The DLCA-RLCA transition arising in 2D-aggregation: simulations and mean field theory

A sticking probability model based on the average cluster lifetime is employed for deducing a kernel capable to describe the kinetics of computer simulated irreversible aggregation processes in two dimensions. The deduced kernel describes not only the time evolution of the cluster size distribution for diffusion limited aggregation (DLCA) and reaction limited aggregation (RLCA) but also for the entire transition region between both regimes. The model predicts a crossover to diffusion limited cluster aggregation for all sticking probabilities at long aggregation times. The time needed for reaching the DLCA limit increases for decreasing sticking probability. Received 16 April 2001 and Received in final form 24 May 2001     

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     

Diffusion limited cluster aggregation with irreversible slippery bonds

Irreversible diffusion limited cluster aggregation (DLCA) of hard spheres was simulated using Brownian cluster dynamics. Bound spheres were allowed to move freely within a specified range, but no bond breaking was allowed. The structure and size distribution of the clusters was investigated before gelation. The pair correlation function and the static structure factor of the gels were determined as a function of the volume fraction and time. Slippery bonds led to local densification of the clusters and the gels, with a certain degree of order. At low volume fractions densification of the clusters occurred during their growth, but at higher volume fractions it occurred mainly after gelation. At very low volume fractions, the large-scale structure (fractal dimension), size distribution and growth kinetics of the clusters was found to be close to that known for DLCA with rigid bonds. Restructuring of the gels continued for long times, indicating that aging processes in systems with strong attraction do not necessarily involve bond breaking. The mean-square displacement of particles in the gels was determined. It is shown to be highly heterogeneous and to increase with decreasing volume fraction.     

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.     

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     

Size and stability of liposomes: A possible role of hydration and osmotic forces

Dynamic light scattering and electrophoretic mobility measurements have been used to characterize the size, size distribution and zeta potentials (ζ-potentials) of egg yolk phosphatidylcholine (EYPC) liposomes in the presence of monovalent ions ( Na⁺ and K⁺). To study the stability of liposomes the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory has been extended by introducing the hydrated radius of the adsorbed ions onto the liposome surfaces. The decrease of liposome size is explained on the basis of the membrane impermeability to some ions which generate osmotic forces, which leads to evacuate water from liposome inside.     

Kinetic Simulation of Fractal Aggregation on Liquid Surfaces

A modified fractal growth model based on the deposition, diffusion, and aggregation (DDA) with cluster rotation is presented to simulate two-dimensional fractal aggregation on liquid surfaces. The mobility (including diffusion and rotation) of clusters is related to its mass, which is given by D_m=D₀ s^-γ_D and θ_m=θ₀s^-γ_θ, respectively. We concentrate on revealing the details of the influence of deposition flux F, cluster diffusion factor γ_D and cluster rotation factor γ_θ on the dynamics of fractal aggregation on liquid surfaces. It is shown that the morphologies of clusters and values of cluster density and fractal dimension depend dramatically on the deposition flux and migration factors of clusters.     

Colloidal aggregation with sedimentation: computer simulations

A computer model for colloidal aggregation is presented that considers both the Brownian motion and the gravitational drift experienced by the colloidal particles and clusters. It is shown that the aggregation crosses over from diffusion-limited aggregation to another type with a higher cluster fractal dimension, a speeding up followed by a slowing down of the aggregation rate, an algebraically decaying cluster size distribution, and a higher concentration required for gelation. Although these findings are in accordance with the experimental results, some interpretations are different.     

Collapse of sodium polyacrylate chains in calcium salt solutions

The sodium salt of polyacrylic acid (NaPA) precipitates in the presence of Ca²⁺-ions. This phase behaviour can be represented by a phase diagram where the critical NaPA concentration is plotted versus the critical Ca²⁺ concentration resulting in a straight line as a phase boundary. The location of this phase boundary is influenced by the presence of an inert monovalent salt like NaCl. The present contribution focuses on the coil dimensions of NaPA chains in dilute aqueous solution corresponding to the one phase region of such a phase diagram. A variety of parameters with which the size and shape of the polyelectrolyte chains can be modulated are revealed. Approaching the phase boundary by decreasing the NaPA concentration at a constant Ca²⁺ content leads to a collapse of the NaPA chains. Combined static and dynamic light scattering suggests a compact spherical shape as the final state of this transition, both in 0.1 M NaCl and in 0.01 M NaCl. In the lower NaCl concentration, indication is presented for the existence of a cigar or pearl necklace like intermediate. Most strikingly, the collapsed chains can be reexpanded by increasing the concentration of inert NaCl at constant content of NaPA and Ca²⁺. Clearly, excessive Na⁺-ions displace the Ca²⁺-ions from the NaPA chains. Received 18 July 2000 and Received in final form 24 August 2000     

Measurement of Aggregate Fractal Dimensions Using Static Light Scattering

Aggregates formed from colloidal particles will vary in shape according to the aggregation regime prevalent. Compact structures are formed when the aggregation is slow, whilst loose tenuous structures are formed when rapid (or diffusion limited) aggregation prevails. These structures can be fractal in nature, that is, there is a relationship between porosity and the number of primary particles making up the aggregate, and is described by the fractal dimension, d_F. Fractal dimensions of hematite aggregates have been measured experimentally using the static light scattering technique. Fractal dimensions varied with aggregation regimes; for the rapid aggregation regime, d_F was found to be 2.8, whilst for conditions in which aggregation was slow (retardation forces prevail), d_F's of 2.3 were measured. For conditions which lead to aggregation in which both diffusion and retardation forces play a part, structures with fractal dimensions such that 2.3 < d_F < 2.8 were found. The effects of adsorbed fulvic acid, a naturally occuring organic acid, on the kinetics of hematite aggregation and on the resulting structure of hematite aggregates were also investigated. The study of aggregate structure shows that the fractal dimensions of hematite aggregates which are partially coated with fulvic acid molecules are higher than those obtained with no adsorbed fulvic acid. The scattering exponents obtained from static light scattering experiments of these aggregates range from 2.83 ± 0.08 to 3.42 ± 0.1. The scattering exponents of greater than 3 indicate that the scattering is the result of objects that contains pores which are bounded by surfaces with a fractal structure, and can be related only to surface fractal dimension. The high fractal dimensions are due to restructuring within the aggregates, which only occured at low coverage by the organic acid.     

Cluster-cluster aggregation in binary mixtures

The structure and aggregation kinetics of three-dimensional clusters composed of two different monomeric species at three concentrations are thoroughly investigated by means of extensive, large-scale computer simulations. The aggregating monomers have all the same size and occupy the cells of a cubic lattice. Two bonding schemes are considered: (a) the binary diffusion-limited cluster-cluster aggregation (BDLCA) in which only the monomers of different species stick together, and (b) the invading binary diffusion-limited cluster-cluster aggregation (IBDLCA) in which additionally monomers of one of the two species are allowed to bond. In the two schemes, the mixed aggregates display self-similarity with a fractal dimension d(f) that depends on the relative molar fraction of the two species and on concentration. At a given concentration, when this molar fraction is small, d(f) approaches a value close to the reaction-limited cluster-cluster aggregation of one-component systems, and when the molar fraction is 0.5, d(f) becomes close to the value of the diffusion-limited cluster-cluster aggregation model. The crossover between these two regimes is due to a time-decreasing reaction probability between colliding particles, particularly at small molar fractions. Several dynamical quantities are studied as a function of time. The number of clusters and the weight-average cluster size display a power-law behavior only at small concentrations. The dynamical exponents are obtained for molar fractions above 0.3 but not at or below 0.2, indicating the presence of a critical transition between a gelling to a nongelling system. The cluster-size distribution function presents scaling for molar fractions larger than 0.2.     

Radar polarimetry of Saturn's rings: Modeling ring particles as fractal aggregates built of small ice monomers

We analyze ground-based radar polarimetric observations of Saturn's rings at a wavelength of 12.6 cm by employing the model of a vertically and horizontally plane-parallel homogeneous slab composed of clumpy particles in the form of fractal aggregates of small ice monomers. Our model takes full account of the effects of polarization, multiple scattering, and coherent backscattering. Using efficient superposition T-matrix and vector radiative transfer codes, we perform computations of the backscattering circular polarization ratio for fractal aggregates generated with a cluster–cluster aggregation model and having the following characteristics: monomer refractive index m=1.78+i0.003; monomer packing density p=0.2; fractal dimensions D_f=2.5 and 3; and overall fractal radii R in the range 4?R?10 cm. In order to obtain physically realistic values of single-scattering properties of the aggregates we perform averaging over an ensemble of clusters generated for the same values of fractal parameters but having different geometrical configurations of the monomers. We conclude that in the framework of the above morphological model of Saturn's rings and the specific cluster–cluster aggregation procedure, it may be problematic to obtain a satisfactory and realistic agreement between theoretical computations and the observed values of the radar circular polarization ratio.     

Simulation of fractal cluster growth on a substrate under ion bombardment

Ion-stimulated growth of a fractal cluster on the substrate is simulated in regimes that are intermediate between diffusion- and reaction-limited aggregations. It is shown that the fractal dimension of a growing cluster and the type of formed epitaxial structure are due to the sequence of ion-stimulated processes with different fractal dimensions. The algorithm for revealing the ion stimulation mechanism of processes based on fractal laws is proposed.     

Fractal Aggregation Under Rotation

By means of the Monte Carlo simulation, a fractal growth model is introduced to describe diffusion-limited aggregation (DLA) under rotation. Patterns which are different from the classical DLA model are observed and the fractal dimension of such clusters is calculated. It is found that the pattern of the clusters and their fractal dimension depend strongly on the rotation velocity of the diffusing particle. Our results indicate the transition from fractal to non-fractal behavior of growing cluster with increasing rotation velocity, i.e. for small enough angular velocity ω; thefractal dimension decreases with increasing ω;, but then, with increasing rotation velocity, the fractal dimension increases and the cluster becomes compact and tends to non-fractal.     

Diffusion in supported lipid bilayers: Influence of substrate and preparation technique on the internal dynamics

The diffusion law of DMPC and DPPC in Supported Lipid Bilayers (SLB), on different substrates, has been investigated in details by Fluorescence Recovery After Patterned Photobleaching (FRAPP). Over micrometer length scales, we demonstrate the validity of a purely Brownian diffusive law both in the gel and the fluid phases of the lipids. Measuring the diffusion coefficient as a function of temperature, we characterize the gel-to-liquid phase transition of DMPC and DPPC. It is shown that, depending on the type of substrate and the method used for bilayer preparation, completely different behaviours can be observed. On glass substrates, using the Langmuir-Blodgett deposition technique, both leaflets of the bilayer have the same dynamics. On mica, the dynamics of the proximal leaflet is slower than the dynamics of the distal leaflet, although the transition temperature is the same for both layers. Preparing bilayers from vesicle fusion in same conditions leads to more random behaviours and shifted transition temperatures.     

Particle aggregation versus cluster aggregation in high dimensions

We distinguish two different types of irreversible aggregation-accretion of individual particles and successive aggregation of clusters of comparable size. In aggregation of particles which follow trajectories of fractal dimensionD ₁, we show that physical limits on the aggregation rate impose a lower bound on the fractal dimensionD ₀ of the aggregate. Ind-dimensional space,D ₀{d–D}₁ + 1. Thus aggregation of ballistic particles, withD ₁ = 1, is not fractal. By contrast, cluster aggregates appear to attain a finite, limitingD ₀ in high dimensions. We present a soluble model with this property, and argue that it should agree with Sutherland's binary aggregation model in high dimensions. For this model,D ₀ depends continuously on a parameter; the exponent is not universal.     

Enhancement Factor in the Light Backscattered by Fractal Aggregated Media

We investigate the phenomenon of the enhanced backscattering of light from soft sediments of fractal clusters. The clusters consist of spherical PMMA particles with the diameter of 0.4 μ, aggregated in aqueous solutions of NaCl. We found that the kinetics of aggregation, which determines the average cluster size in sediments, is controlled by the salt concentration and that the sediments are mutually self-similar media. In comparison to uniform random media, specific features for the enhancement peaks are revealed. It is found that the peak line-shape reflects the particularities of the density of scatterers in a fractal-like medium. It is shown experimentally that the enhancement factor in the light backscattered by fractal aggregated media is sensitive to the average cluster size. On this basis, we suggest a possible way to distinguish between mutually self-similar media.     

Aggregation kinetics of CeO<Subscript>2</Subscript> nanoparticles in KCl and CaCl<Subscript>2</Subscript> solutions: measurements and modeling

To characterize the environmental transport and health risks of CeO₂ nanoparticles (NPs), it is important to understand their aggregation behavior. This study investigates the aggregation kinetics of CeO₂ NPs in KCl and CaCl₂ solutions using time-resolved dynamic light scattering (TR-DLS). The initial hydrodynamic radius of CeO₂ NPs measured by DLS was approximately 95 nm. Attachment efficiencies were derived both from aggregation data and predictions based on the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory. The deviations of the DLVO predictions were corrected by employing the extended DLVO (EDLVO) theory. The critical coagulation concentration (CCC) of CeO₂ NPs at pH = 5.6 is approximately 34 mM for KCl and 9.5 mM for CaCl₂. Furthermore, based on the EDLVO theory and the von Smoluchowski’s population balance equation, a model accounting for diffusion-limited aggregation (DLA) kinetics was established. For the reaction-limited aggregation (RLA) kinetics, a model that takes fractal geometry into account was established. The models fitted the experimental data well and proved to be useful for predicting the aggregation kinetics of CeO₂ NPs.     

Growth of fractal patterns during irradiation-induced amorphization in nano-sized Ni–W multilayers

In the Ni–W system, a uniform amorphous Ni–W phase was obtained by ion irradiation of the nano-sized Ni–W multilayers at liquid-nitrogen temperature. Interestingly, before undergoing complete amorphization, fractal patterns were observed at a relatively low irradiation dose (3×10¹⁴ Xe⁺/cm²), and the patterns were characterized to consist of crystalline grains of Ni-enriched solid solution. The fractal dimension was measured to be about 1.68±0.05, which was very close to that expected by the cluster diffusion-limited aggregation model. Received: 30 January 2002 / Accepted: 2 February 2002 / Published online: 3 May 2002     

Theory of XMCD spectra at the L2,3 absorption edges of mixed valence Ce and Yb compounds in high magnetic fields

We examine the structure of aggregates formed due to DNA interaction with dipalmitoylphosphatidylcholine (DPPC) in presence of Ca²⁺ and Zn²⁺ using small-angle synchrotron X-ray diffraction (SAXD) and neutron scattering (SANS). SAXD shows structural heterogeneity as a function of the cation concentration and temperature: At low cation concentration (∼1 mM), aggregates show two DPPC phases, one with a lateral segregation of DNA and cation, while higher cation concentration improves the DNA packing and the condensed lamellar phase is observed in DNA+DPPC+20mMion²⁺ aggregates. The SANS detected the dissolution of the condensed lamellar phase into unilamellar DPPC+Zn²⁺ vesicles due to gel ↦ liquid-crystal phase transition in DNA+DPPC+20mM Zn²⁺ aggregates with the short fragmented salmon sperm DNA.