SCIENCE/TECHNOLOGY
Fractals Offer Mathematical Tool for Study of Complex Chemical Systems Mathematical language that uses noninteger dimensions to describe irregular shapes can be used to model catalysis, kinetics, colloid aggregation Stu Borman, C&EN Washington
-ATLANTA Fractal geometry is a mathematical language that can be used to describe irregular forms in nature— rivers, mountains, trees, coastlines, and clouds—as well as complex shapes associated with scientific phenomena. With the language of fractals, "one can broach a range of problems, some very pure and mathematical and others to a greater or lesser degree applied or theoretical," says Michael E. Fisher of the Institute for Physical Science & Technology at the University of Maryland.
Distribution of potential around fractal cluster formed by random aggregation of small particles (right) shows that growth tends to occur at extremities owing to spatial screening of interior. Fractal snow flake (center) is also formed by aggregation, except that a deterministic rather than a random growth rule is used. "Sierpinski carpet" (far right), a fractal formed by an iterative process (a through d), is an exact fractal, exactly self-similar at any length scale. Fractals found in nature are only statistically self-similar because they are formed by random processes. Graphics are courtesy of Emory's Fereydoon Family 28
April 22, 1991 C&EN
Scientific use of fractal geometry began in physics but more recently has been growing in other fields such as biology, geology, and economics, in addition to chemistry. In recognition of the increasing importance of fractals in chemistry, a multisession, five-day symposium on the topic was held last week at the ACS spring meeting in Atlanta. According to symposium organizer Fereydoon Family, professor of physics at Emory University, the session was held "to introduce the chemical community to fractal and scaling concepts and to show that a number of chemists are already using these ideas effectively in their research in chemistry." Fractal mathematics makes it possible to describe complex objects, especially random objects, in terms of noninteger dimensions. A fractal object, says Family, is "infinitely rich and extremely irregular—the most irregular thing you can imagine." The concept of fractals origi-
nated with Benoit B. Mandelbrot of IBM, a feat that has won him many honors. Family says, "The most profound property of fractals is self-similarity, which means that a fractal has no characteristic length scale. If you expand a piece of a fractal, it looks the same as the original picture, the same as the whole fractal. Every piece of a fractal looks like the whole. There is no fixed length scale in the system." Although iterated fractals that are perfectly self-similar can be constructed, self-similarity, says Family, "is true of real fractals found in nature only in a statistical sense. Natural objects are random, and a piece will not look exactly like the whole. But if you calculate a correlation function that tells you how the object is put together geometrically, you'll see the same behavior in a small piece of the object as in a much larger piece. This is the essence of fractal geometry."
Another difference between natural and ideal fractal objects is that a natural fractal is not self-similar over all length scales. There are both upper and lower size limits beyond which a natural object is no longer fractal. A major strength of fractal analysis is the ability to characterize complex objects in terms of their fractal dimension. Unlike regular one-, two-, or three-dimensional geometric objects, fractal objects exhibit noninteger dimensionality. Whereas a straight line is one-dimensional, a randomly coiled polymer chain in solution has a dimension somewhere between 1 and 2. A square or triangle is two-dimensional, but the surface of a solid catalyst usually has a dimension between 2 and 3. Chemistry professor David Avnir of Hebrew University, Jerusalem, in his widely cited book, 'The Fractal Approach to Heterogeneous Chemistry: Surfaces, Colloids, Polymers" [John Wiley & Sons, New York, 1989], explains that "the main contribution of fractal geometry to modern scientific thinking is not so much through the development of useful mathematical technical tools, as in helping many scientists to overcome the psychological barriers of treating problems [that] involve very complex geometries." Fractals are common in chemistry on almost all scales larger than those associated with the size of small
molecules. "As soon as you get into the realm of macromolecules, gels, porous materials, or rough surfaces, you simply keep on encountering fractals time and time again," says research leader Paul Meakin of Du Pont. "I've never actually gone looking for fractals. They've always appeared in a very natural way in the work that I'm doing. In fact, it's amazing in many respects the frequency with which you run into fractals." The fractal concept has "enormously large descriptive powers for describing shapes," says Fisher. "In a very deep sense—and I think this is a point that [Cornell University chemist] Roald Hoffmann has also made often—chemistry is about shape, how things look, how things fit together, and how shapes change. The advantage of the fractal concept—and the reason why you can certainly say that every scientifically educated person should know something about it—is that it gives us a whole range of new tools and ways of thinking about the shapes of everything." Fisher adds, "Where we have regular shapes in chemistry—simple crystals, well-defined buckminsterfullerenes, and things like that—we don't need fractals. The classical descriptions of shape are fine. As soon as we have things that look random, that grow through aggregation processes—colloids, condensing clus-
ters, and such—then fractals are going to be there whether we like it or not. So it befits us to understand the concept of fractals, what the fractal language tells us, and what fractal dimension means." As with a lot of scientific research, he says, "some of the questions turn out not to be terribly profound, or even in some sense the wrong questions. The fractal dimension of some of the objects studied may have little to do with underlying mechanisms. In other places it turns out that if you understand the fractal dimension you really get to the root of things." The fractal concept comes into its own, says Fisher, when "you're faced with something with an interesting, complex shape, but you don't know what to do about it. The first rule of good science in many ways is not to be just descriptive, but to try to find some quantitative measure. The great importance of Mandelbrot's concept was in some sense merely to show how this could be done in a general way." A widely cited example of fractal dimension, says Fisher, "responds to the question, 'What is the length of the coastline of Great Britain?' Mandelbrot points out that that's not a well-defined concept, because the apparent length of the coastline depends upon the scale on which you measure it—that is, on the length of your measuring rod."
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April 22, 1991 C&EN 29
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