D. Weaire

Soap, Cells and Statistics-Random Patterns in Two Dimensions

Random two-dimensional patterns crop up in a wide variety of scientific contexts. What do they have in common? How can they be classified or analysed? These questions are underlined, and partly answered, by a survey of such patterns, paying particular attention to soap cell networks, metallurgical grain structures and the Giants Causeway.

A counter-example to Kelvin's conjecture on minimal surfaces

Abstract Kelvins conjecture, that a b.c.c. arrangement of his minimal tetrakaidecahedron divides space into equal cells of minimum surface area, has stood for over one hundred years. We have found a counter-example, in the form of a structure analogous to that of some clathrate compounds and also related to the β-tungsten structure. Its surface area is approximately 0.3% less than that of Kelvins structure.

The foam drainage equation

The drainage of liquid in a foam may be described in terms of a nonlinear partial differential equation for the foam density as a function of time and vertical position. We review the history and recent development of this theory, analysing various exact and approximate solutions and relating them to each other.

Computer simulation of a two-dimensional soap froth

Abstract We have developed a computer program by means of which a two-dimensional soap froth may be simulated. This system is of interest in connection with theories of grain growth in polycrystalline solids, but only limited and somewhat contradictory experimental statistical data have hitherto been available. We present preliminary results for the evolution in time of a froth of 100 cells, with periodic boundary conditions.

The kinetics of cellular patterns

Many materials, including soap froths, polycrystalline alloys, ceramics, lipid monolayers and garnet films, have structures composed of either two- or three-dimensional polygonal domains separated by well defined boundaries. Usually, the surface energy of these boundaries makes the pattern unstable, causing certain grains to shrink and eventually to disappear. Thus the pattern coarsens continuously unless other factors arrest the motion of the boundaries. The authors review recent theoretical, computational and experimental progress in their understanding of the asymptotic scaling laws that describe coarsening. In most cases the elementary expectation, that the mean grain radius scales with the square root of time, is confirmed. They pay particular attention to the history of the field, to understand why this elementary result has remained in doubt until now.

Some remarks on the arrangement of grains in a polycrystal

Abstract Aboavs observation of a correlation between the number of sides of a grain in a section of polycrystal and the number of sides of neighboring grains may be simply explained using Eulers Theorem. It is unreasonable to infer from this correlation that the grains are not disposed at random.

Relaxed continuous random network models: (I). Structural characteristics☆

Abstract Two elastic-energy-relaxed continuous random network (Polk) models for tetrahedrally bonded amorphous semiconductors have been obtained: a 201-atom model built entirely at Yale and a 519-atom model relaxed from a structure built by Polk and Boudreaux which originated at Harvard. In relaxing the coordinates to minimize the total energy the Keating potential was used for the interatomic interactions. The models are analyzed in terms of density, elastic distortion energy, elastic constants, numbers of five-, six- and seven-fold rings, distribution of dihedral angles, and radial distribution functions. We find that, despite their different origins, the models have essentially identical characteristics. Our principal conclusions are as follows: (a) The density of the CRN model is, to within 1%, that of diamond cubic. (b) The bulk modulus is about 3% lower than that for the diamond cubic structure and the shear modulus lies between the two diamond cubic shear moduli. (c) There are, to within ± 10% (and with corrections for surface effects), 0.38 five-fold, 0.91 six-fold and 1.04 seven-fold rings per atom. (d) For a reasonable value of the bond bending force constant, rms bond length distortions are about 1.0% and bond angle distortions are about 7.0°. (e) The radial distribution function agrees very well with experiment for all four principal peaks.

On the Road Again: Metal Foams Find Favor

Lightweight yet stiff, metal foams are experiencing a resurgence of interest for applications ranging from automobiles to dental implants.

Stress and strain in liquid and solid foams

Both liquid and solid foams, together with analogous cellular materials, have distinctive mechanical or rheological properties which find many applications. This review concentrates on the search for a basic understanding of the underlying mechanisms, in terms of specific structural models. Computer simulations play an increasing role and are beginning to be applied to three-dimensional models.

The physics of foam

Some recent progress in the study of liquid foams is reviewed in outline. Calculations of foam conductivity are presented, which further improve upon the approximation recently proposed by us, in accounting for the nonlinearity in the dependence of conductivity on liquid fraction.