The Geometry of Bubbles and Foams

Abstract

Abstract. We consider mathematical models of bubbles, foams and froths, as collections of surfaces which minimize area under volume constraints. The resulting surfaces have constant mean curvature and an invariant notion of equilibrium forces. The possible singularities are described by Plateau's rules; this means that combinatorially a foam is dual to some triangulation of space. We examine certain restrictions on the combinatorics of triangulations and some useful ways to construct triangulations. Finally, we examine particular structures, like the family of tetrahedrally close-packed structures. These include the one used by Weaire and Phelan in their counterexample to the Kelvin conjecture, and they all seem useful for generating good equal-volume foams.

Efficient Time Integration of Maxwell's Equations with Generalized Finite Differences

“…To achieve grids with such angles, we recall the concept of TCP structures described by Frank and Kasper [18]. In all of these structures, there are four combinatorial types of Voronoi cells, which all have only pentagonal and hexagonal faces, with no adjacent hexagons [51]. In the literature, the most common TCP structures are A15, C15, and Z lattices.…”

“…However, it is not obvious that the TCP structures are better for Yee-like schemes. For example, the edge lengths differ less in the BCC tetrahedra than in the TCP structures [51]. The A15, C15, and Z structures are illustrated in Figure 3, and the vertices of each TCP structure are listed in Table 1.…”

“…The listed x-coordinates of Table 1 should then be multiplied by √ 3 to obtain the correct tiling. The replicable dual structure consists of six 12-hedra, four 14-hedra and four 15-hedra [51].…”

Fractal Growth of Carbon Schwarzites

“…Density functional @F) calculations in the local density approximation for nanotubes of variable radius R (where H = l/(ZR)andK=0)[36,37JandCso(whereH= l/RandK= 1/R2) [36]permittoextractthevaIuesx=3.1 eVand F = 1.7 eV. Consistently a value of = 1.5 eV can be extracted form the available calculations of the cohesive energy of schwarzites [11,19,22] by means of the Gauss-Bonnet theorem [32]:…”

“…where A is the (portion of the) surface which the total energy refers to, y = 2.82 eVIA2 is the energy for unitaryflaf surface (a graphite sheet) [36], and the mean and gaussian curvatures, respectively, with R, and R2 the principal radii of curvature. Minimal surfaces are characterized by R, = -RR~ at all positions.…”