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.     

Cluster formation and rheology of photoreactive nanoparticle dispersions

We show how photocrosslinking of small nanoparticles within a very dilute colloidal dispersion leads to the formation of large fractal particle clusters, which have a strong influence on the viscosity of the dispersion although the overall solid content is well below 5 wt %. Furthermore, the solvent plays an important role because of its function as an optical filter, for example, in toluene only photocrosslinking but no photocleavage takes place. Therefore, a diffusion-controlled cluster growth mechanism, leading to clusters with low fractal dimension, is expected; on the other hand, in tetrahydrofuran the photoreaction is partially reversible. Therefore, the cluster growth in this case is reaction controlled, leading to more compact clusters with higher fractal dimension, which therefore only have a negligible effect on the rheological properties of the solvent. In this context, we will briefly discuss the possibility to use our nanoparticle system as opto-rheological switch.     

Driven Brownian coagulation of polymers

We present an analysis of the mean-field kinetics of Brownian coagulation of droplets and polymers driven by input of monomers which aims to characterize the long time behavior of the cluster size distribution as a function of the inverse fractal dimension, a, of the aggregates. We find that two types of long time behavior are possible. For 0≤a<1/2 the size distribution reaches a stationary state with a power law distribution of cluster sizes having exponent 3/2. The amplitude of this stationary state is determined exactly as a function of a. For 1/2相似文献   

Application of fractals and kinetic equations to cluster formation

We describe a rigorous connection between a coalescence growth model employing the Smoluchowski equation and the log-normal distribution which has been used for decades to classify empirically small particle distributions. The model assumes that only binary collisional events occur, there is conservation of monomers over the lifetime of the process, the clusters can be represented using a mass fractal approach and there are no “magic numbers”. Numerically, it is easy to account for the effects of evaporation, magic numbers, other inhomogeneities and possibly a non-conservative process. The model correctly incorporates the existence of multiple kinetic pathways for producing almost all cluster sizes. The properties of elemental cluster size distributions can apparently be related to the nature of the monomers as represented by the Periodic Table. The model classifies cluster size distributions on the basis of a single scaling parameter which itself is a function of the dimensionality of the space in which the coalescence process occurs, the fractal dimensionality of the clusters, the fractal dimensionality of the trajectories of the agglomerating species between collisions and the scaling of the cluster velocities with increasing cluster size.     

A small-angle scattering study on equilibrium clusters in lysozyme solutions

We use small-angle scattering experiments to investigate the structural properties of aqueous lysozyme solutions under conditions where the existence of equilibrium clusters has recently been demonstrated (Nature 2004, 432, 492). We also discuss the possible emergence of a low angle scattering contribution, which recently attracted interest due to its appearance in solutions of various proteins. We demonstrate that in lysozyme solutions under our experimental conditions such rising low q intensities can only be observed under special circumstances and can thus not be attributed to the existence of a universal long-range attraction. We then focus on the structural properties of the equilibrium clusters as a function of protein concentration, temperature, and ionic strength. We show that the experimental structure factors obtained from the scattering measurements exhibit the typical cluster-cluster peak q(c) reflecting the mean distance between charged clusters as well as a monomer-monomer peak q(m), which represents the nearest neighbor shell of monomers within a single cluster. The underlying principle for the formation of these structures is the coexistence of two opposing forces, a short-range attraction and a long-range repulsion due to residual charges. We can quantitatively analyze our scattering data by applying a simple equilibrium cluster model and calculate an average cluster aggregation number, N(c). The thus obtained cluster aggregation number increases linearly with volume fraction. We also observe an increasing N(c) as temperature decreases and as the screening of residual charges increases. We point out the importance of the existence of equilibrium clusters and the universality of this phenomenon for self-assembling processes observed in nature. Finally, we discuss the limitations of our simple globular cluster model in view of recent findings from computer simulations.     

Mass-fractal clustering and power-law decay of cluster size in 1-propanol aqueous solution

Mesoscale structure of 1-propanol aqueous solutions with propanol mole fraction xp ranging from 0.1 to 0.33 has been studied by means of small angle neutron scattering (SANS) and large-scale reverse Monte Carlo (RMC) technique. Analysis of the SANS intensities in terms of a fractal model shows that the fractal dimension df of mesoscale structure of the solution is about 1.8-1.9 for water-rich solution and about 1.5 for propanol-rich solution. Percolation analysis on the RMC results reveals that the water molecules and the propanol molecules cluster, respectively, as a mass fractal, the dimension dM of which is about 2.3-2.5 for both clusters for water-rich solution. Furthermore, the distribution of the cluster size is expressed by a simple power law with an exponent tau of about 1.35-1.5 for the propanol clusters and 1.05-1.2 for the water clusters. These results imply that the current solution is characterized by polydisperse mass fractals. In fact, a theoretical relation for polydisperse system of mass fractals, df = dM(2-tau), holds well in the current solution. The characteristic change in df from 1.8-1.9 to 1.5 described above is attributed to the crossover between the water-rich regime and the propanol-rich regime. Most of the water molecules and the propanol molecules are located on the interface between clusters, and the water molecules form thin layers of about 10 A thick irrespective of 1-propanol content studied.     

Fractal growth of rotating DLA‐clusters

A theoretical model for fractal growth of DLA‐clusters in two‐ and three‐dimensional Euclidean space is proposed. This model allows to study some statistical properties of growing clusters in two different situations: in the static case (the cluster is fixed), and in the case when the growing structure has a nonzero rotation around its germ. By the direct computer simulation the growth of rotating clusters is investigated. The fractal dimension of such clusters as a function of the rotation velocity is found. It is shown that for small enough velocities the fractal dimension is growing, but then, with increasing rotation velocity, it tends to the unity.     

Aggregation-fragmentation in a model of DNA-mediated colloidal assembly

We present results from an off-lattice Monte Carlo simulation of DNA-mediated colloidal assembly. In this simulation, the aggregation-fragmentation of a binary mixture of DNA-coated colloidal particles is studied through a simplified model of base-pair hybridization. Bonding between monomers is modeled as a simple temperature-sensitive A/B-type interaction, where type A and B monomers can bond to only the opposite type (no A/A or B/B attachments are allowed). The actual chemistry of base-pair hybridization is not included in the model. The morphological structures of the clusters formed as well as the kinetics of growth are analyzed in our 2D simulations. The fractal dimension and kinetic growth exponents for clusters formed near the DNA "melting" temperature agree with those seen previously for 2D diffusion-limited cluster aggregation (DLCA) models. The clusters appear more compact, exhibiting signs of local order at intermediate temperature values. At higher temperatures, the formation of large clusters is not favorable under the action of temperature-dependent fragmentation, and the system eventually reaches a steady state as a collection of small aggregates. The temperature profile for this dissolution of the colloidal assembly is sharp, indicating that the selective hybridization process provides a highly sensitive measurement tool. At high temperatures, we analyze the steady-state behavior of the average cluster size in terms of an aggregation-fragmentation model.     

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.     

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.     

Analysis of the structure of very large bacterial aggregates by small-angle multiple light scattering and confocal image analysis

This work aims at developing a more accurate measurement of the physical parameters of fractal dimension and the size distribution of large fractal aggregates by small-angle light scattering. The theory of multiple scattering has been of particular interest in the case of fractal aggregates for which Rayleigh theory is no longer valid. The introduction of multiple scattering theory into the interpretation of scattering by large bacterial aggregates has been used to calculate the fractal dimension and size distribution. The fractal dimension is calculated from the form factor F(q) at large scattering angles. At large angles the fractal dimension can also be computed by considering only the influence of the very local environment on the optical contrast around a subunit. The fractal dimensions of E. coli strains flocculated with two different cationic polymers have been computed by two techniques: static light scattering and confocal image analysis. The fractal dimensions calculated with both techniques at different flocculation times are very similar: between 1.90 and 2.19. The comparison between two completely independent techniques confirms the theoretical approach of multiple scattering of large flocs using the Mie theory. Size distributions have been calculated from light-scattering data taking into account the linear independence of the structure factor S(q) relative to each size class and using the fractal dimension measured from F(q) in the large-angle range or from confocal image analysis. The results are very different from calculations made using hard-sphere particle models. The size distribution is displaced toward the larger sizes when multiple scattering is considered. Using this new approach to the analysis of very large fractal aggregates by static light multiple scattering, the fractal dimension and size distribution can be calculated using two independent parts of the scattering curve.     

Experimental investigation of colloidal gel structures

We investigate experimentally the structural properties of colloidal gels, formed under both diffusion-limited and reaction-limited aggregation conditions, using light scattering measurements and compare the results with the literature Monte Carlo (MC) simulations. The scattering structure factors have been measured for the two classes of gels in the range of the particle volume fractions between 0.02 and 0.07. From these, the corresponding fractal dimension values have been estimated. These have been found to be in good agreement with those estimated from the structure factors computed from MC simulated gels. On the basis of our previous research (Lattuada et al. Langmuir 2003, 19, 6312), this confirms that the scattering structure factor of a gel provides erroneously a small fractal dimension value, which decreases as the particle volume fraction increases. Furthermore, it is observed that the average size of the fractal clusters is larger in real gels than in simulated gels.     

Detailed model of the aggregation event between two fractal clusters

A model has been developed for describing the aggregation process of two fractal clusters under quiescent conditions. The model uses the approach originally proposed by Smoluchowski for the diffusion-limited aggregation of two spherical particles but accounts for the possibility of interpenetration between the fractal clusters. It is assumed that when a cluster diffuses toward a reference cluster their center-to-center distance can be smaller than the sum of their radii, and their aggregation process is modeled using a diffusion-reaction equation. The reactivity of the clusters is assumed to depend on the reactivity and number of their particles involved in the aggregation event. The model can be applied to evaluate the aggregation rate constant as a function of the prevailing operating conditions by simply changing the value of the particle stability ratio, without any a priori specification of a diffusion-limited cluster aggregation, reaction-limited cluster aggregation, or transition regime. Furthermore, the model allows one to estimate the structure properties of the formed cluster after the aggregation, based on the computed distance between the aggregating clusters in the final cluster.     

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.     

Monomer, clusters, liquid: an integrated spectroscopic study of methanol condensation

We have combined static pressure, spectroscopic temperature, Fourier transform infrared spectroscopy (FTIR), and small angle X-ray scattering (SAXS) measurements to develop a detailed picture of methanol condensing from a dilute vapor-carrier gas mixture under the highly supersaturated conditions present in a supersonic nozzle. In our experiments, methanol condensation can be divided into three stages as the gas mixture expands in the nozzle. In the first stage, as the temperature decreases rapidly, small methanol n-mers (clusters) form, increase in concentration, and evolve in size. In the second stage, the temperature decreases more slowly, and the n-mer concentrations continue to rise. Thermodynamic and FTIR experiments cannot, however, definitively establish if the average cluster size is constant or if it continues to increase. Finally, when the vapor becomes supersaturated enough, liquid droplets form via nucleation and growth, consuming more monomer and reducing the concentration of clusters. At the point where liquid first appears, cluster formation has already consumed up to 30% of the monomer. This is significantly more than is predicted by a model that describes the vapor phase as an equilibrium mixture of methanol monomer, dimer, and tetramer. An energy balance suggests that a significant fraction of the cluster population is larger than the tetramer, while preliminary SAXS measurements suggest that these clusters contain, on average, 6 monomers.     

Characterization of Particle Size and Shape in an Ageing Bismuth Titanate Sol Using Dynamic and Static Light Scattering

The ageing characteristics of a metal-alkoxide solution, used to prepare thin films of ferroelectric bismuth titanate, Bi₄Ti₃O₁₂, were carefully scrutinized using laser light scattering. Dynamic light scattering experiments revealed a bimodal distribution of hydrodynamic diameters, with larger molecular clusters increasing in size from 4 to 68 nm over the course of 60 days ageing in a sealed vial. Static light scattering measured the radius of gyration to be 71 nm. Comparison of these two results suggests that these clusters are approximated as thin cylindrical (or chainlike) structures. Using ellipsoid scattering theory, plots of scattered light intensity versus q (scattering) vector also suggest a long cylindrical type cluster with a length of 350 nm and a diameter of 4 nm. Some deviation of experimental data from the theoretical curves probably is an indication of some flexibility and/or fractal nature of the molecular clusters. Such anisotropy in oligomeric cluster shape apparently has a strong influence on final crystallographic texturing of ceramic thin films made by spin-coating of these solutions on flat substrates.     

Relation between aggregation kinetics and the structure of kaolinite aggregates

Dynamic and static light scattering were applied to the determination of the stability ratio and fractal dimension of kaolinite (KGa-2) at different kaolinite or/and electrolyte concentrations at pH 9.5. Dynamic light scattering was used to measure the kinetics of early stage aggregation to determine the stability ratio, W, as well as the cluster sizes which determine the fractal regime. Static light scattering was used to measure the fractal dimension, D(f). Results show that the two classes of "universality" (Lin et al. Nature 1989, 339, 360) characterizing the diffusion- and reaction-limited regimes of cluster-cluster aggregation do apply to colloidal kaolinite as limit cases when W approximately 1 or W > 100, respectively. In the intermediate regime where 5 < W < 100, the growth of the aggregate radius showed a power-law behavior similar to diffusion-limited cluster aggregation. For the intermediate aggregation regime, a scaling relation between fractal dimension and stability ratio, reflecting a continuous increase in particle packing density in the aggregate as the sticking probability of particles was reduced, was demonstrated.     

剪应力下弱作用势胶体颗粒聚团的特点

探讨了不同剪应力下，具有Ｌｅｎｎａｒｄ－Ｊｏｎｅｓ势的胶体颗粒聚团的结构特性，包括簇团的大小分布，径向分布函数，分形维数和原子配位数。研究表明，在弱作用力下，胶体簇团的分布随剪应力的增加而趋向小簇团一边；径向分布函数曲线随剪应力的增加而降低，胡在近程距离内降低得最多；分形维数随剪应力的增加表现为先增加后减小的趋势，其值随模拟条件的不同而在１．９－２．４之间变动。剪应力“场”对分形维数的大小没有太大     

Vapor Nucleation and Droplet Growth: Cluster Distribution Kinetics for Open and Closed Systems

A theory based on cluster distribution kinetics for single-monomer addition and dissociation is presented as a framework for homogeneous and heterogeneous vapor nucleation and growth dynamics. For continuous cluster and monomer distributions in a well-mixed non-steady-state flow system, population (mass) balance equations yield moment equations for the cluster mass moments. Nuclei are either homogeneously generated or heterogeneously seeded, and subsequent cluster growth occurs by reversible condensation of vapor monomers. The zeroth moment is the number (or moles) of clusters, the first moment is cluster mass, and the second moment gives cluster-size variance. Solutions are proposed for steady-state flow (open) and non-steady-state batch (closed) systems. Experimental data are interpreted by recognizing that droplets typically observed in nucleation experiments have grown much larger than their nuclei. This allows resolution of the large temperature-dependent discrepancy between experiment and classical nucleation theory. 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.