Fractal Growth Phenomena

The following sections are included:

• Foreword

• Preface

• Contents

During the last decade it has widely been recognized by physicists working in diverse areas that many of the structures common in their experiments possess a rather special kind of geometrical complexity. This awareness is largely due to the activity of Benoit Mandelbrot (1977, 1979, 1982, 1988), who called attention to the particular geometrical properties of such objects as the shore of continents, the branches of trees, or the surface of clouds. He coined the name fractal for these complex shapes to express that they can be characterized by a non-integer (fractal) dimensionality. With the development of research in this direction the list of examples of fractals has become very long, and includes structures from microscopic aggregates to the clusters of galaxies…

Our present knowledge of fractals is a result of an increasing interest in their behaviour. Some of the basic properties of objects with anomalous dimension were noticed and investigated at the beginning of this century mainly by Hausdorff (1919) and Besicovich (1935). The relevance of fractals to physics and many other fields was pointed out by Mandelbrot, who demonstrated the richness of fractal geometry and presented further important results in his recent books on the subject (Mandelbrot 1975, 1977 and 1982). The purpose of this chapter is to give an introduction to the basic concepts, properties and types of fractals.

In the previous chapter such complex geometrical structures were discussed which could be interpreted in terms of a single fractal dimension. The present chapter is mainly concerned with the development of a formalism for the description of the situation when a singular distribution is defined on a fractal. As we shall see, the structure of fractals plays an essential role in the physical processes they are involved in and, as a result, one needs infinitely many dimension-type exponents to characterize these distributions as was first recognized by Mandelbrot (1974). This idea was later developed further by others, including Hentschel and Procaccia (1983), Benzi et al (1984), Frisch and Parisi (1984) and Halsey et al (1986)…

When one tries to determine the fractal dimension of growing structures in practice, it usually turns out that the direct application of definitions for D given in the previous two chapters is ineffective or can not be accomplished. Instead, one is led to measure or calculate quantities which can be shown to be related to the fractal dimension of the objects.

Three main approaches are used for the determination of these quantities: experimental, computer and theoretical. Experiments represent a standard way of examining phenomena in every field of physics and they have been playing an important role in the development of research concerning fractal growth as well. The situation is less typical in the case of the other two approaches. Since the physics of fractal growth lacks a unified theoretical description, most of the investigations prompted by theoretical motivations are based on computer simulation. The only theoretical principle which seems to be applicable to a relatively wide range of growth processes is renormalization which will be discussed in the last Section following a discussion of the experimental and numerical methods for determining D.

In Part II. models based on growing structures made of identical particles will be treated. While in Chapter 2. mostly artificial examples were discussed, here we shall concentrate on more realistic models which are constructed in order to reflect the essential features of specific growth phenomena occurring in nature. Various models allowing exact or numerical treatment have been playing an important role in the studies of growth. Because of the complexity of the phenomena it is usually a difficult task to decide which of the factors affecting the growth plays a relevant role in determining the structure of the growing object. In a real system the number of such factors can be relatively large, and this number is decreased to a few by appropriate model systems. Thus, the investigation of these models provides a possibility to detect the most relevant factors, and demonstrate their effects in the absence of any disturbance…

Many of the growth processes in nature are governed by the spatial distribution of a field-like quantity which is inherently non-local, i.e., the value of this quantity at a given point in space is influenced by distant points of the system, in addition to its immediate neighbourhood. For example, such behaviour is exhibited by the distribution of temperature during solidification, the probability of finding a diffusing particle or cluster at a given point, and electric potential around a charged conductor…

Many growth processes lead to space filling objects with a trivial dimension coinciding with the dimension of the space d in which the growth takes place. However, the surface of these objects may exhibit special scaling behaviour. The three basic possibilities are the following: the surface may be i) smooth, having a trivial dimension d_s = d − 1, ii) fractal with D < d and iii) self-affine, characterized by an anisotropic scaling of the typical sizes. In this chapter cluster growth models producing the third kind of interfaces will be discussed…

Aggregation of microscopic particles diffusing in a fluid medium represents a common process leading to fractal structures. If the density of the initially randomly distributed particles is larger than zero, the probability for two “sticky” particles to collide and stick together is finite. It is typical for such systems that the resulting two-particle aggregate can diffuse further and may form larger fractal clusters by joining other aggregates (Friedlander 1977). As a result the mean cluster size increases in time and, in principle, after a sufficiently long period all of the particles in the finite system become part of a single cluster. In many cases the force between two particles is of short range and it is strong enough to bind the particles irreversibly when they contact each other. For example, such behaviour can be observed for iron smoke aggregates formed in air (Forrest and Witten 1979) or in aqueous gold colloids (Weitz and Olivera 1984)…

The formation of complex patterns by moving unstable interfaces is a common phenomenon in many fields of science and technology. During pattern formation in real systems the surface tension of the boundary between the growing and the surrounding phases plays an important role. This is in contrast to the case of cluster growth models discussed in Part II., where such effects were not taken into account. Thus we shall use the term pattern formation for growth processes in which surface tension is essential…

The large number of relevant new results obtained by various theoretical approaches and computer simulations has stimulated an increased interest in the experimental systems exhibiting fractal pattern formation. It has turned out that there are many publications and unpublished results on the desks of scientists which, in the light of the recent theoretical progress, may represent good starting points for further investigations. In particular, the formation of random branching structures in thin solid films was observed some time ago in several laboratories, but it was the theoretical framework of fractal geometry which revived the interest in the related experiments…

The following sections are included:

• Appendix A Algorithm for growing diffusion-limited aggregates

• Appendix B Construction of a simple Hele-Shaw Cell

• Appendix C Basic concepts underlying multifractal measures

• Author Index

• Subject Index