Fractal Characterization of Silica Sol Prepared by the Sol-Gel Method: From the Sol Formation to the Flocculation Process

The fractal characterization of silica particles prepared by the sol-gel method was obtained; from the beginning of the sol-gel synthesis to the aggregation process of these particles by adding metal ions in solution, the fractal dimension was determined. At the beginning of the sol-gel process, unstable structures were formed due, essentially, to the auto-catalytic nature of the sol-gel condensation reactions; these particles are fractal structures with a fractal exponent corresponding to a reaction limited aggregation regime. As the time proceeds, the reactants are consumed approaching the system to equilibrium, stabilizing the size of the silica particles. The silica sol can be flocculated by adding metal ions in solution. The fractal exponent for the aggregation process was determined, obtaining a value corresponding to a diffusion limited aggregation regime.     

Model of bound water formation in fractal clusters resulting from reversible aggregation of colloidal particles

A model is proposed for the formation of bound water in fractal clusters that arise in the course of the reversible aggregation of colloidal particles. The dependence of the specific content of bound (nonfreezing) water on the concentration of a colloidal solution is theoretically determined. It is found that this value is related to the concentration of the colloidal solution via a power law. It is shown that the exponent is determined by the total surface area and morphology of fractal clusters, as well as by the mechanism of their aggregation and disintegration.     

Influence of natural organic matter and ionic composition on the kinetics and structure of hematite colloid aggregation: implications to iron depletion in estuaries

The stability and aggregation behavior of iron oxide colloids in natural waters play an important role in controlling the fate, transport, and bioavailability of trace metals. Time-resolved dynamic light scattering experiments were carried out in a study of the aggregation kinetics and aggregate structure of natural organic matter (NOM) coated hematite colloids and bare hematite colloids. The aggregation behavior was examined over a range of solution chemistries, by adjusting the concentration of the supporting electrolyte-NaCl, CaCl2, or simulated seawater. With the solution pH adjusted so that NOM-coated and bare hematite colloids were at the same zeta potential, we observed a significant difference in colloid stability which results from the stability imparted to the colloids by the adsorbed NOM macromolecules. This enhanced stability of NOM-coated hematite colloids was not observed with CaCl2. Aggregate form expressed as fractal dimension was determined for both NOM-coated and bare hematite aggregates in both NaCl and CaCl2. The fractal dimensions of aggregates formed in the diffusion-limited regime indicate slightly more loosely packed aggregates for bare hematite than theory predicts. For NOM-coated hematite, a small decrease in fractal dimension was observed when the solution composition changed from NaCl to CaCl2. For systems in the reaction-limited regime, the measured fractal dimensions agreed with those in the literature. Colloid aggregation was also studied in synthetic seawater, a mixed cation system to simulate estuarine mixing. Those results describe the important phenomena of iron oxide aggregation and sedimentation in estuaries. When compared to field data from the Mullica Estuary, U.S.A., it is shown that collision efficiency is a good predictor of the iron removal in this natural system.     

Spontaneous fractal aggregation of gold nanoparticles and controlled generation of aggregate-based fractal networks at air/water interface

The spontaneous fractal aggregation of as-prepared cetytrimethylammonium bromide (CTAB)-capped gold nanoparticles was found to happen at the air/water interface after spreading their chloroform solution on water surfaces. Aided by Langmuir-Blodgett techniques, these fractal aggregates can be interconnected with each other to form aggregate-based fractal networks.     

Fractal Morphology and Breakage of DLCA and RLCA Aggregates

Latex aggregates, formed in 1 M McIlvaine buffer solution and 0.2 M NaCl solution, have been characterized in terms of aggregate size distribution and fractal morphology. This was achieved using three sizing techniques (image analysis, laser scattering, and electrical sensing) in which size distributions and fractal properties of the aggregates were measured. Estimates of fractal dimensions were made using the two-slope method based on dimensional analysis and the small-angle light scattering method. Aggregate suspensions were prepared using both water and a mixture of heavy water/ water as the solvent. The latter essentially eliminated sedimentation, which was observed after one day of aggregation when water alone was used as a solvent. Latex aggregates formed by diffusion-limited colloid aggregation (DLCA) and reaction-limited colloid aggregation (RLCA) had fractal dimensions close to 1.8 and 2.1, respectively. As observed through image analysis, DLCA aggregates possessed a loose tenuous structure, whereas RLCA aggregates were more compact. Disruption of both DLCA and RLCA aggregates has been investigated in laminar flow and turbulent capillary flow. The shear forces introduced by a laminar shear device with a shear rate up to 1711 s(-1) were unable to bring about aggregate breakup; shearing facilitates aggregate growth in the case of DLCA. However, latex aggregates were significantly disrupted after passage through a turbulent capillary tube at 95209 s(-1). Copyright 2000 Academic Press.     

A modified CCA model describing gelation processes

In order to simulate a gelation process that occurs by condensation-polymerization reactions in solution, the cluster–cluster aggregation (CCA) model is modified by introducing four- and two-functional units as monomers. The modified CCA model shows a clear dependence of the fractal dimension of the resulting gel structure on the ratio between the numbers of the four- and two-functional units. The model should be applicable to various real systems. For example, it explains the behavior of experimentally found fractal dimension that depends on the amount of water added in the sol–gel aggregation process of SiO₂ systems.     

Hierarchical assembly of diphenylalanine into dendritic nanoarchitectures

Highly ordered, multi-dimensional dendritic nanoarchitectures were created via self-assembly of diphenylalanine from an acidic buffer solution. The self-similarity of dendritic structures was characterized by examining their fractal dimensions with the box-counting method. The fractal dimension was determined to be 1.7, which demonstrates the fractal dimension of structures generated by diffusion limited aggregation on a two-dimensional substrate surface. By confining the dendritic assembly of diphenylalanine within PDMS microchannels, the self-similar dendritic growth could be hierarchically directed to create linearly assembled nanoarchitectures. Our approach offers a novel pathway for creating and directing hierarchical nanoarchitecture from biomolecular assembly.     

AN ATOMIC FORCE MICROSCOPY STUDY ON THE AGGREGATION OF ISOTACTIC POLY（METHYL METHACRYLATE）

Aggregation process of isotactic poly(methyl methacrylate) (i-PMMA) has been studied extensively for many years, and considerable progress has been made in both experimental and theoretical studies. They are, however, seldom sustained by real-space observations of the underlying morphology. In this paper, the aggregation process of i-PMMA in concentrated acetone solutions and the fractal structure of the resulting three-dimensional clusters were characterized on the basis of real-space AFM observations of their two-dimensional projection. It was found that spherical multiple-chain particles formed upon collapse and aggregation of the involving chains as a whole during quenching the solution to room temperature. By keeping the solution at room temperature, the initially formed particles stick together upon contact to form larger particles through reassembling very slowly. The succeeding collision of the enlarged spherical particles leads to the formation of small clusters. These newly formed small clusters grow when they meet with other clusters or single Brownian particles. This leads to the formation of large clusters with fractal dimension of 1.95$±0.05, which suggest a reaction-limited cluster aggregation of i-PMMA in a concentrated acetone solution. This is in accordance with the conclusion obtained by light scattering measurements.     

Effect of Escherichia coliform on the biomineralization of calcium bilirubinate in mimic systems

The influence of Escherichia coliform (E. coli), especially the proteins it secretes, on the nucleation, growth and aggregation processes of calcium bilirubinate (CaBR) in different mimic systems, such as NaCl aqueous solution and model bile, is studied in this paper. The results show that in NaCl aqueous solution the morphology of calcium bilirubinate (CaBR) changes from amorphous sheet structure to a highly self-organized and highly self-replicated fractal structure that is accumulated by rhombic microcrystals after the addition of E. coli. In model bile with the existence of E. coli, CaBR also forms a fractal structure, but the fractal structure is staghorn-like. Meanwhile, the composition of the prepared CaBRs is nonstoichiometric, and the crystallization is greatly improved with the existence of E. coli. Besides, formation mechanism of CaBRs affected by E. coli is explored, showing that the proteins secreted by E. coli interact with the Ca(2+) ions to provide nucleation sites for CaBRs, and the conformation of the proteins becomes more ordered, resulting in the improvement of the crystalline of CaBRs. On the other hand, the interaction of proteins and the Ca(2+) ions also promote the aggregation state of the CaBRs.     

Estimation of the dynamic size of fractal aggregates

A model is presented for an aggregation act occurring between two aggregates of any mass and fractal dimension. The kinetics of aggregation is also analyzed, as well as some previous works concerning the structure of fractal aggregates. As a result, a generalized curve is derived describing the normalized dynamic radius of clusters of spherical character as dependent on both the aggregate fractal dimension and the space dimension. It is shown how the curve may be utilized to determine the dynamic size of anisotropic aggregates. The obtained dependence can be used to estimate the dynamic size of fractal aggregates, to evaluate the prefactor in mass–radius relation and to model the aggregation kinetics.     

Measurement of Fractal Aggregates of Polydisperse Particles Using Small-Angle Light Scattering

The effect of primary particle polydispersity on the structure of fractal aggregates has been investigated through the salt-induced, diffusion-limited aggregation of mixtures of hematite. The fractal dimension was determined experimentally using three independent methods: q dependence of static light scattering, kinetic scaling, and correlation of aggregate mass and linear size both determined from Guinier scattering. The fractal dimensions D(f) obtained were 1.75+/-0.03, 1.76+/-0.03, and 1.70+/-0.05, respectively. The use of a previously derived fractal mean particle size was validated in allowing data collapse to master curves for the aggregation kinetics data. The fractal mean particle size is shown to have general utility by taking a number weighting to describe polydisperse aggregation kinetics and a mass weighting to describe small q scattering behavior. Copyright 2000 Academic Press.     

Hierarchical organization in aggregates of protein molecules

The aggregation of proteins into small clusters is studied by atomic force and electron microscopy. Scaling laws and fractal behaviour in the growth of the aggregates and in the correlation between aggregates is seen. A phase diagram of the aggregation process where the protonic concentration of the solution and the density of protein are varied shows the existence of specific growth processes resulting in different branch-like structures. The resulting structures are strongly influenced by the shape of each protein molecule. Lysozyme and ribonuclease are found to form spherical structures with a narrow size distribution.     

The growth of compact layers at the electrode interface: Part IV. Fractal character of uracil films at the mercury-aqueous electrolyte interface

Compact layers of uracil films grown on the mercury electrode-aqueous solution interface were studied by means of time-resolved FFT impedance spectroscopy. The films are characterized by the fractal dimension which evolves with time. The results are discussed in terms of the cluster-cluster aggregation model and the hole formation mechanism proceeding in the compact layer.     

Large Scale Porous Structure in Gels and Relationship with Their Plastic Behavior

Silica gels (classical aerogels and composite aerogels) have been prepared by classical gelation and addition of silica soot in the gelifying solution before gelation. Due to the aggregation mechanisms, these structures are characterized by a fractal organization. The fractal network previously described in the literature (1–100 nm) which results from the aggregation mechanism of the organosiloxane is affected by the addition of the silica soots. Ultra Small Angle X-ray Scattering (USAXS) experiments (done at ESRF) shows that besides the fractal network built by the organosiloxane, the silica soots are forming another porous structure at a higher scale.The mechanical properties seem to be dependent on this large pore structure. Under isostatic pressure, aerogels display an irreversible shrinkage caused by plastic deformation. As a consequence of this plastic shrinkage it is possible to densify, by the pore collapse tending towards the silica glass. The densification mechanism is different from the one obtained by a sintering at high temperature. The pore collapse mechanism is favored by the large pores structure of the composite aerogels, in contrast to viscous sintering.     

Light-scattering study of the aggregation of syndiotactic poly(methyl methacrylate) in solution.

Syndiotactic poly(methyl methacrylate (s-PMMA) may undergo aggregation in n-butyl chloride (n-BuCl) at temperatures below the theta temperature. The aggregation behavior of the s-PMMA with weight-average molecular weight M(w) =6.06 x 10(5) g mol(-1) was studied by a combination of static and dynamic laser-light-scattering experiments. A solution of concentration 1.12 x 10(-4) g mL(-1) was quenched from 50 degrees C (above the theta temperature in n-BuCl, 35 degrees C to 12 degrees C, and the aggregation process was measured over 60 h. The time dependence of M(w) the root-mean-square z-average radius of gyration < R(g) >, and the average hydrodynamic radius were used to monitor the growth of the aggregates, with the result M(w) approximately < R(g) > d(f) (where d(f) = 1.98 +/- 0.02), which implies the formation of a fractal aggregate. The observed fractal dimension, d(f), is close to that expected for a reaction-limited cluster aggregation for which d(f) = 2.1. In addition, atomic force microscopy was used to image the aggregates.     

Self‐assembled supramolecular microgels: Fractal structure and aggregation mechanism

Multifunctional molecules were designed to produce microgels with specific structures. Both static light scattering and dynamic light scattering were employed to determine the fractal dimension of the microgels. The protein, avidin, was strongly bound to four biotin moieties. Biotin was attached covalently to specifically engineered peptide nucleic acid (PNA) oligomers. Three designed DNA oligomers self‐assembled to produce a trifunctional three‐way junction (TWJ) with single‐stranded ends that were complementary to the PNA sequence. The sizes of the supramolecular aggregates were characterized by dynamic light scattering. The fractal dimension was obtained from the angular dependence of the scattered intensity when the microgels were large enough. When the microgels were formed via cooling from a temperature above the melting point of the PNA–DNA helices, reversible structures with a fractal dimension of approximately 1.86 were formed, which is consistent with a cluster–cluster aggregation mechanism. When the microgels were formed by the slow addition of biotinylated PNA bound to the TWJ to a solution of avidin at room temperature, the observed fractal dimension approached 2.6, which is consistent with a point–cluster aggregation mechanism. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 3037–3046, 2003     

Fractal analysis of a discotic texture

The dendritic-type texture displayed by low molar mass discotic charge transfer systems possessing a columnar hexagonal ordered phase has been subjected to a fractal analysis. The diffusion limited aggregation approach was used to simulate the growth of the texture. The results are that the texture displays a fractal geometry and that the prominent features of the discotic texture are correctly reflected by the simulated clusters, including the fractal dimension. The conclusion is that the texture is conditioned by the lattice structure of the discotic phase and the kinetic processes, the sticking rules in the terms of diffusion limited aggregation model, responsible for the growth of the texture.     

Growth of Fractal Aggregates in the Intermediate Regime of Particle Accommodation

The growth dynamics of fractal aggregates was studied within the framework of continuum model in the self-consistent mean field approximation. The regime that is intermediate between the diffusion-limited aggregation and reaction-limited aggregation was considered. The dependence of aggregate fractal dimension on the attachment probability of particles during their collisions with an aggregate was obtained. In the limiting cases, the values of fractal dimension coincide with those determined earlier. The domain of the values of attachment probability was revealed where several regions characterized by their own values of fractal dimension were specified in the structure of growing cluster. Physical nature of the emergence of various regions in the aggregate structure was discussed.