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.     

Rotational diffusivity of fractal clusters

The rotational diffusion behavior of fractal clusters generated through an off-lattice cluster-cluster aggregation algorithm in both diffusion-limited cluster aggregation and reaction-limited cluster aggregation conditions is investigated. The extended Kirkwood-Riseman theory (Garcia de la Torre et al., Macromolecules, 1987) is used to estimate the cluster rotational diffusion tensor. The three eigenvalues of this tensor, which correspond to the three main rotational diffusivity values of the cluster, have been computed for each generated cluster. Once the eigenvalues have been sorted in ascending order, each of them has been averaged over several thousands of clusters. It is found that one of the three main average rotational diffusivities is substantially larger than the other two, indicating significant anisotropy of fractal clusters. Moreover, a rotational hydrodynamic radius Rh,r has been determined on the basis of the mean value of the three average rotational diffusivities, which is about 25% larger than the mean translational hydrodynamic radius Rh calculated through the same Kirkwood-Riseman theory. Finally, the obtained Rh,r values have been applied to interpret dynamic light scattering data from aggregating colloidal systems and to investigate the reliability of the assumption, Rh = Rh,r, typically made in the literature.     

Hydrodynamic radius of fractal clusters

The Kirkwood-Riseman theory has been used to derive an analytical formula for the evaluation of the hydrodynamic radii of fractal clusters. The proposed relation is based on knowledge of the particle-particle correlation function and can be applied to clusters containing any number of particles larger than 4. The calculated values of the hydrodynamic radius are compared with the results of other theoretical approaches proposed in the literature, as well as several experimental data. Finally, the developed relation has been used in connection with a population balance equation model that computes the clusters' mass distribution to estimate the average hydrodynamic radius for a population of fractal clusters of different sizes. The obtained results have been compared to suitable experimental data for a silica colloidal suspension aggregating under reaction-limited conditions.     

Nonlinear optics of metal fractal clusters

Linear and nonlinear optical properties of metal fractal clusters are studied. Experimental evidence is presented proving that clusters possess unique optical nonlinearities, characterized by rapid response, broadbandness and enormously high values of nonlinear susceptibility. A 10⁶ times enhancement of the DFWM signal has been observed experimentally in silver particles aggregated into clusters. For the first time frequency-and polarization-selective photomodification of fractal silver clusters has been realized.     

Elastic properties of networks of fractal clusters

Elastic properties of casein gels were followed by dynamic measurements for various volume fractions of casein micelles far above the percolation threshold. Elastic and loss storage moduli versus volume fractions obey a power law with an exponent close to 3. The previous theoretical models for networks made of a homogeneous packing of fractal flocs do not predict the observed behavior. A model is proposed to estimate the vectorial contribution to the network elasticity. This model is in agreement with our experimental results. The value of the fractal dimension of the flocs derived from this analysis is discussed as a function of the volume fraction of primary particles.     

Light induced aggregation of metal clusters

The influence of light of the Mie plasmon wavelength on the aggregation of 10nm-gold clusters in aqueous colloidal systems is studied by spectral extinction measurements and electron microscopy. The detected significant acceleration of the aggregation process is attributed to additional Van der Waals like forces between the particles. Thermal and photochemical effects are discussed and excluded.     

Optical properties of fractal clusters of small metallic particles

In this paper we describe a method to calculate the optical properties of deterministic fractal clusters. Our method takes advantage of the fact that deterministic fractals can be constructed by an iterative rule. We calculate first the optical properties of a small cluster that describes one stage in the iteration. The optical properties of this cluster are then assigned to an “effective particle”. A small number of these produce the next stage in the fractal construction. We performed the calculations for metallic particles with dielectric functions described by the hydrodynamic model. Results in the dipolar and quadrupolar approximations for a cluster at the second fractal stage shows the double extinction peak often seen in experimental studies.     

Proposed connection between critical exponents and fractal dimensions in the Ising model

Scaling properties near the critical point indicates the existence of self-similarity behavior for the critical phenomena. Although the system considered here is not a truly dynamic one, we propose a specific set of relations between fractal dimensions and critical exponents in the Ising model of statistical mechanics. In particular, we put forward, corresponding to six critical exponents for the Ising model, six fractal dimensions. Assuming the latter proposals, we can then derive relationships between such fractal dimensions.     

A density functional study on the aggregation of alumina clusters

A calculation has been performed to explore the mechanism of aggregation reaction between two small molecular clusters [(Al₂O₃) _n1 and (Al₂O₃) _n2] by the density functional theory method. Five pathways of aggregation reactions between two different (Al₂O₃)₁ clusters isomers were identified. The detailed process of (Al₂O₃)₁ interaction with (Al₂O₃)₂ were also obtained. All the products of the aggregation reactions between two cage structures are cage-dimer structures. We calculated the thermodynamic properties of all the aggregate clusters. The Gibbs free energy changes of aggregation reactions in 0–1000 K were negative and increased with the temperature increase. The energy changes of enthalpy and entropy were also obtained.     

Formation of antimony oxide clusters by gas aggregation

Sb_xO_y clusters are produced by using a gas aggregation technique. Antimony vapor is mixed with He/O₂ or He/N₂O and cooled in a reaction channel. After photoionisation with a KrF (248 nm) or ArF (193 nm) excimer laser the products are mass analyzed in a time of flight mass spectrometer. In the presence of N₂O no oxide clusters besides SbO⁺ can be detected, while with oxygen under similar experimental conditions dramatic changes can be observed. At low oxygen partial pressure the obtained spectra are dominated by the pure Sb _x ⁺ clusters with low intensity of Sb_xO _y ⁺ , whereas at high oxygen partial pressure antimony oxides following the general sequence SbO⁺(Sb₂O₃)n are most abundant. The same stable species can furthermore be produced via aggregation of vaporised solid antimony oxide (Sb₂O₃). Within these experiments another new Series of antimony oxides tentatively assigned to (Sb₂O₃) _n ⁺ appeared in the mass spectra.     

Fractals: giant impurity nonlinearities in optics of fractal clusters

A theory of nonlinear optical properties of fractals is developed. Giant enhancement of optical susceptibilities is predicted for impurities bound to a fractal. This enhancement occurs if the exciting radiation frequency lies within the absorption band of the fractal. The giant optical nonlinearities are due to existence of high local electric fields in the sites of impurity locations. Such fields are due to the inhomogeneously broadened character of a fractal spectrum, i.e. partial conservation of individuality of fractal-forming particles (monomers). The field enhancement is proportional to theQ-factor of the resonance of a monomer. The effects of coherent anti-Stokes Raman scattering (CARS) and phase conjugation (PC) of light waves are enhanced to a much greater degree than generation of higher harmonics. In a general case the susceptibility of a higher-order is enhanced in the maximum way if the process includes “substraction” of photons (at least one of the strong field frequencies enters the susceptibility with the minus sign). Alternatively, enhancement for the highest-order harmonic generation (when all the photons are “accumulated”) is minimal. The predicted phenomena bear information on spectral properties of both impurity molecules and a fractal. In particular, in the CARS spectra a narrow (with the natural width) resonant structure, which is proper to an isolated monomer of a fractal, is predicted to be observed.     

Nonlinear optical properties of organic molecular assemblies and fractal metallic clusters

The paper overviews progress in the development of molecular media based on organic compounds possessing a high third-order nonlinear optical susceptibility. Such media are useful in the design of elements for optical data processing devices. The_X ⁽³⁾ values of polyconjugated polymers and organic dyes are presented; these depend on the film production method and molecular assembly organization. The review discusses nonlinear optical properties of metallic fractal clusters. The nonlinear response of a molecular medium is shown to be greatly enhanced by aggregation of molecules and by their location in a local field of a metallic cluster.Novosibirsk Institute of Organic Chemistry, Siberian Branch, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 34, No. 6, pp. 90–105, November–December, 1993.Translated by L. Smolina     

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.     

Mass spectrometric study of MgO clusters produced by the gas aggregation technique

Gas phase (MgO) _n ⁺ and (MgO) _n Mg⁺ clusters were produced in a gas aggregation source and studied by using laser-ionization time-of-flight mass spectrometry. A MgO molecule apparently serves as the nucleus for cluster growth, to which Mg and O atoms add. The heat generated by the formation of metal-oxygen bonds, and that added to the cluster by ionization leads to the production of clusters with the stoichiometry of the stable high-temperature oxide. The abundance maxima observed in the mass spectra indicate that the clusters form compact cubic structures similar to pieces of the MgO crystal lattice. The primary fragmentation channel responsible for the observed patterns is probably the loss of MgO monomers.     

Thermal restructuring of fractal clusters: the case of a strawberry-like core-shell polymer colloid

Thermal restructuring of fractal styrene-acrylate copolymer clusters dispersed in water has been investigated experimentally in the temperature range between 313 and 363 K. The particles constituting the clusters are of strawberry-like core-shell structure with a soft core and a rigid shell grafted on the core polymer chains. Due to the incomplete coverage of the core, the rather soft core may "flow out" through the open areas of the shell, leading to coalescence with the neighboring particles. The clusters were generated under diffusion-limited cluster aggregation conditions, and the restructuring kinetics was monitored by small-angle light scattering. Two sets of thermal restructuring experiments have been performed at various temperatures: (1) restructuring of growing clusters during aggregation and (2) restructuring of preformed clusters in the absence of aggregation. It is found that restructuring occurs only at temperature values above 323 K. In the absence of aggregation, restructuring leads to an increase of the fractal dimension and a decrease of the radius of gyration of the clusters. At sufficiently long times, both quantities reach a plateau value due to the presence of the grafted rigid shell, which constrains the coalescence of the soft core. A simple model, based on coalescence theory of liquid droplets and accounting for the incomplete coalescence and its dependence on temperature, has been developed to interpret the restructuring kinetics in the absence of aggregation. It is found that the proposed model can represent the measured experimental data well.     

Nucleation,growth, and fractal dimensions of icosahedral and polyicosahedral clusters

In this paper, the concept offractal geometry was applied to the growth of icosahedral and polyicosahedral clusters. Fractal dimensions were calculated for high-frequency icosahedral casters, vertex-sharing polyicosahedral clusters, and linked polyicosahedral clusters. These cluster growth pathways were compared with the fractal growth mechanisms of colloidal particles. Close similarities were found despite the tremendous differences in particle sizes and the forces governing their nucleation and growth processes.Dedicated to Prof. Lawrence F. Dahl on his 65th birthday; based in part on a lecture presented by BKT at the Dahl symposium held at the University of Wisconsin, Madison, Wisconsin on September 17, 1994.     

A simple model for the structure of fractal aggregates

Structural properties of small aggregates containing up to 100 particles have been studied through detailed Monte Carlo cluster-cluster aggregation simulations in both diffusion-limited and reaction-limited conditions. First, the radius of gyration, the radius of the smallest sphere encompassing the cluster, and the particle-particle correlation function, g(r), have been computed based on the positions of all the particles in the cluster, and their fractal scaling has been analyzed. Then, an empirical model has been developed to simulate the g(r) function for aggregates of any size and used to determine the corresponding structural properties and scattering structure factors. Finally, in order to illustrate the application of the structural properties thus computed, two experiments on diffusion-limited aggregation have been performed, and the average scattering structure factors have been measured as a function of time using a small-angle light-scattering device. The obtained average scattering structure factors have been simulated using the Smoluchowski population balance equations, using the single aggregate structural properties and scattering structure factor predicted by the developed empirical g(r) model.     

Electron-gas clusters: the ultimate jellium model

The local spin-density approximation is used to calculate ground- and isomeric-state geometries of jellium clusters with 2 to 22 electrons. The positive background charge of the model is completely deformable, both in shape and in density. The model has no input parameters. The resulting shapes of the clusters exhibit breaking of axial and inversion symmetries; in general the shapes are far from ellipsoidal. Those clusters which lack inversion symmetry are extremely soft against odd-multipole deformations. Some clusters can be interpreted as molecules built from magic clusters. The deformation produces a gap at the Fermi level. This results in a regular odd-even staggering of the total energy per electron and of the HOMO level. The strongly deformed 14-electron cluster is semimagic. Stable isomers are predicted. The splitting of the plasmon resonance due to deformation is estimated on a classical argument.