Pilots, parachutes and a drunk's walk

Fractals, Scaling and Growth Far From Equilibrium
September 25, 1998

The concept of fractal surfaces is familiar to anyone who has studied the surface of the sea from different heights. If you are in an aeroplane gaining height over inhabited terrain, various features such as cars, houses, roads, and fields gradually look smaller and smaller as you climb higher, so that with experience pilots can estimate their height from their appearance. Over the sea the same trick does not work. As you go higher, waves of a given size indeed appear smaller, but at the same time you become aware of larger waves that visually take their place, thus in large measure compensating the effect of increasing distance. The net result is that over a large range of heights the appearance does not change very much. Pilots shot down over the sea during the second world war were aware of this effect: if they were tempted to release their parachute early for fear of being trapped underneath it, they had to remember that they could not easily tell whether they were 20 feet or 200 feet above the sea, and the only safe rule was not to release their parachute before their feet were wet.

Surfaces that look similar at different magnifications are called fractal. In Fractals, Scaling and Growth Far from Equilibrium Paul Meakin gives a masterly survey of the mathematical tools available for describing such surfaces, summarises models of growth and relevant experimental studies, and then applies the mathematics of fractals and scaling to the growth of surfaces and interfaces in materials. This book provides an authoritative account for graduate students and researchers studying the geometry of surfaces and interfaces. In the preface Meakin relates the relevance of this to long-standing observations such as the hexagonal patterns with which snowflakes grow.

The key concepts are introduced early on. Self-similar surfaces are ones where all aspects scale equally when the magnification is changed, whereas self-affine surfaces are ones where as the magnification is changed the vertical aspects scale differently from the horizontal features. Self-affine scaling is described by a Hurst exponent, defined as the power to which the scaling of one parameter must be raised to give the scaling of another parameter. For example, in Brownian motion the distance moved by a particle is proportional to the square root of the time elapsed. This is sometimes known as the drunken man's walk, since if a man is so drunk that the direction in which he takes each step bears random relation to the direction of the previous step, then his distance from the pub where he started will be proportional to the square root of the time for which he has been walking. The Hurst exponent for Brownian motion (and the drunken man's walk) is thus 1/2.

Fractal dimensionality is illustrated by the well-known Sierpinski gasket. This can be constructed by taking a solid equilateral triangle of side L, and removing an inverted equilateral triangle of side L/2 from its middle, thus forming a pattern of three triangles each with side L/2. The procedure is then repeated for each of these triangles with side L/2, to form nine triangles of side L/4, and so on. After the nth stage the sub-triangles would have sides of length t = L/2n. If the process is continued an infinite number of times, the gasket acquires fractal properties. To see how remarkable these are, consider how the amount of material varies when you increase the length scale by a factor of two. If you had a simple filled triangle, the area would increase by 22=4. But with the Sierpinski gasket the amount of material increases by a factor of only three when the scale is doubled. This somewhat counterintuitive result arises from the fractal nature of the gasket, or to put it another way from the fact that we continued removing material for an infinite number of steps.

All this is elementary to experts in the field, but this book does not remain elementary for many pages. Perhaps one of the most remarkable chapters is on the growth of interfaces. A number of simple and not-so-simple growth models are considered, and it is shown how these can lead to various rough surfaces that cannot be described by simple Euclidean shapes. There are extensive comparisons between model and experiment, and in this context Meakin illustrates how atomic-force microscopy has a particularly valuable contribution to make. As all PhD students writing their literature surveys know, when you are reviewing work at the cutting edge of a field it is a challenge both to be comprehensive and to maintain a story line, so that the whole is more than simply a series of summaries strung like beads on a thread. Meakin succeeds remarkably well, and he is also not restrained when he feels critical appraisal is required.

Some 20 years ago, Brian Pippard wrote two splendid volumes on the physics of vibration, in which he included an account of what happens when a mass subject to the force of gravity is placed on a vibrating platform. A practical experiment to illustrate this would be to lie a loudspeaker on its back with a small stone on the cone, and attach the coil to a signal generator. At very small amplitudes the stone would simply move up and down with the cone of the loudspeaker, but as the amplitude was increased, the stone would lose contact with the cone during part of the cycle, and its motion would no longer be sinusoidal. At some amplitudes the frequency of its motion would be an integer fraction of the loudspeaker frequency, because the cone could exercise more than one cycle while the stone was out of contact. At other amplitudes the motion of the stone could become utterly chaotic, resulting in an irritating rattle.

It happened that this analysis was relevant to some experiments performed in collaboration between Oxford and Lausanne in which we measured the motion of the tip of tiny cantilever, which was so small that the contact between the tip and the vibrating surface was only a few atoms across. I sent a description of the results to Pippard, saying that this must be the smallest example ever of the phenomenon that he had analysed. He wrote back saying how much he wished he had had all the mathematical apparatus now available for describing the dynamics of nonlinear systems and the chaotic behaviour they can exhibit as they are driven further and further from equilibrium. This book provides a definitive account of the mathematical description of shapes of systems far from equilibrium, and richly deserves inclusion in the prestigious Cambridge Nonlinear Science series. As if this were not enough, among the 1,356 references Meakin promises two further volumes, also to be published by Cambridge University Press. I was frustrated to learn that understanding the growth of snowflakes remains a challenge. Perhaps one of his future volumes will contain the answer.

Andrew Briggs is reader in materials, University of Oxford.

Fractals, Scaling and Growth Far From Equilibrium

Author - Paul Meakin
ISBN - 0 521 45253 8
Publisher - Cambridge University Press
Price - £75.00
Pages - 674

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