Roughing It Michael Marder atomic scale, although it is overall a straight line. Viewed through Mandelbrot's micro- scope, the increase of roughness with in- creasing magnificationappearsto occur in an orderly fashion, as M0.75. There is a variety of different ways to measure roughness, some of which eive different answers at small and - large scales, and the computer captures these variations as well. although it cannot en- - compass the large span of magnification Ideal solid-uch as the tetrahedron, cube, Kalia and co-workers (1) are now showing scales accessible to experiment. and sphere-have long provided models of that this mechanism leads in large-scale The simulations represent a considerable the natural world. Yet the more closely one computer simulations to fractal fracture sur- technical achievement. Calculations ofsuch looks at a natural object, such as a round faces with quantitative resemblance to those a size are possible only with a parallel com- stone, the more resolutely it differs from a in experiments. puter in which large numbers of separate sphere. Seen sufficiently close, most surfaces Their simulations take place in a variety processing units work simultaneously. All resemble mountainous landscapes, jagged of virtual materials, including graphite and such machines are still a bit experimental, surfaces be rough at all, and why shbuld the jaggedness grow in such a predictable way under the microscope? One possibility, strongly supported by the numerical simulations by Kalia et d., is that such roughness arises naturally .in the act of breaking two surfaces apart. Almost all means of exposing a new surfaceto the world are types of breaking, from the erosion of a tiny chip of stone from a cliff face to the dropping of a drinking glass. Breaking up is hard to do in anything but a rough and messy way because crack tips go unstable. Think of a crack tip racing through a solid, cutting its atoms apart. Experiments have long shown (5) that when the tip be- wmes too energetic, the material rips rather than dividing cleanly. The simulations by The author is in the Department of Physics and Cen- ter for Nonlinear Dynamics. The University of Tex- as at Austin, Austin. TX 78712, USA. E-mail: marder@Xhaos.ph.utexas.edu and irregular (see figure, part A): ~ ali -- et d. have found through their com- puter simulations (1 ) the ways in which roughness can arise spontaneously in the most primitive act of creating new surfaces, the process of fracture. The subversive idea that natural forms are better described by irregular fractal shapes than by Euclid's five regular polyhedra is Mandelbrot's (2). Not only did he assemble the math- ematics needed to describe primitive forms of irregularity and dictate the terminology of a new field, he also car- ried out detailed observations of natu- ral Phenomena in varied One roken silicon crystal, about 1 .! these case studies (3) that the pm across. [Courtesy of U. Purbach, J. Hauch, and A. De Lozanne] (middle and right) Computer simula- surface of shattered steel is self-afine; tion of the fracture surface of a broken piece of silicon nitride. [Courtesy of P. Vashishta and R. Kalia] that is, seen from a distance, it may appear smooth, but viewed under a micro- silicon nitride. The computations typically but this group is roughing it even more than scope at magnification M, the apparent involve more than a million atoms, and be- most, having built its own supercomputer roughness in the field of view grows as M0.75. cause it is out of the question for the pro- by hooking together the innards of 40 work- This simple law has continued to receive grammers to specifythe initial positions of all stations. They set a high standard for the confirmation in experimental studies (4), of these atoms individually, the descriptions realism built into the atomic interactions, but where does it come from? Whv should of how the simulationswere ~rmared have a which are derived as thorouehlv as ~ossible a a peculiar ring of real-world experiments. To create noncrystalline silicon nitride, the re- searchers cut small spherical clusters out of a perfect crystal, ram them together under pressure, heat them to 2000 K, wait for them slowly to become more dense, and finally cool the resulting solid down to room tem- perature. To study breakage, the computer saws a small notch into one side of the block, grabs two opposite sides in imaginary jaws, and pulls them apart until the block snaps in two (see figure, part B). Without fractal geometry, there would be no language to describe the results, except to say that they look like a mess. As the crack jumps ahead, its head splits repeatedly into multiple cracks, most of which proceed a short distance sideways into the solid and then die. Between searching for weak spots in the solid and fendingoff powerful acoustic waves racing around inside the sample at high speeds, it is hardly surprising that the main crack path is quite irregular on the -, . from other types of computation and from experimental data. Such large-scale compu- tations are still too new to say whether the results are realistic and compare well with experiment on a detailed level. The neces- sary experimentshave not even been formu- lated, much less performed. Some experi- mental aspects of silicon nitride, such as highly elongated crystallites, have not yet been incorporated into the simulation. Still, this first rough cut at replicating reality in the computer looks promising. It is messy, and in the right way. References 1. R. K. Kalia etal., Phys. Rev. Left. 78.21 44 (1 997); A. Omeltchenko etal., ibid., p. 2148. 2. B. B. Mandelbrot, The Fractal Gmtryof Nature (Freeman, New York, 1982). 3. B. B. Mandelbrot eta/., Nature 308,721 (1984). 4. E. Bouchaud et a/., Phys. Rev. 8 48,2917 (1993); V. Y. Milman et aL, Prog. Mater. Sci. 38, 425 (1994). 5. M. J. Doyle. J. Mater. Sci. 18, 687 (1983); J. Rneberg etal.. Phys. Rev. 8 45,5146 (1992). www.sciencemag.org SCIENCE VOL. 277 1 AUGUST 1997