# Fractals in the Natural Sciences

M. Fleischmann
D. J. Tildesley
R. C. Ball
Pages: 208
https://www.jstor.org/stable/j.ctt7ztsgz

1. Front Matter
(pp. [i]-[iii])
(pp. 1-2)
3. Fractal geometry: what is it, and what does it do?
(pp. 3-16)
B. B. Mandelbrot

Instead of attempting to introduce and link together the papers that follow in this Discussion Meeting, we prefer to ponder the question, ‘What is fractal geometry?’ We write primarily for the comparative novice, but have tried to include tidbits for the already informed reader.

Before we tackle what afractalis, let us ponder what a fractalis not. Take a geometric shape and examine it in increasing detail. That is, take smaller and smaller portions near a point P, and allow every one to be dilated, that is, enlarged to some prescribed overall size. If our shape belongs to...

4. [Illustrations]
(pp. None)
5. Fractals, phase transitions and criticality
(pp. 17-34)
R. B. Stinchcombe

Typical fractals are characterized by structure on all scales of length (Mandelbrot 1977). Simple idealized examples much as the recursively constructed Cantor bar, Koch curve, or Sierpiński gasket (figure 1) go into (parts of) themselves under discrete scale changes, so having the property of ‘discrete’ self-similarity. Regular fractals with this property can occur naturally, for example in the spectra of incommensurate systems (Hofstadter 1976). However, real fractals are more often random and typically statistically self-similar. An example is the percolation network (figure 2), which will be discussed in §2.

The property of continuous, or discrete, or statistical, self-similarity is sufficient...

6. Fractals and phase separation
(pp. 35-54)
D. W. Schaefer, B. C. Bunker and J. P. Wilcoxon

Fractal patterns typically appear in systems that develop far from equilibrium. These patterns are often interpreted by kinetic models, the properties of which are usually known only through computer simulation. In their simplest form, kinetic models do not allow for structural rearrangement and therefore do not yield minimum energy configurations. By contrast, the more common thermodynamic models describe growth phenomena close to equilibrium. These models involve parameters for the surface and bulk energy, and predict compact non-fractal structures.

The traditional models for phase separation, nucleation and growth, and spinodal decomposition, are thermodynamic models. These models successfully describe the essential experimental...

7. Experiments on the structure and vibrations of fractal solids
(pp. 55-70)
E. Courtens and R. Vacher

Aerogels are monolithic solid materials with an extremely tenuous microscopic structure (Fricke 1985; Vacheret al. 1989a). The most thoroughly investigated aerogels are made of silica. These can be prepared with a porosity as high as 99%. In consequence, they exhibit unusual physical properties, making them suitable for a number of technical applications, such as Cerenkov radiators, supports for catalysts, or thermal and acoustic insulators. Suitably prepared aerogels are also excellent examples of fractal solids (Vacheret al. 1988a). Thus it is both of fundamental and technical interest to determine the structure of aerogels, to investigate the mechanisms of their...

8. Universality of fractal aggregates as probed by light scattering
(pp. 71-88)
M. Y. Lin, H. M. Lindsay, D. A. Weitz, R. C. Ball, R. Klein and P. Meakin

The aggregation of colloids has been the subject of scientific investigations for over 100 years. The past few years have seen considerable progress in our understanding of the complex physics that govern this process. A key to this recent success is the recognition that the structure of the colloidal aggregates exhibits scale invariance or dilation symmetry, and can be described as a fractal (Weitz & Oliveria 1984). This has afforded a quantitative description of the highly disordered structure of the clusters, which has in turn afforded a more detailed description of the kinetic growth process that forms these aggregates.

The class...

9. Light-scattering studies of aggregation
(pp. 89-102)
J. G. Rarity, R. N. Seabrook and R. J. G. Carr

An early theory of the kinetics of aggregation, coagulation and precipitation is that of Smoluchowski (1916). Many studies of aggregation (Reerink & Overbeek 1954; Ottewill & Shaw 1966) have made use of this theory to model the initial stages of the reaction. In recent years there has been an upsurge of interest in the study of aggregation because of the discovery of the fractal nature of the aggregates produced in these reactions (Schaefferet al. 1984; Weitz & Oliveira 1983). Large-scale computer simulations of aggregation reactions (Meakin 1983; Kolbet al. 1983) have indicated some departure from ‘Smoluchowski’ kinetics. This has stimulated the...

10. Time-series analysis
(pp. 103-122)
D. S. Broomhead and R. Jones

Mass action kinetics give rise to nonlinear evolution equations for systems of chemical reactions; similarly, fluid flows are governed by nonlinear laws in all but special limiting cases. Even stock markets, it seems, when controlled by computer program, evolve as deterministic nonlinear dynamical systems. The realization that nonlinearities have important consequences on the dynamics of real systems is not new (except, perhaps, in the case of the stockmarket). Over the past century (Poincaré 1899), and particularly over the past two decades (see, for example, Guckenheimer & Holmes 1983; Bergéet al. 1984) there has been a strong interest in the mathematics...

11. Diffusion-controlled growth
(pp. 123-132)
R. C. Ball, M. J. Blunt and O. Rath Spivack

The key element relating different analogues of diffusion-controlled growth (or aggregation) is that the local growth rate is determined by the flux of a field that obeys Laplace’s equation, with the growth surface presenting an equipotential boundary condition. This is evidently realized by true material diffusion in the limit that the growth advances slowly enough that the concentration field ahead of it is quasi-static, and the extreme case of this is represented by the original diffusion limited aggregation (dla) computer model of Witten & Sander (1981) where only one particle approaches the aggregate at any one time.

It is valuable to...

12. Diffusion-limited aggregation
(pp. 133-148)
P. Meakin and Susan Tolman

Pattern-formation processes have been of considerable scientific interest and practical importance for many decades. Interest in the growth of complex structures under non-equilibrium conditions has been stimulated by several recent developments. The dissemination of the concepts of fractal geometry (Mandelbrot 1982) and related ideas have provided us with ways of describing a very broad range of irregular structures in quantitative terms. It has been shown that even very simple nonlinear systems and models (see, for example, May 1976; Lorenz 1963; Feigenbaum 1978) can lead to complex, often chaotic, behaviour that can frequently be described in terms of fractal geometry. In...

13. Electrodeposition in support: concentration gradients, an ohmic model and the genesis of branching fractals
(pp. 149-158)
D. B. Hibbert and J. R. Melrose

Electrodeposition has been of much current interest as an experimentally controllable example of growth governed by diffusion. In the ideal experiment the field controlling transport of the depositing species obeys, in the quasi-stationary approximation, Laplace’s equation (Ball 1986). Experimental realizations with this field being the concentration (Brady & Ball 1984; Kaufmanet al. 1987) and also an electric field (Matsushitaet al. 1984, 1985; Grieret al. 1986; Swadaet al. 1986) have been reported. The fractals of the diffusion-limited extreme (Witten & Sander 1983) have been grown (Brady & Ball 1984; Kaufmanet al. 1987; Kahanda & Tomkiewicz 1988). This extreme has been...

14. [Illustrations]
(pp. None)
15. Flow through porous media: limits of fractal patterns
(pp. 159-168)
R. Lenormand

The purpose of this paper is to provide better understanding of the relevant mechanisms that control the displacement of a wetting fluid by a non-wetting fluid in a porous medium when both capillary and viscous forces are present.

By using experiments on micromodels and computer simulations, we have previously demonstrated the existence of three types of basic displacements:

(a) capillary fingering when capillary forces are very strong compared to viscous forces;

(b) viscous fingering when a less viscous fluid is displacing a more viscous one;

(c) stable displacement in the opposite case.

We have also shown how these displacements can...

16. Fractal bet and fhh theories of adsorption: a comparative study
(pp. 169-188)
P. Pfeifer, M. Obert and M. W. Cole

Since the first exploration of fractal surface properties of solids at molecular scales (Avnir & Pfeifer 1983) experimental investigations have uncovered a wealth of materials with a well-defined fractal surface dimensionD(see, for example Avniret al. 1984, 1985; Schaeferet al. 1987; Pfeifer 1987; Schmidt 1988; Farin & Avnir 1989), as determined by a wide variety of techniques. Properties that are sensitive toDinclude small-angle X-ray and neutron scattering (Bale & Schmidt 1984; Wonget al. 1986; Martin & Hurd 1987), multiple scattering and absorption of light (Berry & Percival 1986), electronic energy transfer between adsorbed molecules (Klafter & Blumen 1984; Evenet...

17. Reactions in and on fractal media
(pp. 189-200)
A. Blumen and G. H. Köhler

Randomness occurs in many areas of modern physics. For example, in the past decade interest has turned increasingly towards the investigation of amorphous solids such as glasses. There are two reasons why this did not happen earlier: on the one hand these substances are now of technical interest; on the other hand, until recently the lack of suitable theoretical methods prevented a satisfactory description of experimental results. It is clear that models developed for crystalline solids are inappropriate, because they are based upon the translational symmetry of the lattice, and glasses are translationally disordered. Glasses show a number of unusual...