## Preface

Fractal geometry provides a powerful approach for the quantitative description of disordered systems. In addition, it is useful for describing the processes that lead to the formation of such complex, highly irregular and random systems and their physical behaviour.1 Fractals treat disorder as an intrinsic phenomenon that is described in terms of a nonintegral dimension with a degree of irregularity that is independent of scale. In weakly disordered systems, the disorder disappears as progressively smaller or larger length scales are probed, whereas there is repetition of the disorder at all length scales in strongly disordered systems. A large number of analytical techniques (light, X-ray and neutron scattering techniques; light, X-ray and electron microscopy; sedimentation and particle counting techniques; etc.) are now available to probe this repetitive disorder.

In the natural environment, there is a great need to describe complex physico-chemical systems and processes quantitatively. By their very nature, environmental systems are disordered and, thus, are perfect candidates for quantitative description using fractal dimensions. Indeed, many natural objects have been shown to be fractal, including star constellations, clouds, coastlines, trees, snowflakes, brain circumvolutions, proteins, cellulose, colloidal aggregates, several minerals and clays, limestones and sandstones, sediments, soils, and their organic, mineral and microbial components. In addition, the physical, chemical and biological properties and processes of natural systems may be described using a fractal approach. This includes the quantification of aggregate structures in air, water, soils and sediments; flow through porous media; distributions of organisms, adsorption phenomena and reaction kinetics.

An 'ideal' or 'regular' fractal structure exhibits 'self-similarity' over all characterization length scales, i.e. the structure can be decomposed into smaller copies of itself, so that when any portion of the structure is magnified it will appear identical to a larger part. Since natural structures tend to be self-similar over only a finite range of length scales,2 they are most often referred to as 'random' fractals.

The notion of fractals can be used to describe very diverse objects according to the property of interest. In the natural sciences, a large number of physical properties and processes will depend upon the scaling behaviour of the mass, surface and pore spaces of the system (Figure 1). If the mass and the surface area scale in the same

1 Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman, New York.

2 Pfeifer, P. and Obert, M. (1989). Fractals: basic concepts and terminology. In The Fractal Approach to Heterogeneous Chemistry, Avnir, D. (ed.). John Wiley and Sons, Ltd.

Figure 1 Different fractals considered in environmental sciences. (a) Surface fractal: colony of Bacillus subtilis on agar; see Figure 8.3 for more details. (b) Mass fractal: aggregate of hematite particles formed at pH 4, ionic strength 150 mM, in the presence of natural organic matter of [C] = 2.8 mg dm-3; see Chapters 4 and 5 for more details on aggregation processes. (c) Pore fractals: soil profile in which a (darker) preferential pathway is visible; see Figure 2.21 for more details.

Figure 1 Different fractals considered in environmental sciences. (a) Surface fractal: colony of Bacillus subtilis on agar; see Figure 8.3 for more details. (b) Mass fractal: aggregate of hematite particles formed at pH 4, ionic strength 150 mM, in the presence of natural organic matter of [C] = 2.8 mg dm-3; see Chapters 4 and 5 for more details on aggregation processes. (c) Pore fractals: soil profile in which a (darker) preferential pathway is visible; see Figure 2.21 for more details.

manner, then the system is considered a mass fractal; if the pore space and the surface follow the same scaling law, then the system is a pore fractal; and if only the surface is fractal, then the system is considered a surface fractal. Each system can be represented by a three-dimensional network that divides the system on the basis of its distribution of mass, surface or pore sites. Although the different fractal dimensions are not always distinguished in the literature (again contributing to some of the confusion), an attempt has been made in this volume to distinguish clearly the different measurements (even when it was not made clear in the original reference!).

As a result of its practical utility for examining natural systems, fractal theory has developed in the geophysical, soil and atmospheric sciences, although little critical discussion has attempted to relate the different fields. The use of fractal concepts by scientists from very different disciplines and backgrounds has resulted in some confusion in the literature with respect to the meaning of the term fractal, nearly always resulting in unwarranted confusion, but also occasionally leading to unsound science. One of the objectives of this book is to reduce the confusion resulting from such a broad use of this important approach to the quantification of disorder. Specifically, the book was written in order: (i) to provide an introduction to the theory of environmental fractals (Chapters 1 and 2); (ii) to summarize the available techniques for quantifying fractal structures in environmental systems (Chapter 3); (iii) to describe how the fractal approach can be employed to describe environmental processes such as coagulation, flocculation, adsorption and desorption (Chapters 4-6); and (iv) and to describe critically a number of important environmental applications of fractal analysis (humic substances, Chapter 7; microorganisms, Chapter 8; aerosols, Chapter 9).

In this volume, as for the other volumes in the series, the goal is to provide a critical review of the literature and a thorough explanation of the most important physico-chemical processes. This book is the result of the efforts of a number of authors, collaborators and students. The International Union of Pure and Applied Chemistry (IUPAC) provided much of the structure and funding, through the Division of Chemistry and the Environment, which allowed this project come to fruition. The role of the series editors, Professor H.P. van Leeuwen and Professor J. Buffle, is also greatly appreciated.

N. Senesi

Dipartimento di Biología e Chimica Agro-Forestale ed Ambientale Università degli Studi di Bari

K.J. Wilkinson

Department of Chemistry, Université de Montréal

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