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.     

火灾烟颗粒的分形结构形状模拟与光散射计算

针对火灾烟颗粒的形状特点，提出并建立火灾烟颗粒分形结构凝团的形状模型，并对烟颗粒扫描电镜(SEM)图像进行分析，获取分形结构模型中的单个凝团中基本颗粒个数、凝团分形维数、基本颗粒半径等参数.利用该模型对火灾烟颗粒的形状进行模拟的结果表明，该模型能够较好反映出烟颗粒的形貌特征.利用形状模型对火灾烟颗粒散射进行初步计算表明，在其他参数相同的情况下，相对于同体积的球形颗粒，分形凝团具有前向散射较弱，后向散射较强的特征.     

Formation of fractal structures by aggregation of anisometric iron (III) hydroxide particles

Coagulation of anisometric iron (III) hydroxide sols has been studied by dynamic light scattering, TEM and ultra-small-angle X-ray scattering. The anisometry of rod-like individual colloidal particles can be characterised with a mean aspect ratio of 10. It was established for the first time, that the aggregates formed by coagulation of anisometric particles show definite fractal structure with a mass fractal dimension of 1.95 ± 0.05.     

Fractal Character of Dynamic Light Scattering of Particles

Dynamic light scattering signals from particles, exhibit fractal characteristics. This feature can be used to determine the particle size. The use of the fractal dimension, as a quantitative method to analyze the properties of dynamic light scattering signals from submicron particles, is presented. The analysis is performed directly on the time‐resolved scattered intensity, and the Box Dimensions of light scattering signals of particles with diameters 100, 200, 500 and 1000 nm. The experimental results show that the fractal dimensions of light scattering signals correlate well with particle size. In the submicron size range, the smaller the particles, the larger their fractal dimensions. Compared with the PCS technique, only several hundreds of samples are required in the fractal method. Therefore, the data processing is easily accomplished. However, this method only provides the mean particle size, but not the particle size distribution.     

Small angle neutron scattering and small angle X-ray scattering studies of platinum-loaded carbon foams

The morphology of carbon nanofoam samples comprising platinum nanoparticles dispersed in the matrix was characterized by small angle neutron scattering (SANS) and small angle X-ray scattering (SAXS) techniques. Results show that the structure of pores of carbon matrix exhibits a mass (pore) fractal nature and the average radius of the platinum particles is about 2.5 nm. The fractal dimension as well as the size distribution parameters of platinum particles varies markedly with the platinum content and annealing temperature. Transmission electron micrographs of the samples corroborate the SANS and SAXS results.      

Dynamical scaling of the structure factor of some non-Euclidean systems

Predictions of nonlinear theories on dynamics of new phase formation have been examined for the hydration of calcium silicates with light water and heavy water. In the case of hydration with light water, reasonable agreement has been observed with dynamical scaling hypothesis with a new measure of the characteristic length. The characteristic length does not follow a power law relation with time. Hydrating mass is found to be mass fractal throughout hydration, with mass fractal dimension increasing with time. But, in the case of hydration with heavy water, no agreement has been observed with the scaling hypothesis. Hydrating mass undergoes transition from mass fractal to surface fractal and finally again to mass fractal. The qualitative features of the kinetics of hydration, as measured in small-angle scattering experiments, are strikingly different for hydration with light water and heavy water.     

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)     

Self—assembly behaviour of amphiphilic diblock copolymer in selective solvents studied by synchrotron small—angle x—ray scattering

The aggregation behaviour of styrene-vinyl benzoic acid (PS_m-b-PVBA_n) amphiphilic diblock copolymers in selective solvents with different m and n was investigated by synchrotron small-angle x-ray scattering (SAXS). We have carried out a detailed analysis of scattering intensity, dimension, shape and microstructure of the diblock copolymers of narrow distribution in water, methanol, ethanol and isopropanol selective solvents, respectively. We have found that the aggregation behaviour of the copolymer depends on the nature of the solvent and the micelle forms flat disc objects with the ratio of radius ω=0.4. The average radius gyration R_g of the copolymer decreases as solvents change from isopropanol to ethanol and to methanol, and increases with increasing pH in aqueous solution, but decreases with the addition of CoCl_2 in ethanol solvent. The scattering intensity of diblock copolymer micelle follows I(h)∝h^{-α} in different selective solutions, suggesting that the PS_m-b-PVBA_n coils have self-similar structure behaviour or a fractal structure in the selective solvents. All of these revealed that the aggregation behaviour of the diblock copolymer changes dramatically with experimental condition in the selective solvent. The increase of mass fractal dimension (D_m) from 2.12 to 2.47 indicates that the copolymer chain changes from a swollen coil to a rather compact disc in the course of changing solvents, decreasing surface fractal dimension (D_s) from 2.98 to 2.58 indicates that the copolymer micelle change from a rather rough surface to a smooth form in the course of increasing pH in aqueous solutions, and increasing D_m and D_s from 2.29 to 2.35 and 2.70 to 2.90, respectively, indicates the shrinkage of copolymer micelle to a rather compact and rough disc form by adding CoCl_2 in ethanol solvents.     

Wave interaction with a random fat fractal: dimension of the reflection coefficient

We study numerically the wave transition through one-dimensional random fat-fractal slabs, which can serve as a model of porous media. It is found that the qualitative behaviour of the scattering data essentially depends on the fractal exponent. In order to characterize the behaviour of the reflection coefficient we introduce its dimension which turns out to be between two and three. We find that the dependence of this dimension versus the fractal exponent is a non-monotonic function.     

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.     

Small angle neutron scattering studies on the interaction of cationic surfactants with bovine serum albumin

The structure of the protein-surfactant complex of bovine serum albumin (BSA) and cationic surfactants has been studied by small angle neutron scattering. At low concentrations, the CTAB monomers are observed to bind to the protein leading to an increase in its size. On the other hand at high concentrations, surfactant molecules aggregate along the unfolded polypeptide chain of the protein resulting in the formation of a fractal structure representing a necklace model of micelle-like clusters randomly distributed along the polypeptide chain. The fractal dimension as well as the size and number of micelles attached to the complex have been determined.      

A highly efficient algorithm for the generation of random fractal aggregates

A technique to generate random fractal aggregates where the fractal dimension is fixed a priori is presented. The algorithm utilizes the box-counting measure of the fractal dimension to determine the number of hypercubes required to encompass the aggregate, on a set of length scales, over which the structure can be defined as fractal. At each length scale the hypercubes required to generate the structure are chosen using a simple random walk which ensures connectivity of the aggregate. The algorithm is highly efficient and overcomes the limitations on the magnitude of the fractal dimension encountered by previous techniques.     

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.     

Measurements of Silica Aggregate Particle Growth Using Light Scattering and Thermophoretic Sampling in a Coflow Diffusion Flame

The evolution of silica aggregate particles in a coflow diffusion flame has been studied experimentally using light scattering and thermophoretic sampling techniques. An attempt has been made to calculate the aggregate number density and volume fraction using the measurements of scattering cross section from 90° light scattering with combination of measuring the particle size and morphology from the localized sampling and a TEM image analysis. Aggregate or particle number densities and volume fractions were calculated using Rayleigh–Debye–Gans and Mie theory for fractal aggregates and spherical particles, respectively. Using this technique, the effects of H₂ flow rates on the evolution of silica aggregate particles have been studied in a coflow diffusion flame burner. As the flow rate of H₂ increases, the primary particle diameters of silica aggregates have been first decreased, but, further increase of H₂ flow rate causes the diameter of primary particles to increase and for sufficiently larger flow rates, the fractal aggregates finally become spherical particles. For the cases of high flame temperatures, the particle sizes become larger and the number densities decrease by coagulation as the particles move up within the flame. For cases of low flame temperatures, the primary particle diameters of aggregates vary a little following the centerline of burner and for the case of the lowest flame temperature in the present experiments, the sizes of primary particles even decrease as particles move upward.     

Diffusive mobility of fractal aggregates over the entire Knudsen number range

We determine the effective mobility radius for fractal aggregate particles. Our method is to use static light scattering to measure the radius of gyration R(g) of the aggregates, and dynamic light scattering to measure the diffusion coefficient hence the mobility radius R(m). The range of our results can be specified by the Knudsen number Kn, which is the mean free path of the medium molecules divided by the radius of the aggregate. Our results apply to the entire range of Kn from the continuum limit (Kn=0) to the free molecular limit (Kn>1). In the continuum regime we find R(m)/R(g)=0.97+/-0.05 when the aggregate fractal dimension is D(f) approximately 2.15, and 0.70+/-0.05 when D(f) approximately 1.75. The latter result is independent of Kn for Kn < or approximately 1.3. The free molecular mobility goes as R(m)=aN(0.44+/-0.03), where a is the monomer radius and N is the number of monomers per aggregate. Since R(g) approximately aN(1/D(f)), R(m)/R(g) is not a constant when Kn is large. We find for all Kn that the functionality of R(m)/R(g) must always begin with the correct N-->1 limit, and this affects experimental observation.     

Effect of monomer geometry on the fractal structure of colloidal rod aggregates

The fractal structure of clusters formed by diffusion-limited aggregation of rodlike particles is characterized over three decades of the scattering vector q, and displays an unexpected dependence on the aspect ratio of the constituent monomers. Monte Carlo simulations of aggregating Brownian rods corroborate the experimental finding that the measured fractal dimension is an increasing function of the monomer aspect ratio. Moreover, increasing the rod aspect ratio eliminates the structural distinction between diffusion- and reaction-limited cluster aggregation that is observed for spheres.     

火灾烟颗粒分形模型和球形模型光散射的比较研究

对烟颗粒的光散射进行模拟计算是研究火灾烟颗粒光散射特性的重要手段,目前对于火灾烟颗粒光散射的数值计算多采用球形或椭球模型.实际上,火灾烟颗粒的形貌与球形和椭球均存在着显著差异.扫描电子显微镜图像表明,烟颗粒具有近似分形的结构.本文利用离散偶极近似方法计算了随机取向的火灾烟颗粒分形凝团以及同体积的球形颗粒的光散射Muller矩阵,并对两者的归一化Muller矩阵元素随散射角的分布进行了比较.研究表明:火灾烟颗粒分形模型和球形模型的归一化矩阵元素F₁₁(θ)/     

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．     

Characterization of Agglomerated and Aggregated Aerosol Particles Using Image Analysis

Accurate particle size characterization of aggregated and agglomerated particles is only possible by analysis of photographs. Both the primary particle size and the morphology of the aggregate are important experimental results. Since standard image analysis techniques for particle size analysis usually recognize only single particles, a new programme, called here Sparse Hough Transformation, was developed for the automated recognition of spherical particles within an aggregate. The method is shown to perform well, even for images with many overlapping particles. The structure of the aggregate is analysed using the fractal dimension, determined from the density-density correlation function. Finite size effects, important when dealing with aggregates containing few primary particles, are taken into account by including a cut-off function.