Vol. 3  Iss. 5  The Chemical Educator  © 1998 Springer-Verlag New York, Inc.

ISSN 1430-4171 http://journals.springer-ny.com/chedr S 1430-4171 (98) 05246-0

Fractals in Chemistry by A. Harrison. Oxford Chemistry Primers, Vol. 22, Oxford Science Publications, Oxford, 1995, ISBN 0-19-855767-1, 90 pp, UK price 5 pounds, 99 pp.

In the plethora of books devoted to fractals, this modest, primer-size book is a welcomed addition, filling in the long-overdue niche of presenting this topic to undergraduate chemistry students. Whereas these students are the primary target of this Oxford series, this book is of value to graduate students as well, not only in chemistry but in neighboring domains such as materials and polymers science, surface and colloid science, geochemistry, and physics.

The main aim of the book is to show the curious student, who has been accustomed to think of nature in terms of perfect geometry of flat planes, exact spheres, ideal pyramids, and so on, that nature is much more complex, and that fractal geometry offers an entry to the study of this complexity and of its effects. The direct relevance to the chemistry student in this area stems from the recognition that many heterogeneous processes are highly sensitive to the details of the geometry that defines the heterogeneity. A celebrated example is catalyst activity, which is intimately dictated by geometric details of a catalyst's structure. This is true for unsupported catalysts, for supported ones, and for the supports themselves. In some notable cases, such as zeolites and well-defined metal crystals, the problem of quantifying both the description of the morphology and its effects on activity is accessible by standard geometrical tools. However, unless special care is taken, production of supports, catalysts, and other materials results in structural disorder and complexity. In industry and in the laboratory, these are actually the majority of cases, leading to questions such as: what is the meaning of surface area when, for convoluted surfaces, it depends on the size of the yardstick molecule used to determine the surface area?

It must be stated clearly that by no means, and rightly so, does Harrison propose that fractal geometry is the general, ultimate solution to structural problems in chemistry; but it is proposed that by adopting and adapting some concepts of fractal geometry, one can illuminate heterogeneous chemistry from a new angle, and provide hitherto unattainable insights on the sensitivity of reactivity to the details of structure in its broadest sense. Indeed it is perhaps in order to comment now on what can and what cannot be claimed by fractal analysis of experimental data from the practical point of view.

Fractal analysis is a tool with which one analyzes how a given property of matter, either extensive (e.g., absolute surface area) or intensive (e.g., density), changes with the resolution of its measurement. In that sense, fractal analysis is a resolution analysis. Such is proven practically useful if a simple relation between property and resolution emerges, allowing both the condensation of data into few parameters and some finite range of predictability. It has been an empirical observation, the roots of which are yet an enigma, that in many cases the scaling relation that is obtained takes the form of a simple power law of the form

where b is a constant that defines the sensitivity to resolution changes. Often then, the exponent has been further interpreted in terms of a simple function of the fractal dimension. Such further interpretation is not always needed, but it has helped researchers to link experimental observations to simulations and to theoretical approaches that have fractal structures at their base. And again, while this power-law scaling relation is not necessarily the only one of interest, it is the focus of this Primer because it has been observed in a wide range of materials, processes, and reactions. A practice has emerged in the scientific literature that whenever one encounters such an empirical relation, one often terms it a "fractal scaling relation." Indeed, the main feature of fractal geometry that made it a priori attractive for approaching the problems of structural complexity has been the resemblance of theoretical objects generated by fractal geometry algorithms to many cases of realistic disorder found in the material world. In particular, the shapes of these objects are characterized by self-similarity, namely by invariance to transformation of scale.

Yet one must stress the clear distinction between the teaching of fractal geometry as a theoretical mathematical tool, and the experimentally observed power laws which have been interpreted in terms of fractal objects. In the former, one should have the scaling relation over very many orders of magnitude (in fact, an infinite number of orders); in practice, however, one usually observes about one order of magnitude, maybe two, rarely spanning beyond it. These limited range is the outcome of the dictates of lower and upper cutoffs inherent in any real system. Is it then justified to call such objects fractal? It is an open debate. The view of this reviewer is that yes, the benefits of analyzing complex structures in terms of a resolution analysis, and in terms of a resulting limited-range power law, greatly outweigh the demands of mathematical rigor. Thus fractal geometry, in its more relaxed and imperfect practical form, has greatly facilitated the revolution in contemporary heterogeneous chemistry which, at last, has embraced the so-called pathological structures as a legitimate child. This revolution is clearly and concisely explained by Harrison in this book, which I highly recommend not only for the bookshelf of practitioners of heterogeneous chemistry, but also to be read and taught.