E. A. Jagla
Centro Atomico Bariloche
8400 Bariloche, Argentina
Alberto G. Rojo (*)
Department of Physics
University of Michigan
Ann Arbor, MI 48109-1120
(734-764-4223) , email@example.com
(*) I will be staying at the Embassy Suites (612)-333-3111, March 19-21
Popular Version of Paper [B25.09]
Monday morning, March 20, 12:36 p.m.
APS March 2000 Meeting, Minneapolis
Manifestations of geometric order in nature display
a diversity of beautiful shapes with different symmetries.
The phenomenon of columnar jointing in some kinds of volcanic rock--especially
basaltic lava flows--is one spectacular example where cracks split
the rock in a set of parallel columns.
The length of the columns is from a few to hundreds of meters, and the cross section shows a distinctive pattern of mostly pentagonal and hexagonal polygons whose sizes vary from a few centimeters to about 4 meters. These columns, whose size places them among the largest geometrical patterns of inanimate nature, are impressive enough to have inspired legends attributing their origin to supernatural beings, as reflected in names like Devil's Postpile in California, and Giant's Causeway in Northern Ireland (see figure below)
This splendid polygonal arrangement presents enduring puzzles, despite the fact that it has been realized for more than a century that columns result from the contraction of the cooling lava. A more recent picture corresponds to the fractures forming at the surface and propagating to the interior. Also, there is evidence that, at the surface, the pattern of fractures consists mostly of 90 degree angles (the `T' junctions), and that the quasi-hexagonal pattern develops as the fractures propagate to the interior. The detailed mechanism underlying this ordering process is unclear, and constitutes the main focus of our work. We propose a mechanism of ordering that explains the observed quasi-hexagonal patterns and which, in contrast with previously proposed models, applies also to desiccating cornstarch where very similar patterns emerge on a length scale thousands of times smaller.
The idea is the following. Lava, as well as cornstarch, starts fracturing at the surface. These incipient fractures are very similar to those seen in mud, were cracks start at points where some critical stress is surpassed, and then propagate rather rapidly, usually along a smooth path. Close to a crack the material remains stressed in the direction parallel to the crack (the stress in the perpendicular direction was relieved by the crack itself). Therefore, newer cracks tends to join older cracks at right angles. (In fact, the quasi-hexagonal pattern is never seen in mud or in any other surface fragmentation process.) In order to understand the developing of this ordered pattern one needs to consider a mechanism of propagation of cracks towards the interior of the material. Our model starts with a random array of cracks that then progresses towards the interior and, through small gradual variations, "explores" configurations of lower energy. The results of computer simulations of our model are shown in the animation below, where time corresponds to depth of the rock. The final equilibrium pattern consists of hexagons as well as a considerable fraction of pentagons and heptagons, showing a close similarity with the observed patterns of lava and cornstarch (quantitative comparisons of the fraction of polygons are also in very good agreement).
In general, the idea of energy minimization giving rise to ordered patterns is not new. For instance, all patterns arising from the crystalline order of solids (from snowflakes to the macroscopic shape of crystals) have their ultimate origin in the tendency of atoms to assume spatial positions that minimize the energy. However, we show that the energy minimization process when fractures penetrate the rock, driving the system to states with ordered polygonal patterns of fractures is fundamentally different from the ordering force in crystals.
Computer simulation of the evolution towards a quasi-hexagonal pattern. Time plays the role of depth within the rock. Note the similarity with the patterns of the Giant's Causeway (shown below in a digitized photograph).
Digitized map of the Giant's Causeway