Deformation of Platonic foam cells: Effect on growth rate

Abstract

The diffusive growth rate of a polyhedral cell in dry three-dimensional foams depends on details of shape beyond cell topology, in contrast to the situation in two dimensions, where, by von Neumann's law, the growth rate depends only on the number of cell edges. We analyze the dependence of the instantaneous growth rate on the shape of single foam cells surrounded by uniform pressure; this is accomplished by supporting the cell with films connected to a wire frame and inducing cell distortions by deforming the wire frame. We consider three foam cells with a very simple topology; these are the Platonic foam cells, which satisfy Plateau's laws and are based on the trivalent Platonic solids (tetrahedron, cube, and dodecahedron). The Surface Evolver is used to model cell deformations induced through extension, compression, shear, and torsion of the wire frames. The growth rate depends on the deformation mode and frame size and can increase or decrease with increasing cell distortion. The cells have negative growth rates, in general, but dodecahedral cells subjected to torsion in small wire frames can have positive growth rates. The deformation of cubic cells is demonstrated experimentally.

Z-cone model for the energy of an ordered foam

“…However, it should be possible to pursue its generalisation, so that 3(f) can be predicted for cases such as that of a typical random foam, in which Z varies with f, similarly to how the model of strictly regular isotropic Plateau polyhedra can give insight into the properties of disordered dry foams. 10 It will, for example, be necessary to take into account the role of near neighbours in determining the Voronoi cell size and hence f. As a rst step, we are in the process of extending the cone model to ordered bidisperse foams with curved bubble-bubble contact areas.…”

“…Nevertheless the estimates which arise from the Z-cone model has proven useful in a broader context. Note that the at-sided cube which is represented in the dry limit is not the same thing as the "isotropic Plateau polyhedron" used by Hilgenfeldt et al 7 and Evans et al 10 The latter represents a single cell in an extended foam, so that it may be given the appropriate Plateau geometry for stability.…”

Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter

“…53,59 These versatile shape metrics have been studied in the field of integral geometry 60 and successfully applied to analyze structure in jammed bead packs, 38,61 bi-phasic assemblies, 62,63 foams, 64 and other cellular structures. 59,64 There is a one-to-one correspondence between this class of Minkowski tensors and the multipole expansion of the surface normal vector distribution ρ(n) of a convex Voronoi polytope F(a). 65,66 For ideal crystals where all Voronoi facets have equal size, the values of the BOO q l and of the MSM q l are the same; these symmetries are fcc, hcp, the icosahedron, and sc (simple cubic).…”

Minkowski tensors of anisotropic spatial structure

“…of copolymer films in electric fields [8]. Anisotropy indices derived from the normal vector distribution of a given shape, similar to the MT, have been used to describe the shape anisotropy of simulated 3D foam cells [42,43] and liquid interfaces [3,44]. An anisotropy measure applicable to porous media is derived from the directional variations of average chord lengths.…”