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.     

Fractal aggregates evolution of methyl red in liquid crystal

The spontaneous formation of dendritic aggregates is observed in a two-dimensional confined layered system consisting of a film composed of liquid crystal, dye and solvent cast above a polymer substrate. The observed aggregates are promoted by phase separation processes induced by dye diffusion and solvent evaporation. The growth properties of the aggregates are studied through the temporal evolution of their topological properties (surface, perimeter, fractal dimension). The fractal dimension of the completely formed structures, when they are coexistent with different types of structures, is consistent with theoretical and experimental values obtained for Diffusion-Limited Aggregates. Under different experimental conditions (temperature and local dye concentration) the structure forms without interactions with other kinds of structures, and its equilibrium fractal dimension is smaller. The fractal dimension is thus not a universal property of the observed structures, but rather depends on the experimental conditions.     

分形凝聚粒子的光散射特性研究

粒子的形状和凝聚对光散射特性有着很大的影响.基于分形生长的受限扩散(DLA)模型,模拟了凝聚粒子的三维空间分形结构,并采用回转半径法计算了凝聚粒子的分形维数.利用离散偶极子近似(DDA)方法研究了纳米石墨凝聚粒子的光散射特性,对于原始粒子数不同的凝聚粒子及分形结构不同的凝聚粒子,数值计算了散射强度和偏振度随散射角的分布...     

水分子凝胶中有机凝胶因子聚集体的分形结构研究

水分子凝胶是一种新型软凝聚体系.是凝胶因子在很低的浓度下在水中聚集、自组装，使水凝胶化形成的凝聚体系.透射电镜(TEM)表明凝胶因子在水中聚集、自组装成细纤维状结构.通过对TEM照片进行数字化处理，采用Sandbox法和密度-密度相关函数法计算的结果表明凝胶因子在聚集组装过程中具有典型的分形特征.根据C++程序计算出分形维数D＝1.814—1.977.以分形理论对凝胶因子的聚集过程以及由此形成的水分子凝聚体系的分形特征进行了讨论.利用小角x射线散射(SAXS)研究进一步表明，凝聚体系的分形结构存在于尺度α 关键词： 分形 凝胶因子 水分子凝聚体系 透射电镜(TEM) 小角x射线散射(SAXS)     

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.     

Annealing two-dimensional diffusion-limited aggregates

We have studied the structure of annealed two-dimensional diffusion-limited aggregates (DLA). The annealing process consists of introducing internal flexibility to the original DLA rigid structure as well as excluded volume interactions between particles. From extensive Monte Carlo simulations we obtained aggregates with fractal dimension slightly higher than that obtained for two-dimensional DLA structures. This is somehow surprising since the fractal dimension of the annealed structure is determined not only by connectivity but also by the competing effects of excluded volume interactions and configurational entropy, whilst in the rigid DLA only diffusion counts for the fractal-dimension value.     

Inelastic collisions and anisotropic aggregation of particles in a nematic collider driven by backflow

We design a nematic collider for controlled out-of-equilibrium anisotropic aggregation of spherical colloidal particles. The nematic surrounding imparts dipolar interactions among the spheres. A bidirectional backflow of the nematic liquid crystal in a periodic electric field forces the spheres to collide with each other. The inelastic collisions are of two types, head to tail and head to head. Head-to-tail collisions of dipoles result in longitudinal aggregation while head-to-head collisions promote aggregation in the transversal direction. The frequency of head-to-head collisions is set by the impact parameter that controls the resulting shape of aggregates, their anisotropy, and fractal dimension.     

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     

随机分布烟尘团簇粒子辐射特性研究

基于分形理论,采用蒙特卡罗方法对随机分布的烟尘团簇粒子结构进行了仿真模拟,利用离散偶极子近似(discrete dipole approximation, DDA)方法研究了随机分布的烟尘团簇粒子的辐射特性,分析讨论了分形维数、原始微粒粒径和数量以及复折射率对随机分布烟尘团簇粒子辐射特性的影响.研究表明,在给定分形维数的情况下,烟尘团簇粒子的辐射特性取决于原始微粒粒径、数量及复折射率;原始微粒较小的团簇粒子,当分形维数较小时,吸收截面变化不明显,但当分形维数大于2时,吸收截面骤然增大,然而,对于具有比较大的原始微粒粒径、数量及复折射率的烟尘团簇粒子,吸收截面随着分形维数的增大而单调递减;随着分形维数的增大,团簇粒子的散射截面、消光截面及单次散射反照率均单调递增;从整体上来讲,团簇粒子的辐射特性与等效球形粒子的辐射特性存在着比较大的差别,并且这种差别随着分形维数的增大而减小.该工作对研究气溶胶粒子的辐射及气候效应具有重要的科学价值. 关键词： 烟尘团簇粒子 辐射特性 离散偶极子近似方法     

Adsorption of Ideal Polymers on an Infinitely Ramified Fractal

We study ideal polymer chains interacting attractively with the borders of the lacunas of an infinitely ramified fractal, the Sierpinski carpet. Ideal chains are simulated on finite stages of construction of this fractal at various temperatures. The mean-square displacement and the mean number of adsorbed monomers of N-step chains are estimated in these lattices, and extrapolations to the fractal limit (infinite lattice) consider the exact forms of finite-size corrections as previously predicted by the series expansion method. In the noninteracting case, a finite fraction of the monomers is adsorbed, and this fraction increases as the temperature decreases. However, there is evidence that the critical exponent v which governs the growth of the chains varies with the temperature in a nonmonotonic way. At high temperatures v increases with decreasing temperature, and thus the chains are more stretched than in the noninteracting case. At an intermediate temperature, v starts to decrease and is still positive at very low temperatures, when the chains grow along the borders of several lacunas, occasionally crossing the bulk between them.     

Study of the structure of aqueous solutions of ionic macromolecules by low-frequency Raman spectroscopy

Using low-frequency Raman spectroscopy, the structure of aqueous solutions of polyacrylic acid as a function of polyelectrolyte concentration and chain length was studied. The fractal dimension of structures in aqueous solutions of polyacrylic acid with different acid concentrations and polyion chain lengths was determined. The size of fractal regions in systems with different structures was estimated.     

Aggregation of a hydrophobically modified poly(propylene imine) dendrimer

The poly(propylene imine) dendrimer DAB-dendr- (NH2)8 was hydrophobically modified with dodecanoyl end groups. The modified dendrimer was deposited onto mica by adsorption from solution and observed by atomic force microscopy. With the decrease of adsorption time, the modified dendrimer varied from continuous film to scattered islands. For the adsorption time of 20s the dendrimer formed a sub-monolayer thin film that contained many fractal aggregates of fractal dimension 1.80 that were > 1 microm in diameter and no more than 0.8nm thick. After 5 months at 1#1 , the initial fractal aggregates transformed into disks and other less-branched shapes with average heights of the domains of 0.6nm and 0.4nm, respectively. Formation of the fractal aggregates is explained by diffusion-limited aggregation. The slow reorganization of dendrimer molecules in the fractal aggregates occurs at a temperature well above the Tg of the dendrimer.     

Thermodynamic and scaling behaviour in finite diffusion-limited aggregation

Rényi's entropies for diffusion-limited aggregates are studied as a function of the number N of particles contained in the aggregates. It is found that Rényi's values increase with log N in a linear fashion, and that the aggregates exhibit multifractal behaviour for finite values of N. When N → ∞, the aggregate has a monofractal structure. Rényi's entropies depend on the fractal dimension of the aggregate. When the fractal dimension increases, the values of K_q decrease for q ? 1> and increase for q > 1.     

水分子凝聚体系的介孔结构分形学

透射电镜研究表明，4,40-双硬脂酰胺基二苯醚在水中聚集、自组装成缠绕细纤维状聚集体，进而使整个体系形成三维网络结构．水分子被包囊在这个网络结构中，形成一种新型的凝聚体系(水分子凝胶)．水分子凝胶是一种典型的纳米介孔物质，其复杂的微孔结构可以用分形维数D来表征，通过气体吸附方法(孔度法和比表面积法)计算，求得水分子凝胶体系的微孔结构的分形维数为2.1?2.2．对于纤维状三维网络结构的分形表征，通过粘度法和Cayley分形树模型得出分形维数为1.98．由此推测其分形网络形成的过程是一个初始成核-生长-枝化的循环过程．     

Monte Carlo simulation of aggregate morphology for simultaneous coagulation and sintering

A model for simulation of the three-dimensional morphology of nano-structured aggregates formed by concurrent coagulation and sintering is presented. Diffusion controlled cluster–cluster aggregation is assumed to be the prevailing coagulation mechanism which is implemented using a Monte–Carlo algorithm. Sintering is modeled as a successive overlapping of spherical primary particles, which are allowed to grow as to preserve overall mass. Simulations are characterized by individual ratios of characteristic collision to fusion time. A number of resulting aggregate-structures is displayed and reveals structure formation by coagulation and sintering for different values of . These aggregates are described qualitatively and quantitatively by their mass fractal dimension D_f and radius of gyration. The fractal dimension increases from 1.86 for pure aggregation to 2.75 for equal characteristic time scales. As sintering turns out to be more and more relevant, increasingly compact aggregates start to form and the radius of gyration decreases significantly. The simulation results clearly reveal a strong dependence of the fractal dimension on the kinetics of the concurrent coagulation and sintering processes. Considering appropriate values of D_f in aerosol process simulations may therefore be important in many cases.     

Quantum mechanics simulation and experimental study of adhesive interaction and aggregation of carbon-silicate nanoparticles—reinforcing fillers of polymer composites

Nanoparticles are widely used as polymer composite-reinforcing additives—fillers. Understanding the interaction mechanisms and regularities responsible for nanoparticle aggregation is of great significance for elucidating the nature of reinforcing of polymer composites. The paper reports on quantum mechanics calculations and full-scale experimental study of adhesive interaction of carbon and silicate adsorption complexes (nanomodels of active filler particles of polymer composites). The quantum mechanics approach allowed describing the adhesive properties of particle aggregates reasoning from nanoscopic structure of their surface. The quantum mechanics data were checked for adequacy on schungite—a natural mineral containing carbon and silicate. Schungite microparticles were milled to nanosizes by colloidal grinding in various disperse liquid media (alcohol, acetone, water) and the structure and properties of aggregated schungite micro- and nanoparticles were studied; fractal analysis of their surface was performed. It is found that smaller aggregates of silicate and carbon particles with higher surface fractal dimension are formed in colloidal grinding with small molecular sizes of disperse media (in our case, ethanol or methanol) and this agrees with the data predicted by quantum mechanics calculations.     

Kinetics of colloidal fractal aggregation by differential dynamic microscopy

We study the kinetics of an aggregation process induced by adding salt to a stable colloidal suspension of 73 nm (diameter) particles. Despite the subdiffraction size of the colloidal particles, the process is monitored via optical microscopy, which is used here to obtain time-resolved scattering information about the colloidal aggregates. The radius of the aggregates is determined as a function of time and their fractal dimension is extracted. Our results are compatible with a diffusion limited aggregation process, as independently confirmed by spectral turbidimetry measurements on the same sample.     

Structure of gels and aggregates of disk-like colloids

The static structure factor (S(q)) of dispersions and gels of disk-like mineral colloids (Laponite) was investigated using time- and ensemble-averaged light scattering. The evolution of S(q) in time after increasing the ionic strength of well-dispersed Laponite suspensions shows that Laponite aggregates and forms fractal clusters. The structure of the aggregates does not depend on the ionic strength, but the rate of growth increases very strongly with the ionic strength. At concentrations below about 3 g/l (0.12% v/v) the aggregates sediment while at higher concentrations space-filling gels are formed. The gels are homogeneous on length scales larger than the correlation length which decreases strongly with decreasing ionic strength and increasing concentration. However, the local structure is the same, independent of the concentration and the ionic strength. Received 6 August 2000 and Received in final form 16 March 2001     

The Use of Small Angle Light Scattering to Study Structure of Flocs

In this paper, a new approach using small angle light scattering to study the structure of large micron sized flocs is described. The flocs were made up from uniformly sized, approximately spherical hematite particles, induced to aggregate using 250 mM KCl to ensure that growth was governed by diffusion limited aggregation alone. The fractal dimensions of the hematite aggregates changed from 1.73 to 2.23 ± 0.05 as the aggregation proceeded, indicating that restructuring occurred during aggregation. The fractal dimensions obtained from the small angle light scattering technique are compared with those obtained from the combination of volume fraction and floc size measurement. The steady state fractal dimensions (2.23 ± 0.05) obtained by small angle light scattering are comparable to the fractal dimensions obtained from determining the slope of the relationship between floc volume fraction and floc size (2.09 and 2.14 ± 0.05). The discrepancy between the fractal dimensions obtained by these techniques during the initial stages of aggregation is due to the fact that the latter method measures the average fractal dimensions, whilst the former method measures the fractal dimensions at any particular instant.     

具有幂次相互作用的磁性粒子凝聚过程的数值研究

在扩散限制凝聚模型的基础上引入粒子的自旋自由度(包括自旋向上和向下),并假设粒子间存在幂次Ising磁相互作用,采用Monte Carlo方法研究了在不同相互作用力程情况下磁性粒子的分形生长规律.模拟结果表明,当粒子间以反铁磁方式耦合时,凝聚体中的粒子自旋交替凝聚.当粒子间以铁磁方式耦合时,凝聚体中粒子的自旋分布与相互作用力程有关:对于短程作用系统,凝聚体中存在大小不同的自旋畴块,即为铁磁生长;而对于长程相互作用系统,凝聚体中的自旋出现反常分布,即中心区域是近似反铁磁生长的结构,其外围后续生长的粒子却保持 关键词： 幂次相互作用 扩散限制凝聚模型 自旋