分形理论及其在聚集生长研究中的应用

分形理论是现代非线性科学的一个重要分支,近年来发展越来越快,应用研究也越来越广泛。本文简要介绍了分形的基本理论及其在聚集生长研究中的应用,最后展望了分形理论的应用前景及其发展前景。     

胶体系统中的非线性生长与聚集

聚苯乙烯胶体可以实验条件控制下成为不同的准二维分形聚集生长系统，由于聚集机制的不同，聚集集团的和聚集过程的动力学表现丰富的多彩的特性，这些生长模拟系统可以分别作为自相似分形生长，自仿射分形生长，渗流集团等多种非线性生长的实验体系。     

纳米颗粒聚集形态对纳米流体导热系数的影响

纳米流体中悬浮的纳米颗粒可以增强其导热性能已经得到广泛认可,然而纳米流体颗粒增强传热的机理目前尚不清楚.研究表明,纳米颗粒的聚集是纳米流体导热系数增大的重要机制,而且纳米颗粒聚集的形态对纳米流体的导热系数有重要影响,但是目前的导热系数模型大多是建立在Maxwell有效介质理论的"静态"和"均匀分散"假设基础上.本文用平衡分子动力学模拟Cu-Ar纳米流体,采用Green-Kubo公式计算导热系数,采用Schmidt-Ott关系式计算不同聚集形态下的分形维数.对比导热系数与分形维数可以发现:在相同体积分数下,较低的分形维数会有更高的导热系数,分析了分形维数与导热系数的定量关系.此外,通过径向分布函数可以看出纳米颗粒紧密聚集与松散聚集的差异,基液分子在纳米颗粒附近的纳米薄层中处于动态平衡状态.研究结果有助于理解纳米颗粒聚集形态对导热系数的影响机理.     

一维随机成核生长模型的边界效应

通过计算机模拟，研究周期边界与固定边界对一维随机成生长模型的聚集生长图形形状及相应的分形维数的影响。     

分形理论在物理实验教学中应用的探索

综述了作者近年来将分形理论引入基础物理实验教学中的尝试与实践，介绍了分形维数的测定方法及其应用，阐述了教师参与适量的科学研究是实现基础物理实验教学现代化的重要保证。     

分子聚集溶液理论及其应用

本文应用经典热力学和分子热力学方法建立了分子聚集溶液理论。该理论具有通用性；它不仅适用于非电解质溶液，而且也可应用于电解质溶液。另外，溶液的分子聚集度参数与组分浓度的相关性规律还可用来描述电解质溶液的传递性质。     

分形理论在光谱识别中的应用

分形理论是研究一类不规则、混乱复杂,但其局部和整体具有相似性体系的科学。分形维数是分形理论中用于描述对象的不规则度和自相似性的基本度量。文章以符合朗伯-比尔定律的光谱信号为研究对象,在概述分形几何基本原理的基础上,提出了以分形维数作为光谱识别特征的方法,运用相空间重构得出了光谱信号的分形维数,通过对光谱信号的分形维数进行比较,达到识别不同光谱的目的,最后举例对该方法进行了说明。     

分形—它的应用与进展讲座第五讲 如何测量分形维数

在传统的欧几里德几何中研究的结构,比如正方形、圆、直线、柱体等,都可以用一定的特征长度来完全描述.比如知道圆的半径,就可以画出来.对于自然界的几何结构,象树、云、山脉、波浪、海岸线等,都是不规则的,没有特征长度,如何来定量特征这些结构呢? Mandelbrot发现在这些不规则结构中有一类非常普遍的结构,从不同的尺度范围来看,它们与整体是自相似的[1,2].比如树,它的一支树枝与整体是自相似的.用放大镜来观察水中的旋涡与不用放大镜时看到的相似.望远镜中看到的云的那部分与不用望远镜时看到的整块也是相似的.对于这种实际的客体,用自相…     

掺杂NaCl水溶液结晶形貌的分形研究

水溶液结晶是研究分形形貌的重要实验之一。本文通过改变溶液浓度,溶质组分和结晶温度等宏观实验参数,用分形维数的方法研究了这些参数对结晶形貌的影响。指出在低浓度下的非平衡态中溶液结晶形貌具有分形结构;不同晶体的生长作用力相互竞争,混乱度最大的情况下形成的分形结构最凸出,生长作用力是影响晶体形貌的主要因素;同时,分子热运动也是影响结晶形貌的因素之一。     

硫酸铜电解沉积的分形与电导率研究

用电化学的方法,制备出具有分形结构的金属铜样品,计算了不同电解电压下铜样品的分形维数,并求出其电导率.发现随着电解电压的增大,分形结构的维数趋于增大,而铜样品的电导率减小.     

Competition Between Self-birth and Catalyzed Death in Aggregation Growth with Catalysis Injection

We propose two irreversible aggregation growth models of aggregates of two distinct species （.4 and B） to study the interactions between virus aggregates and medicine efficacy aggregates in the virus-medicine cooperative evolution system. The A-species aggregates evolve driven by self monomer birth and B-species aggregate-catalyzed monomer death in model I and by self birth, catalyzed death, and self monomer exchange reactions in model II, while the catalyst B-species aggregates are assumed to be injected into the system sustainedly or at a periodic time-dependent rate. The kinetic behaviors of the A-species aggregates are investigated by the rate equation approach based on the mean-field theory with the self birth rate kernel IA（k） = Ik, catalyzed death rate kernel JAB（k） = Jk and self exchange rate kernel KA （k, l） = Kkl. The kinetic behaviors of the A-species aggregates are mainly dominated by the competition between the two effects of the self birth （with the effective rate I） and the catalyzed death （with the effective rate JB0）, while the effects of the self exchanges of the A-species aggregates which appear in an effective rate KAo play important roles in the cases of I 〉 JBo and I = JBo. The evolution behaviors of the total mass M1^A（t） and the total aggregate number MA（t） are obtained, and the aggregate size distribution ak（t） of species A is found to approach a generalized scaling form in the case of I ≥ JBo and a special modified scaling form in the case of I 〈 JB0. The periodical evolution of the B-monomers concentration plays an exponential form of the periodic modulation.     

Competition Between Self-birth and Catalyzed Death in Aggregation Growth with Catalysis Injection

We propose two irreversible aggregation growth models of aggregates of two distinct species (A and B) to study the interactions between virus aggregates and medicine efficacy aggregates in the virus-medicine cooperative evolution system. The A-species aggregates evolve driven by self monomer birth andB-species aggregate-catalyzed monomer death in model I and by self birth, catalyzed death, and self monomer exchange reactions in model II, while the catalyst B-species aggregates are assumed to be injected into the system sustainedly or at a periodic time-dependent rate. The kinetic behaviors of the A-species aggregates are investigated by the rate equation approach based on the mean-field theory with the self birth rate kernel I_A(K)=Ik, catalyzed death rate kernel J_AB(k)=Jk and self exchange rate kernel K_A(k,l)=Kkl. The kinetic behaviors of the A-species aggregates are mainly dominated by the competition between the two effects of the self birth (with the effective rate I) and the catalyzed death (with the effective rate JB₀), while the effects of the self exchanges of the A-species aggregates which appear in an effectiverate KA₀ play important roles in the cases of I>JB₀ and I=JB₀. The evolution behaviors of the total mass M^A(t)₁ and the total aggregate number M^A(t)₀ are obtained, and the aggregate size distribution a_k(t) of species A is found toapproach a generalized scaling form in the case of I ≧ JB₀ and a special modified scaling form in the case of I0. The periodical evolution of the B-monomers concentration plays an exponential form of the periodic modulation.     

Fractal Aggregation Under Rotation

By means of the Monte Carlo simulation, a fractal growth model is introduced to describe diffusion-limited aggregation (DLA) under rotation. Patterns which are different from the classical DLA model are observed and the fractal dimension of such clusters is calculated. It is found that the pattern of the clusters and their fractal dimension depend strongly on the rotation velocity of the diffusing particle. Our results indicate the transition from fractal to non-fractal behavior of growing cluster with increasing rotation velocity, i.e. for small enough angular velocity ω the fractal dimension decreases with increasing ω, but then, with increasing rotation velocity, the fractal dimension increases and the cluster becomes compact and tends to non-fractal.     

直线加速器中的非线性空间电荷效应和发射度增长

直线加速器中束团的非线性空间电荷效应是引起发射度增长的重要原因之一．根据计算在屏蔽筒中非均匀分布京团的空间电荷效应的普遍方法，推导了直线加速器圆波导中几种非均匀分布束团的非线性场能公式，并给出数值计算结果．讨论了由非线性所引起发射度增长．     

Fractal Aggregation Under Rotation

By means of the Monte Carlo simulation, a fractal growth model is introduced to describe diffusion-limited aggregation (DLA) under rotation. Patterns which are different from the classical DLA model are observed and the fractal dimension of such clusters is calculated. It is found that the pattern of the clusters and their fractal dimension depend strongly on the rotation velocity of the diffusing particle. Our results indicate the transition from fractal to non-fractal behavior of growing cluster with increasing rotation velocity, i.e. for small enough angular velocity ω; thefractal dimension decreases with increasing ω;, but then, with increasing rotation velocity, the fractal dimension increases and the cluster becomes compact and tends to non-fractal.     

Random aggregation and random-walking center of mass

Two random aggregation models are used in demonstrating the properties of the random displacementsr _i of the center of mass of aggregating particles. It is found that r _i is a randomly decreasing sequence that scales with the cluster size (steps)s and _i ^=1/s r _i s ^1/D , whereD is the fractal dimension. The center-of-mass random walk is a consistent representation of the dynamics of aggregation.     

平行吸引力场中的凝聚生长

利用DLA模型,研究平行力场对凝聚物生长的影响.结果表明:力场使凝聚物的生长失去对称性,粒子凝聚密度增加,且引力场越强,密度值越大;同时凝聚生长出现不可再生长区域,凝聚物生长迅速由生长区过渡到饱和区.     

Diffusion-Limited Aggregation with Polygon Particles

Diffusion-limited aggregation (DLA) assumes that particles perform pure random walk at a finite temperature and aggregate when they come close enough and stick together. Although it is well known that DLA in two dimensions results in a ramified fractal structure, how the particle shape influences the formed morphology is still unclear. In this work, we perform the off-lattice two-dimensional DLA simulations with different particle shapes of triangle, quadrangle, pentagon, hexagon, and octagon, respectively, and compare with the results for circular particles. Our results indicate that different particle shapes only change the local structure, but have no effects on the global structure of the formed fractal cluster. The local compactness decreases as the number of polygon edges increases.     

Diffusion-Limited Aggregation with Polygon Particles

Diffusion-limited aggregation (DLA) assumes that particles perform pure random walk at a finite temperature and aggregate when they come close enough and stick together. Although it is well known that DLA in two dimensions results in a ramified fractal structure, how the particle shape influences the formed morphology is still unclear. In this work, we perform the off-lattice two-dimensional DLA simulations with different particle shapes of triangle, quadrangle, pentagon, hexagon, and octagon, respectively, and compare with the results for circular particles. Our results indicate that different particle shapes only change the local structure, but have no effects on the global structure of the formed fractal cluster. The local compactness decreases as the number of polygon edges increases.     

Crystal Growth and Nonlinear Optical Properties of 4－Br－4＇－Methoxychalcone（BMC）

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