects which may occur during the diffusion-limited cluster-cluster aggregation process, especially when the viscosity of the solvent is changed in time and/or a directed transport of clusters is present in the system. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Flocculation and percolation in reversible cluster-cluster aggregation

Off-lattice dynamic Monte-Carlo simulations were done of reversible cluster-cluster aggregation for spheres that form rigid bonds at contact. The equilibrium properties were found to be determined by the life time of encounters between two particles (t_e). t_e is a function not only of the probability to form or break a bond, but also of the elementary step size of the Brownian motion of the particles. In the flocculation regime the fractal dimension of the clusters is d_f=2.0 and the size distribution has a power law decay with exponent τ=1.5. At larger values of t_e transient gels are formed. Close to the percolation threshold the clusters have a fractal dimension d_f=2.7 and the power law exponent of the size distribution is τ=2.1. The transition between flocculation and percolation occurs at a characteristic weight average aggregation number that decreases with increasing volume fraction.     

X-ray diffraction characterization of a spatially coherent crystalline nanocomposite obtained from a melt of KCl, KBr and KI salts

Crystalline bulks, obtained by the Bridgman technique from a melt that was previously prepared by mixing equal molar fractions of the salts KBr, KCl and KI were characterized by powder and single-plate X-ray diffraction. The crystalline bulks are formed by a composite material consisting of a highly textured aggregation of crystallites of two different face-centred-cubic solid solutions with corresponding unit-cell sizes of 7.020(2) and 6.456(3) Å. The relative molar fraction concentrations of these solid solutions in the composite are about 0.348 and 0.652, respectively. From these solutions, the first is identified as the binary KBr(9.7%):KI(90.3%) mixed phase and the second is discussed to be the ternary KBr (45.9%):KCl (51.1%):KI (3.0%) mixed phase. The lattices of most of the crystallites forming the composite, no matter the phase they belong to, are spatially coherent to each other within the crystalline bulk.     

From colloidal aggregation to spinodal decomposition in sticky emulsions

Aggregation mechanisms of emulsions at high initial volume fractions () is studied using light scattering. We use emulsion droplets which can be made unstable towards aggregation by a temperature quench. For deep quenches and , the aggregation mechanism is identified as diffusion-limited cluster aggregation (DLCA). An ordering of the clusters, which is reflected by a peak in the scattering intensity, is shown to result from the intercluster separation, exhibiting different scaling than that observed at lower volume fractions. This manifests an increasing similarity to spinodal decomposition observed as is increased. For and shallow quenches, different mechanisms, closer to spinodal decomposition, are observed. These results allow the subtle boundaries between DLCA and spinodal decomposition to be explored. Received: 7 April 1998 / Revised: 19 August 1998 / Accepted: 21 August 1998     

Finite size effect of harmonic measure estimation in a DLA model: Variable size of probe particles

A finite size effect in the probing of the harmonic measure in simulation of diffusion-limited aggregation (DLA) growth is investigated. We introduce a variable size of probe particles, to estimate harmonic measure and extract the fractal dimension of DLA clusters taking two limits, of vanishingly small probe particle size and of infinitely large size of a DLA cluster. We generate 1000 DLA clusters consisting of 50 million particles each, using an off-lattice killing-free algorithm developed in the early work. The introduced method leads to unprecedented accuracy in the estimation of the fractal dimension. We discuss the variation of the probability distribution function with the size of probing particles.     

Unusual cluster shapes and directional bonding of an fcc metal: Pt/Pt(111)

Small clusters of Pt adatoms grown on Pt(111) exhibit a preference for the formation of linear chains, which cannot be explained by simple diffusion-limited aggregation. Density functional theory calculations show that short chains are energetically favorable to more compact configurations due to strong directional bonding by d(z)(2)-like orbitals, explaining the stability of the chains. The formation of the chains is governed by substrate distortions, leading to funneling towards the chain ends.     

Fractal aggregates of particles

Recent work to understand aggregation of particles using the concept of fractal is reviewed. We introduce all the tools issued from the theory of fractals, and which are useful for our study. Then we define and detail the two main models: the particle-cluster process where individual particles come to stick on large clusters, and the cluster-cluster process where clusters diffuse and stick when they collide. The important parameters of these models are varied and their influence on the resulting geometrical disordered structure is discussed. All along the paper, the connections with existing experiments are given.     

Angular gaps in radial diffusion-limited aggregation: two fractal dimensions and nontransient deviations from linear self-similarity

When suitably rescaled, the distribution of the angular gaps between branches of off-lattice radial diffusion-limited aggregation is shown to approach a size-independent limit. The power-law expected from an asymptotic fractal dimension D = 1.71 arises only for very small angular gaps, which occur only for clusters significantly larger than M = 10(6) particles. Intermediate size gaps exhibit an effective dimension around 1.67, even for M--> infinity. They dominate the distribution for clusters with M<10(6). The largest gap approaches a finite limit extremely slowly, with a correction of order M(-0.17).     

Analysis of gas-phase condensation of nickel nanoparticles

Gas-phase condensation of 8000 nickel atoms is examined by molecular dynamics simulation with a tight-binding potential. A detailed study of the evolution of the system cooled at a constant rate from 1000 K to 77 K is presented. The results are used to identify four distinct stages of the evolution from a hot atomic gas to a few synthesized particles. An analysis of possible nanoparticle formation mechanisms suggests that cluster-cluster aggregation is the dominant one. The simulation shows that there two stages of cluster formation are of primary importance with regard to aggregation. At the first stage, spherical liquid clusters nucleate with uniform size distribution. The second stage is characterized by a distinct transition from uniform to bimodal size distribution due to aggregation of relatively large clusters. The particles obtained by gas-phase synthesis are analyzed by the CNA method [25]. It is found that most nanoparticles produced in the simulation have either icosahedral or mixed FCC/HCP structure.     

Silver fractal networks for surface-enhanced Raman scattering substrates

Based on diffusion-limited aggregation process, a convenient nanotechnique is demonstrated to obtain large silver fractal networks for a surface-enhanced Raman scattering (SERS)-active substrate. The silver fractal networks are of high SERS enhancement factor, large dynamical range. The observed SERS efficiency can be explained in terms of strongly localized plasmon modes relative to the single particle plasmon resonance.