Studies of the collision frequency of ideal gas particles with the surfaces of the objects created using the ballistic deposition technique

This paper is a continuation of our studies of the collision frequency of ideal gas particles with the rough/fractal surfaces. Here, we applied a more realistic surface growth model, i.e. ballistic deposition for creation of fractal objects. We found that the collision frequency with irregular surfaces is the linear function of pressure and this frequency per unit pressure is quite a complicated function of the surface fractal dimension as well as the diameter of colliding particle. The collision frequency with rough surfaces cannot be exactly described by the analytical formula called the Langmuir-Hertz equation. However, we have stated that the deviations of the true collision frequency from the Langmuir-Hertz prediction are not huge and in typical catalytic studies the error introduced by replacing the true frequency by the Langmuir-Hertz prediction can be safely neglected. We have also studied the probability of finding on the surface an atom which has been hit a certain number of times by a gas particle. This probability reveals an interesting behaviour for small gas particles, i.e. it perfectly correlates with the number of directions from which the surface atom is accessible from the gas phase. We have also estimated the evolution of the adsorption energy distribution with the increasing fractal dimension of the surface in the ballistic deposition.     

Influence of fractal surface dimension on the dissolution process of sparingly soluble CaCO<Subscript>3</Subscript> microparticles

The dissolution process of sparingly soluble CaCO₃ microparticles and how the fractal surface dimension of the particles changes during dissolution is analyzed. The particles and the dissolution process are studied using scanning electron microscopy, X-ray diffraction, nitrogen adsorption, laser diffraction and conductance measurements. Ball milling of the particles is shown to maintain the particle crystallinity, and to introduce an increased fractal surface dimension in the 1–10 μm size range. Dissolution is found to increase the surface dimension of initially smooth particles and to maintain the fractal surface roughness of milled particles. The dissolution process increases the relative number of small particles (50 nm–1 μm) whereas the larger ones decrease in size. The solubility of the milled fractal particles was ∼1.8 times higher than that for the initially smooth ones. The presented findings show that developing methods for increasing the fractal surface roughness of particles should be of interest for improving the solubility of poorly soluble drug candidates.     

A Particle Resistance Model for Flow through Porous Media

A particle model for resistance of flow in isotropic porous media is developed based on the fractal geometry theory and on the drag force flowing around sphere. The proposed model is expressed as a function of porosity, fluid property, particle size, fluid velocity （or Reynolds number） and fractal characters D f of particles in porous media. The model predictions are in good agreement with the experimental data. The validity of the proposed model is thus verified.     

Scale-dependent nature of the surface fractal dimension for bi- and multi-disperse porous solids by mercury porosimetry

The surface fractal dimension was calculated by using a mathematical model and mercury intrusion data for a variety of bi- and multi-disperse porous solids including silica gels, alumina pellets, and building stones. The mathematical model was obtained by modifying the well-established scaling relation published previously [Ind. Eng. Chem. Res., 34 (1995) 1383-1386]. It was also verified by comparing with the theoretical surface fractal dimensions for regular fractal structures (Skerpinski tetrahedron and Menger sponge) and the calculated surface fractal dimensions for silica gel and alumina particles via the linear fitting method established previously. The calculation results for various bi- and multi-disperse porous solids have demonstrated that the scale-dependent nature of the surface fractal dimension is ubiquitous. The difference in the surface fractal dimension between pore size intervals usually exists. The estimation of the surface fractal dimension on an average stand may lead to erroneous results.     

Nanodroplets on rough hydrophilic and hydrophobic surfaces

We present results of Molecular Dynamics (MD) calculations on the behavior of liquid nanodroplets on rough hydrophobic and hydrophilic solid surfaces. On hydrophobic surfaces, the contact angle for nanodroplets depends strongly on the root-mean-square roughness amplitude, but it is nearly independent of the fractal dimension of the surface. Since increasing the fractal dimension increases the short-wavelength roughness, while the long-wavelength roughness is almost unchanged, we conclude that for hydrophobic interactions the short-wavelength (atomistic) roughness is not very important. We show that the nanodroplet is in a Cassie-like state. For rough hydrophobic surfaces, there is no contact angle hysteresis due to strong thermal fluctuations, which occur at the liquid-solid interface on the nanoscale. On hydrophilic surfaces, however, there is strong contact angle hysteresis due to higher energy barrier. These findings may be very important for the development of artificially biomimetic superhydrophobic surfaces.     

Boundary Fractal Dimensions from Sections of a Particle Profile

The fractal dimension of particles is commonly evaluated from complete particle boundaries. In this work, a study has been made of the self-similar nature of complete and incomplete boundary profiles of a range of morphologically different copper powders. Boundary images were captured from SEM micrographs of particle boundaries at a range of magnifications up to nearly 14000X. An algorithm was developed to compute the fractal dimension of boundary segments. This algorithm was tested against the Koch Island fractal, and was found to give excellent estimates of the fractal dimension. For the particle system studied, the boundary fractal was found to be sensitive to magnification with appreciable drops in value at high magnification. This demonstrates that the particles studied did not have true fractal boundaries and the use of fractal theory to study particle surface roughness must be used with caution.     

Surface pore tension and adsorption characteristics of polluted sediment

Most natural sediment particles have numerous pores and a complex surface texture which facilitates their adsorption of contaminants. Particle surface structure, therefore, is an important instrumental factor in the transport of contaminants, especially in water environments. This paper reports on the results of adsorption-desorption experiments to analyze polluted sediment surface pore tension characteristics performed on samples from the bottom of Guanting Reservoir. In our analysis, the Frenkel-Halsey-Hill (FHH) equation is applied to calculate the fractal dimensions of particles to quantify the surface roughness and pore tension characteristics. The results show that the surface fractal dimensions of sediment particle surfaces normally measure from 2.6 to 2.85. The volume of pores smaller than 10 nm changes significantly after being contaminated with pollutants and the fractal dimension decreases because the pores adsorb the contaminants. Supported by the National Science Fund for Distinguished Young Scholars (Grant No. 50325929) and the National Key Technology R&D Program during the 11th Five-Year Plan Period (Grant No. 2006BAB05B05)     

Dielectric constant of porous ultra low-k dielectrics by fractal-Monte Carlo simulations

The effective dielectric constant of porous ultra low-k dielectrics is simulated by applying the fractal geometry and Monte Carlo technique in this work. Based on the fractal character of pore size distribution in porous media, the probability models for pore diameter and for effective dielectric constant are derived. The proposed model for the effective dielectric constant is expressed as a function of the dielectric coefficient of base medium and the volume fractions of pores and base medium, fractal dimension for pores, the pore size, as well as random number. The Monte Carlo simulations combined with the fractal geometry are performed. The predictions by the present simulations are shown in good accord with the available experimental data. The proposed technique may have the potential in analyzing other properties such as electrical conductivity and thermal conductivity in porous ultra low-k dielectrics.     

Molecular dynamics study of the equilibrium flux of gas molecules to a fractal/rough surface: Effects of gas atom diameter

The frequency of collisions of ideal gas molecules (argon) with a rough surface has been studied. The rough/fractal surface was created using the random deposition technique. By applying various depositions the surface roughness was controlled and, as a measure of irregularity, the fractal dimensions of the surfaces were determined. The surfaces were next immersed in ideal gas and the numbers of collisions with these surfaces were counted. The calculations were carried out using the simplified molecular dynamics simulation technique (only hard core repulsions were assumed). The calculations were performed for various ratios of gas phase atoms diameter to the surface substrate atoms diameter. The results obtained showed that the size of a gas phase atom has crucial influence on the relation between the frequency of collision and the surface fractal dimension     

KDP晶体单点金刚石车削表面形貌分形分析

 分别使用2维和3维分形方法对单点金刚石车削加工的KDP晶体表面形貌进行了分析，并对表面的3维分形维数和3维粗糙度表征参数进行了比较，分析了二者对表面形貌表征的差异。使用2维轮廓分形方法计算了KDP晶体表面圆周各方向上的分形维数。通过分析得出：3维分形维数与表面粗糙度值成反比关系；使用单点金刚石车削方法加工KDP晶体会形成各向异性特征明显的已加工表面，在一定程度上容易形成小尺度波纹；已加工表面是否具有明显的小尺度波纹特征与表面粗糙度值并无直接关系，但与其表面轮廓分形状态分布密切相关；KDP晶体表面2维功率谱密度与其分形状态具有相近的方向性特征。     

Influence of surface roughness on superhydrophobicity

Superhydrophobic surfaces, with a liquid contact angle theta greater than 150 degrees , have important practical applications ranging from self-cleaning window glasses, paints, and fabrics to low-friction surfaces. Many biological surfaces, such as the lotus leaf, have a hierarchically structured surface roughness which is optimized for superhydrophobicity through natural selection. Here we present a molecular dynamics study of liquid droplets in contact with self-affine fractal surfaces. Our results indicate that the contact angle for nanodroplets depends strongly on the root-mean-square surface roughness amplitude but is nearly independent of the fractal dimension D(f) of the surface.     

Fractal structure of digital speckle patterns produced by rough surfaces

We report on the fractal analysis of digital speckle patterns experimentally generated using an optical setup to record the light scattered from metallic rough surfaces in the normal direction. Using the differential box counting technique, we have calculated the fractal dimension of digital speckle patterns for six samples with different roughness. Our results show a quadratic dependence between the surface roughness and the fractal dimension of the corresponding digital speckle pattern. As an application a method to determine the surface roughness of metallic surfaces is proposed.     

Application of Fractal Geometry in Quantitative Characterization of Aerosol Morphology

The aerosol morphologies of six types of particles plus one other from a different source were characterized quantitatively in terms of fractal dimensions based on boundary, projected area and surface roughness. Except for the particle with a smooth boundary, multifractal features were observed on the projected boundary of aerosols. It was found that the fractal dimensions of the aerosol boundaries in the high-resolution region ranged from 1.00 to 1.13, whereas those in the low-resolution region were distributed from 1.04 to 1.91. The compactness of a particle could be examined in terms of the fractal dimension of its projected area such that a solid particle gave a value of 2.00, a hollow particle with pores inside gave a value of 1.93 and a chain agglomerate gave an even lower value of 1.81. Aerosol topological features retrieved from the gray levels of a microimage were used to illustrate the difference in surface roughness or distinguish particle deposition on the substrate from that of agglomeration. A value of 2.49 was found for a hollow sphere in contrast to a value of 2.26 for a round pollen particle. Finally, a coordinate system was established by employing the computed fractal dimensions as axes to accommodate the particles characterized in this work and the Euclidean distance of a point from the origin was shown to be a potential composite index for aerosol morphology.     

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.     

On direction-invariance of fractal dimension on a surface

It is shown that fractal dimensions along line transects of a stationary, stochastic surface are all identical to one another, save that in one special direction the dimension may be less than in all others. We discuss the implications of this result for practical measurement of surface roughness and for interpretation of fractal dimension. It is argued that the result provides significant new motivation for employing fractal dimension as a canonical measure of surface roughness. Nevertheless, in view of the virtual direction-invariance of fractal dimension, measurement of scale assumes new significance. This matter is addressed.     

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.     

Morphology of Nanostructured Films for Environmental Applications: Simulation of Simultaneous Sintering and Growth

A sequential Brownian dynamics approach was used to establish the morphological evolution of a nanostructured particle deposit accounting for random diffusion, particle–particle and particle–surface interactions through van der Waals forces, and sintering of deposited particles. Monodisperse (30nm radius) titanium dioxide particles were used in the simulations. A linear sintering law rate expression was used to account for the decrease in total surface area of the deposit. Characteristics such as packing thickness, total surface area, and fractal dimension are reported as a function of time during the deposition process. Sintering resulted in higher fractal dimensions (as defined) for the deposits, and elevated temperatures resulted in more compact deposits.     

The Fractal Characteristic of Particles in Granular Material Flows and Its Effect on Effective Thermal Conductivity

According to the fact that many pulverized particles possess fractal characteristic, a fractal model for studying fine particles in granular material flows is first proposed. An expression of particles' fractal distribution is derived to describe the relationship between the particle fractal dimensions and particle velocity distribution function. In accordance with this model, the theoretical particle effective thermal conductivity is derived. The analytical results show that for the small Biot-Fourier number, the effective thermal conductivity increases with the square root of the granular temperature. For very large Biot-Fourier number, the effective thermal conductivity linearly increases with the granular temperature. Numerically calculated results show that the thermal conductivity increases with the particle size fractal dimensions and decreases with the particle surface fractal dimensions.     

规则表面形貌的分形和多重分形描述

以6种具有典型特征的生成元构造了6个具有相同rms粗糙度的规则表面,用变分法计算了这些表面的分形维数,结果表明,分形维数可以将具有相同rms粗糙度的表面区分开来,它定量表征了表面的总体形貌。进一步将多重分形的方法应用到对这些表面的分析中,发现多重分形谱可以全面反映表面概率的分布特征。多重分形谱的宽度可以定量表征表面的起伏程度,多重分形谱最大、最小概率子集维数的差别可以统计表面最大、最小概率处的数目比例。 关键词： 粗糙度 分形维数 多重分形谱