Ballistic Aggregation Research Articles

Properties of evaporated target foils studied by computer simulation of ballistic aggregation

Target foils often show defects during evaporation, floating, or mounting. These defects depend in subtle ways on the evaporation conditions including: angle(s) of incidence of evaporant on substrate, residual gas pressure and composition, rate of evaporation, and temperature of substrate. Some progress has been made in understanding these effects using a computer program that simulates the buildup of such foils during evaporation. The program runs on a microcomputer and models simple ballistic aggregation of sticky atoms on a square lattice. Although real atoms move in three dimensions and presumably relax after collision to positions of minimum potential energy, two-dimensional simulations illustrate many of the properties of real aggregations. A striking behavior of such simulations is the columnar structure in the deposited film and the dependence of that structure upon the distribution of angles of incidence of the evaporant atoms. Atoms incident from a single direction produce foils having narrow fibrous structures, while atoms incident from several directions produce treelike patterns having lower average density and greater horizontal extent. Simulated deposited films show variations in density in the direction perpendicular to the substrate; these variations may be responsible for the tendency of some foils to curl or crumple while being floated.

Percolation on ballistic aggregates.

We have studied site percolation on ballistic aggregates on a two-dimensional square lattice with a single seed particle. A percolating path is defined as the path connecting the seed to a surface particle. The fraction of the connected surface particles is found to obey a scaling relation. The percolation threshold is obtained at p\ensuremath{\approxeq}0.97. This gives us an upper threshold for the impurity tolerance to preserve the connectivity of the structure in thin films with columnar growth where ballistic aggregation is known to be relevant.

Thin film deposition profiles using a ballistic aggregation model

Thin film deposition profiles using a ballistic aggregation model

Ballistic deposition with nonuniform deposition densities: singular density distributions

Ballistic aggregation and deposition with non-uniform deposition probability densities have been investigated. For the case of a singular power law, the density distributions can be described by the strength parameter alpha . Simple theoretical arguments and computer simulations indicate that h approximately sv// and w approximately sv perpendicular to where h is the cluster height, w is the cluster width and s is the cluster size. The exponents v/sub /// and vperpendicular to are given by v/sub ///=(d- alpha )/d and vperpendicular to =1/d where d is the dimensionality of the space in which ballistic aggregations are occurring.

Simple models for the restructuring of three-dimensional ballistic aggregates

The effects of some simple restructuring processes in which pairs of rigid clusters can readjust their positions with respect to each other after collision have been investigated using the three-dimensional hierarchical cluster-cluster aggregation model. In these models we assume that contacts between pairs of particles once formed cannot be broken but may move on the surface of particles in one cluster which contact particles in a different cluster immediately after contact has been made. Such structural readjustment processes increase the fractal dimensionality from a value of about 1.89 without readjustment by an amount which depends on model details. The highest fractal dimensionality obtained in the models investigated here was about 2.13. For ballistic aggregation with a zero impact parameter the fractal dimensionality can be raised from 2.04 to 2.21.

Surface structure of random aggregates on the Cayley tree

Growth through ballistic aggregation and biased diffusion-limited aggregation is investigated on the Cayley tree. For a general branching ratio it is shown that the surface width of a ballistic aggregate remains finite (of order one) as the cluster mass goes to infinity. DLA with an attractive bias is treated approximately. For non-zero bias strength the surface width is asymptotically finite. In the limit of isotropic diffusion (zero bias) a roughening transition occurs which can be described in terms of a single diverging length scale.

Models for Colloidal Aggregation

Computer simulations have been used for more than a quarter of a century to develop a better understanding of non equilibrium growth and aggrega­ tion processes. Important early examples include VoId's ballistic deposi­ tion model for aggregation and sedimentation (1, 2), Sutherland's ballistic cluster-cluster aggregation model for floc formation (3), and Eden's surface growth model for the generation of cell colonies (4). Because of limited computer resources these early simulations were carried out on a small scale using simple models. Since these early pioneering efforts computer speed, storage capabilities, and availability have increased enormously and the task of writing computer programs has become much easier and more reliable. These developments have allowed two almost divergent research directions to develop. On the one hand, models have been developed that incorporate as much as possible of what is known about the physics and chemistry of aggregation processes, and on the other hand, a variety of very simple models have been introduced that allow us to generate the very large structures needed to investigate asymptotic (large size limit) geometric scaling relationships. The exploration of very simple aggregation models has been stimulated by the realization (5-10) that real aggregation processes and simple aggregation models frequently lead to structures that can be described in terms of the concepts of fractal geometry (11). In particular, the recent interest in nonequilibrium growth and aggregation models was initiated by the discovery by Witten & Sander (6) that a simple diffusion-limited aggregation (DLA) model in which particles are added, one at a time, to a growing cluster or aggregate of particles leads to

Dynamical scaling of the surface of finite-density ballistic aggregation.

We study the roughness of aggregates that are formed by ballistic deposition with nonzero flux density of incoming particles. The flux density is controlled by a parameter 0\ensuremath{\le}p\ensuremath{\le}1. The scaling behavior of the interfacial width \ensuremath{\xi} does not depend on p and is the same as \ensuremath{\xi} found in a variety of other, related models. For short times \ensuremath{\xi}\ensuremath{\propto}${t}^{1/3}$ for a one-dimensional substrate and \ensuremath{\xi}\ensuremath{\propto}${t}^{0.22}$ for two-dimensional ones. The latter is not consistent with various theoretical predictions. Rough numerical estimates for the long-time exponent are also presented. In addition, we derive a relation between the width of the geometrical boundary of the aggregate and the width of the active region of growth. This relation, true for models in which the active zone is asymptotically correlated to the surface of the aggregate, is verified by our simulations.

The effects of restructuring on the geometry of clusters formed by diffusion-limited, ballistic, and reaction-limited cluster–cluster aggregation

The effects of simple restructuring mechanisms on the geometry of clusters formed by diffusion-limited, ballistic, and reaction-limited cluster–cluster aggregation have been investigated using three-dimensional off-lattice models. In these models restructuring is allowed to take place immediately after two clusters have contacted each other, but no subsequent restructuring is allowed. Restructuring takes place in three distinct stages and the process can be stopped after each stage. In the case of diffusion-limited cluster–cluster aggregation, the fractal dimensionality is increased from about 1.8 to about 2.1 after the first stage. Complete restructuring increases the fractal dimensionality to about 2.2. Similar but somewhat smaller increases in the fractal dimensionality are found for ballistic and reaction-limited aggregation.

Structural transitions in ballistic aggregation simulation of thin-film growth

Computer simulations of thin-film growth were performed by random ballistic aggregation of hard two-dimensional disks. A simple model incorporating sputtering, surface relaxation, and redeposition of sputtered atoms was used to simulate the effects of ion bombardment during growth. Without ion bombardment the deposits had a voided columnar structure typical of a zone 1 growth. With increasing bombardment flux, a densification of the deposit and an elimination of continuous voids were observed. These results reproduce the general features of a bombardment induced microstructural transition from zone 1 to zone T, and also demonstrate that redeposition of sputtered atoms is essential to this transition.

Growth and Structure of Pyrogenic Silica

ABSTRACTFractal analysis is used to interpret small-angle X-ray and neutron scattering data on commercial fumed silica powders. Rough primary particles are found in high surface area powders. A model based on ballistic polymerization, sintering, and diffusion-limited aggregation is proposed to explain the observed powder structure.

Dynamics of infinite-range ballistic aggregation

The authors calculate the width of the growing interface of ballistic aggregation in the limit in which the range of the sticking interaction between the particles becomes infinite. They derive a scaling form for the width, and they compute the short- and long-time exponents finding nu =3/4 and alpha =1/2. Furthermore, they find that the crossover exponent defining the argument of the scaling function is gamma =1/2. They compare these exact results with computer simulations, finding excellent agreement. They also discuss the relation of these results to those of ordinary finite-range ballistic aggregation. Finally, they present a simple expression for the density of all ballistic aggregation clusters, regardless of the range of the interaction, which agrees with known results and interpolates between the infinite- and finite-range cases.

Growth oscillations in ballistic aggregation

The authors demonstrate the existence of growth oscillations in single clusters of particles grown stochastically according to a synchronous (finite density) version of ballistic aggregation, which is a model of relevance to a variety of experimental situations. They find clear evidence for these unexpected dynamically induced oscillations both in the propagation of the interface and in the resulting density of the cluster. They describe the general features of the spectra of the oscillations, and briefly discuss possible experimental systems in which growth oscillations might be observed.

Effects of random-walk size on the structure of diffusion-limited aggregates.

The step size of random walks is shown to exhibit an important role in controlling the structure of clusters produced by the diffusion-limited aggregation (DLA) processes. In this study, the structural compactness is found to increase significantly as the random-walk size is increased. The step-size effects are precisely quantified by the radius of gyration evaluated during the cluster growth process. It is also observed that by using a very long random walk the DLA model approaches the Vold-Sutherland ballistic aggregation model.

Scaling and shadowing effects in ballistic aggregation.

We have studied the scaling and shadowing properties of two-dimensional off-lattice ballistic aggregation on a seed. The computer simulations show that the limiting semivertical cone angle is about 15.5\ifmmode^\circ\else\textdegree\fi{}. The density as a function of distance from the seed tends to a constant value showing a basic two-dimensional nature of the aggregates and has a correction with a metadimension of about 0.56. The density as a function of angle near the edges of the cone is found to obey a scaling relation similar to the on-lattice case. The probabilities for the fingers of different lengths which shadow the particle near the edge are determined in the computer experiment and also determined analytically.

Model for the formation of nonequilibrium clusters.

We propose a new and simple model for the formation of kinetic clusters. The model appears to have two critical points. The first one corresponds to a site percolation threshold, at which point a fractal structure appears for the first time, and below which the clusters stop growing after a finite time. The second critical point appears to be exactly the same as the site percolation threshold of the lattice in the directed percolation process, and at which point a qualitative change in the structure of the cluster appears. Various critical properties of the model are computed in two dimensions and they appear to be slightly but consistently different from those of random percolation. The clusters reduce to the Eden model in a special limit, and they appear also to have close connection with ballistic aggregation and cellular automata.

Toward quantification of thin film morphology

Before any meaningful quantitative understanding of thin film growth morphology can be developed, a sufficiently detailed model must be established. For the case of films deposited under low mobility conditions, the model of the internal void networks, which are the morphology, is a fractal construction based upon the random competition for cone growth. The origin of the development of cones is a natural consequence of the random ballistic aggregation process. Both current and future approaches to quantitative measurement of this morphological model are discussed and include a series of direct (scanning electron, transmission electron, and field ion and optical microscopy) and indirect (spectroscopic ellipsometry, image enhancement and analysis of morphology photographs, small angle x-ray and electron scattering, and computer simulations of film growth) characterization techniques. It is shown how quantitative preparation–property relations in thin films will have to first start with a quantitative description of the film morphology.

Geometry of thin-film morphology

The columnar morphologies commonly found in all vapor-deposited thin films prepared under low mobility conditions have been classified by several variations of what have been termed structure zone models. Such morphological structures are found to have a strong similarity in shape and form over six orders of magnitude in film thickness and three orders of magnitude in magnification for films of a given thickness. Thick (45-mm) pyrolytic graphite films are shown to be a good demonstration of the continuous growth evolution of conical-shaped units. Due to competition for growth each cone eventually goes through a death stage. A model based upon these general structural observations is presented and is shown to be a geometric construction similar to a Sierpinski gasket. The origin of this morphology seems to be the natural clustering which occurs due to the random process of ballistic aggregation.

Are thin film physical structures fractals?

The columnar physical structures commonly found in vapor-deposited thin films have been classified by several variations of what have been termed structure zone models. Perhaps the most interesting feature of these various models is their universal application to apparently all film materials and deposition processes. A structural self-similarity over a wide range of film thicknesses, preparation conditions, and materials types appear to be pointing toward a common origin. Reasons will be presented as to why thin films may be fractals and the consequence of this suggestion to understanding the origin and evolution of thin film physical structure. In particular, recent ballistic aggregation computer models provide a promising avenue of research.

Are thin film physical structures fractals?

The columnar physical structures commonly found in vapor-deposited thin films have been classified by several variations of what have been termed structure zone models. Perhaps the most interesting feature of these various models is their universal application to apparently all film materials and deposition processes. A structural self-similarity over a wide range of film thicknesses, preparation conditions, and materials types appear to be pointing toward a common origin. Reasons will be presented as to why thin films may be fractals and the consequence of this suggestion to understanding the origin and evolution of thin film physical structure. In particular, recent ballistic aggregation computer models provide a promising avenue of research.