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.     

Dynamical fractal equilibrium in diffusion limited reorganization

The evolution of a compact structure under surface evaporation, diffusion and subsequent re-deposition is studied numerically in d = 2. After a transitory regime, the evolution leads to the spontaneous apparition of a dynamic fractal morphology exhibiting a dimension of order Df = 1.6. The final morphology plays the role of a statistical dynamic attractor. This effect indicates that fractal structures can result from the irreversible evolution of dynamical processes towards internal equilibrium.     

Dependence of fragmentation behavior of colloidal aggregates on their fractal structure

The fragmentation dynamics of aggregate of non-Brownian particles in shear flow is investigated numerically. The breakup behaviors of aggregates having the same connectivity but the different space-filling properties are examined. The Lagrangian particle simulation in a linear flow field is performed. The effect of surrounding fluid on the motion of multiple particles is estimated by Stokesian dynamics approach. The inter-particle force is calculated from the retarded van der Waals potential based on the Lifshitz theory. The results obtained in this work indicate that the fragmentation behavior of colloidal aggregates depends on their fractal structure. However, if the resultant aggregate size is smaller than the critical one, the fragmentation behavior shows the universality regardless of their original structure. Furthermore, the restructuring of aggregate in shear flow and its effect on the fragmentation process are also discussed.     

A model to describe the settling behavior of fractal aggregates

A model to predict fractal dimension from sedimentating fractal aggregates has been successfully developed. This model was developed using the settling rate and size data of fractal aggregates. In order to test the validity of the model, a purpose-built settling rig, equipped with lens with magnification of 1200x, which can capture images of particles/flocs down to 2 microm in diameter was used. The performance and technique of the settling rig were validated by comparing the measured settling rates of 30- and 50.7-microm standard particles with their theoretical settling rates calculated using Stokes' law. The measured settling rates were within 10% agreement with the calculated Stokes' velocities. The settling rates and sizes of the particles/flocs were analyzed using image analysis software called WiT 5.3. The maximum temperature gradient across the settling column was 0.1 degrees C, which effectively eliminated convective currents due to temperature differences in the settling column. A total of 1000 calcium phosphate flocs were analyzed. Calcium phosphate flocs with fractal dimensions varying from 2.3 to 2.8 were generated via orthokinetic aggregation. Measurements of fractal dimensions, using light scattering, were done simultaneously with the settling experiments and they were found to be constant. The fractal dimensions calculated using the model agreed with those obtained by light scattering to within 12%.     

Prediction of three-dimensional fractal dimensions using the two-dimensional properties of fractal aggregates

Fractal dimension analysis using an optical imaging analysis technique is a powerful tool in obtaining morphological information of particulate aggregates formed in coagulation processes. However, as image analysis uses two-dimensional projected images of the aggregates, it is only applicable to one and two-dimensional fractal analyses. In this study, three-dimensional fractal dimensions are estimated from image analysis by characterizing relationships between three-dimensional fractal dimensions (D(3)) and one (D(1)) and two-dimensional fractal dimensions (D(2) and D(pf)). The characterization of these fractal dimensions were achieved by creating populations of aggregates based on the pre-defined radius of gyration while varying the number of primary particles in an aggregate and three-dimensional fractal dimensions. Approximately 2000 simulated aggregates were grouped into 33 populations based on the radius of gyration of each aggregate class. Each population included from 15 to 115 aggregates and the number of primary particles in an aggregate varied from 10 to 1000. Characterization of the fractal dimensions demonstrated that the one-dimensional fractal dimensions could not be used to estimate two- and three-dimensional fractal dimensions. However, two-dimensional fractal dimensions obtained statistically, well-characterized relationships with aggregates of a three-dimensional fractal characterization. Three-dimensional fractal dimensions obtained in this study were compared with previously published experimental values where both two-dimensional fractal and three-dimensional fractal data were given. In the case of inorganic aggregates, when experimentally obtained three-dimensional fractal dimensions were 1.75, 1.86, 1.83+/-0.07, 2.24+/-0.22, and 1.72+/-0.13, computed three-dimensional fractal dimensions using two-dimensional fractal dimensions were 1.75, 1.76, 1.77+/-0.04, 2.11+/-0.09, and 1.76+/-0.03, respectively. However, when primary particles were biological colloids, experimentally obtained three-dimensional fractal dimensions were 1.99+/-0.08 and 2.14+/-0.04, and computed values were both 1.79+/-0.08. Analysis of the three-dimensional fractal dimensions with the imaging analysis technique was comparable to the conventional methods of both light scattering and electrical sensing when primary particles are inorganic colloids.     

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.     

Image processing of the fractal aggregates composed of nanoparticles

The image-processing algorithms are described for chain branched aggregates composed of spherical particles. The processing procedure is based on setting off circles corresponding to the primary particles (spherules) in the images, followed by the construction of a digital model representing a set of diameters and coordinates of all spherules and their centers. The computational procedures are considered in which the digital model is used to calculate a series of parameters characterizing the aggregate’s structure and morphology, including fractal dimension for an individual aggregate and average dimension for a set of aggregates. The “clearance radius” of the aggregate is calculated as a half of geometric mean of the aggregate’s length L and width W. To determine L and W, the algorithm based on searching for the minimum-area (LW) rectangle circumscribing the aggregate’s contour is proposed.     

A first-principles description of proton-driven spin diffusion

Herein we design a reduced Liouville space for the simulation of proton-driven spin diffusion. Using this approach, the experimentally observed carbon-13 polarisation transfer in a powder sample undergoing magic-angle spinning is quantitatively described, directly from crystal geometry and without any adjustable parameters.     

Viscosity of nanofluids: particle shape and fractal aggregates

Colloidal dispersions of nanoparticles in thermal base fluids are known to alter their spherical shapes thereby affecting their surface properties. This aspect has been investigated with respect to the effective viscosity of nanofluids presuming the particles to acquire the shape of prolate spheroid. Also, the contributions of the interfacial layer formed around these particles and their possible agglomeration has been taken in to account. The analysis has been carried out by modifying the Krieger and Dougherty model. The relative viscosity of these nanofluids has been computed as a function of volume fraction, particle size and the eccentricity of the particle. The model also incorporates the concept of fractal dimensions. The results thus obtained compare significantly well with the available experimental data and reaffirms an improvement over earlier models.     

On techniques for the measurement of the mass fractal dimension of aggregates

A review is presented of a number of techniques available for the characterisation of the structure of aggregates formed from suspensions of sub-micron particles. Amongst the experimental techniques that have been commonly used are scattering (light, X-ray or neutron), settling and imaging and these are the focus of this work. The theoretical basis for the application of fractal geometry to characterisation of flocs and aggregates is followed by a discussion of the strengths and limitations of the above techniques. Of the scattering techniques available, light scattering provides the greatest potential for use as a tool for structure characterisation even though interpretation of the scattered intensity pattern is complicated by the strong interaction of light and matter. Restructuring further complicates the analysis. Although settling has long been used to characterise particle behaviour, the absence of an accurate permeability model limits the technique as a means of determining the porosity of fractal aggregates. However, it can be argued that the determination of fractal dimension is relatively unaffected. The strength of image analysis lies in its ability to provide a great deal of information about particle morphology and the weaknesses lie in the difficulties with image processing and sample size as this is a particle counting technique. There are very few papers which compare the fractal dimension measured by more than one technique. Light scattering potentially provides a useful tool for checking settling results. However, further work is required to develop proper models for aggregate permeability and flow-through effects.     

Spectroscopic studies of fractal aggregates of silver nanospheres undergoing local restructuring

We present an experimental spectroscopic study of large random colloidal aggregates of silver nanoparticles undergoing local restructuring. We argue that such well-known phenomena as strong fluctuation of local electromagnetic fields, appearance of "hot spots" and enhancement of nonlinear optical responses depend on the local structure on the scales of several nanosphere diameters, rather than the large-scale fractal geometry of the sample.     

Fractal structure of asphaltene aggregates

A photographic technique coupled with image analysis was used to measure the size and fractal dimension of asphaltene aggregates formed in toluene-heptane solvent mixtures. First, asphaltene aggregates were examined in a Couette device and the fractal-like aggregate structures were quantified using boundary fractal dimension. The evolution of the floc structure with time was monitored. The relative rates of shear-induced aggregation and fragmentation/restructuring determine the steady-state floc structure. The average floc structure became more compact or more organized as the floc size distribution attained steady state. Moreover, the higher the shear rate is, the more compact the floc structure is at steady state. Second, the fractal dimensions of asphaltene aggregates were also determined in a free-settling test. The experimentally determined terminal settling velocities and characteristic lengths of the aggregates were utilized to estimate the 2D and 3D fractal dimensions. The size-density fractal dimension (D(3)) of the asphaltene aggregates was estimated to be in the range from 1.06 to 1.41. This relatively low fractal dimension suggests that the asphaltene aggregates are highly porous and very tenuous. The aggregates have a structure with extremely low space-filling capacity.     

三种金属硫蛋白聚合物静电效应的研究

考察了三种金属硫蛋白(大鼠金属硫蛋白亚型II,兔肝金属硫蛋白亚型I和兔肝金属硫蛋白亚型II)的单体、二聚体和三聚体在pH为5.6-8.5和10.6两种缓冲条件下的静电势分布。其中大鼠金属硫蛋白亚型II的结构直接来自于晶体数据,兔肝金属硫蛋白亚型I和II的结构则通过同源蛋白模型搭建。三种金属硫蛋白的静电势通过有限差分方法求解Poisson-Boltzmann方程得到。对于三种金属硫蛋白的二聚体,pH为5.6-8.5时,单体和单体之间的静电势分布具有明显的互补性;但pH≥10.6时,这种互补性会大大削弱。对于三种金属硫蛋白的三聚体,单体和二聚体之间主要表现为静电排斥,而且pH在10.6下的静电排斥力明显强于pH为5.6-8.5时的静电排斥。     

An alternative description of the paulingite structure

The paulingite structure can be described as an invariant transformation of the gismondine structure. Packets of a fourling construction of the gismondine structures, which is closely related to the merlinoite structure, are used to build the structure of paulingite as an interpenetrating sixling.     

Criteria of parameter selection for quantum chemical simulation of the electronic structure of rare-earth ions by the scattered wave cluster method

Electronic structure calculations of the Ce³⁺ ion in terms of a molecular cluster model in a quasirelativistic approximation are reported. The influence of the self-interaction correction and variations of muffintin parameters on the calculation results is discussed. A method is proposed for estimating the 4f→5d transition energy in the cerium ion using the SCF-SW formalism. Translated fromZhurnal Strukturnoi Khimii, Vol. 41, No. 3, pp. 449-456, May–June, 2000.     

Unlinked cluster effects in limited CI calculations of molecular properties

The unlinked cluster effects in limited CI calculations of dipole moments and polarizabilities are estimated by using the approximate corrections due to Davidson and Siegbahn. The results obtained for FH, H₂O, NH₃, and CH₄ indicate that the limited CI results for molecular electric properties need to be corrected for the erratic treatment of unlinked clusters.     

Atomic view of cluster diffusion on metal surfaces

Field ion microscope measurements show a strong correlation between the mobility and shape of small clusters on fcc(100) metal surfaces. For self-diffusion on Rh(100) this correlation leads to an oscillatory behavior in the activation energy of surface diffusion as a function of cluster size. Comparison of the measured activation energies to previous theoretical calculations indicate that the mechanism of cluster diffusion involves individual displacements of edge atoms (i.e., perimeter diffusion). The rate-determining step in the migration of clusters is the partial detachment of one of the perimeter atoms. The relative ease of adatom motion along straight edges of stationary clusters also permits measurements of diffusion barriers at steps — measurements useful in the interpretation of fractal vs. compact island growth on fcc metal surfaces.     

The origin of anomalous enhancement of electromagnetic fields in fractal aggregates of metal nanoparticles

It is shown that the anisotropy of the environment of metal nanoparticles with plasmon absorption in fractal aggregates is the most important and universal characteristic underlying their unique electrodynamic properties. It is noted that it is this morphological feature, but not the fractal distribution of particles in aggregates as such, that plays the dominant role in the manifestation of the enhancement of a local field. In this case, fractal aggregates possess the ability to enhance local electromagnetic fields only owing to their inherent local anisotropy; macroscopic characteristics of aggregates do not markedly influence their electrodynamic interactions with the external field. The quantitative characteristic of local anisotropy is introduced. Statistical correlation between the factor of local anisotropy and fractal dimension D of aggregates is established within the range 1.6 < D < 2.8. It is disclosed that the local anisotropy is independent of the fractal dimension within the wide range (1.6 < D < 2.5) except for the range D > 2.5 corresponding to aggregates with close-packed particles where the factor of local anisotropy tends toward zero. Strong correlation in the spatial arrangement of particles with the largest local anisotropy of the environment in aggregates and the strength of the local electromagnetic field is established using aggregates of silver nanoparticles as an example for the spectrum in the visible range; a polarization dependence of this correlation is revealed. It is noted that parameters of local anisotropy can be used to determine the degree of imperfection of colloidal crystals via optical methods.     

A review of anomalous diffusion phenomena at fractal interface for diffusion-controlled and non-diffusion-controlled transfer processes

This article reviewed anomalous diffusion phenomena coupled with facile and sluggish charge-transfer reactions at fractal interface. Firstly, the generalised diffusion equation (GDE) involving a fractional derivative which describes diffusion towards and from fractal interface was briefly introduced. And then, anomalous diffusion coupled with facile charge-transfer reaction at fractal interface, i.e., diffusion-controlled transfer process across fractal interface, was mathematically examined by the generalised Cottrell, Sand, Randles-Sevcik and Warburg equations theoretically derived from the analytical solutions to the GDE under the semi-infinite boundary condition. Finally, in order to provide a guideline in analysing anomalous diffusion coupled with sluggish charge-transfer reaction at fractal interface, i.e., non-diffusion-controlled transfer process across fractal interface, this review covered the recent researches into the effect of surface roughness on non-diffusion-controlled transfer process within the intercalation electrodes.