Scaling behavior of surface irregularity in the molecular domain: From adsorption studies to fractal catalysts

For an unexpected variety of solids, the surface topography from a few up to as many as a thousand angstroms is very well described by fractal dimension,D. This follows from measurements of the number of molecules in surface monolayers, as function of adsorbate or adsorbent particle size. As an illustration, we present a first case, amorphous silica gel, whereD has been measured independently by each of the two methods. (The agreement, 3.02±0.06 and 3.04±0.05, is excellent, and the result is modeled by a heavy generalized Menger sponge.) The examples as a whole divide into amorphous and crystalline materials, but presumably all of them are to be modeled as random fractal surfaces. The observedD values exhaust the whole range between 2 and 3, suggesting that there are a number of different mechanisms by which such statistically self-similar surfaces form. We show that fractal surface dimension entails interfacial power laws much beyond what is the source of theseD values. Examples are reactive scattering events when neutrons of variable flux pass the surface (this is of interest for locating fractal substrates that may support adlayer phase transitions); the rate of diffusion-controlled chemical reactions at fractal surfaces; and the fractal implementation of the traditional idea that the active sites of a catalyst are edge and apex sites on the surface.     

Single random walker on disordered lattices

Random walks on square lattice percolating clusters were followed for up to 2×105 steps. The mean number of distinct sites visited (S _N > gives a spectral dimension ofd _s = 1.30±0.03 consistent with superuniversality (d _s =4J3) but closer to the alternatived _s = 182/139, based on the low dimensionality correction. Simulations are also given for walkers on anenergetically disordered lattice, with a jump probability that depends on the local energy mismatch and the temperature. An apparent fractal behavior is observed for a low enough reduced temperature. Above this temperature, the walker exhibits a crossover from fractal-to-Euclidean behavior. Walks on two- and three-dimensional lattices are similar, except that those in three dimensions are more efficient.Supported by NSF Grant No. DMR 8303919 and Nato Grant No. SA 5205 RG 295J82.     

Existing biorefinery operations that benefit from fractal-based process intensification

Ion exchange, adsorption, and chromatography are examples of separation processes frequently used in today's biorefineries. The particular tasks for which these technologies can be successfully applied are highly influenced by capital cost and efficiency. There exists a potential for significantly increasing the efficiency of these processes whereas simultaneously decreasing their size and capital cost. This potential for process intensification can be realized with the use of engineered fractal equipment. The cost savings potential and the possibilities for broadening the use of fractal-based separation technologies in future biorefinery concepts is illustrated by examples of full-scale implementation in the sugar and sweetener industries.     

高分子混合体系相分离过程相结构的自相似与分形

本工作用激光光散射和光学显微镜手段研究了聚甲基丙烯酸甲醋(PMMA)和聚醋酸乙烯酯[PVAc]共混体系相分离过程。结果表明系统的相态结构呈现较强的自相似,结构函数在大波矢处具有q~(-6)的渐近行为,相结构具有分形特征,其分形维数[Fractal Dimension]D约为1,64,初步认为结构函数的标度[Scaling]行为是由于相结构的分形特征所致。     

Microstructure Evolution of Nonhydrolytic Alumina Gels

Nonhydrolitic sol-gel processes of aluminum chloride and aluminum bromide with isopropyl ether and aluminum sec-butoxide were performed at various temperatures. Based on the Arrhenius type variation of the gelation time with temperature, activation energies for the gelation were found to be in the range 19–25 Kcal/mol range. The energies were found to be sensitive to the nature of the aluminum ligands and the chemical scheme. Due to the large activation energy, it is possible to stop the reaction at any time before gelation by cooling the sol to room temperature. Small angle X-ray scattering (SAXS) of sols from the AlClAlCl₃/Pr O system shows unique development of a fractal like structure with nanometer scale order, demonstrated by discrete peaks in the SAXS data. A fractal dimension D = 1.64 was found. An aggregation scheme is proposed to explain this phenomenon. A fractal dimension of 2.4 without small scale ordering found for xerogels prepared from the AlCl₃/ASB system reflects the effect of the different precursors on the microstructure of nonhydrolytic gels.     

Lacunars: Nonfractal molecules with fractional mass dimension

Mass dimension D has been studied analytically and by computer simulation within the framework of the atomic cluster model for large numbers of lacunary molecules (lacunars) from the class of polycyclic hydrocarbons and polyphenyls. These molecules have a central framework and radial peripheral fragments. Due to the presence of lacunas (voids) between the peripheral fragments and because of growth of lacunas with increasing length of these fragments, these lacunary molecules, which are not self-similar fractal objects, have fractional dimension 1 < D _c < 2 in the region of cores. It is established that D _c(q) depends on the number q and length of peripheral fragments, on the core shape and the local symmetry and local anisotropy of the core, and on the character of space occupation by the molecule. For both classes of molecules, analytical dependences D _c(q) have been obtained that explain the results of computer simulation. Similarity and differences between lacunars and fractals are discussed.     

激光大气闪烁的分形分析

 采用分形理论分析了激光大气闪烁的统计特征。研究结果表明：在弱起伏条件下，激光大气闪烁的分形维和奇异性随光强起伏的增强而增大，而其长期相关性则减小；不同Fresnel尺度下具有相同闪烁指数的激光大气闪烁的分形特征存在着明显的差别；在强起伏条件下，有限的数据中尚未发现分形维有饱和现象，因此可以用来描述激光大气闪烁的强度。     

蛋白质互作网络的分形分析

通过盒计数法,对一系列最新研究得到的蛋白质互作网络进行分形分析．研究了这些互作网络的度指数,配对系数,相关性谱等与分形性质密切相关的属性．这些分析指出,按照现有的数据构建的蛋白质互作网络很可能还不具备分形的拓扑结构．     

煤焦分形维数及其对比热容的影响研究

比热容是煤炭热物理性质之一,在煤矿的矿井防火、防止煤与瓦斯突出、井下降温设计及煤炭加工利用(如煤炭的燃烧、气化、焦化、液化等)等方面是关键参数之一,对提高煤炭热能利用率、提高经济效益、减少环境污染等,具有非常重要的意义。比热容的影响因素很多,如煤化程度、水分质量分数、热解温度等,煤焦微观结构的影响也是其中很重要的一方面。分形几何由Mandelbrot 1982年创立,是定量描述自相似或自相关等不规则形体的工具。研究表明,煤焦微观结构具有分形特征。在煤焦分形的研究中,常用的实验技术方法为吸附法、小角度X射线散射法和孔度法,采用扫描电子显微镜和数字图象处理方法研究煤焦的分形结构,能更加深入地理解其分形维数与性能的关系。