Deformation of Platonic foam cells: Effect on growth rate

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.

Figure 1

Experimental and simulation snapshots of foam cells with cube topology suspended from wire frames that are “undeformed” (top) and “stretched” (bottom).

Figure 2

Top: A cell in a 2D foam, which has mean curvature within the edges and mean curvature concentrated at the vertices. Bottom: A cell in a 3D foam, which has mean curvature within the faces as well as mean curvature concentrated along the sharp edges. Only mean curvature within the smooth edges (2D) and faces (3D) contributes to gas diffusion.

Figure 3

The Constructible IPPs $P_4$, $P_6$, and $P_{12}$ are regular polyhedra with spherical-cap faces. Their flat-faced counterparts are the tetrahedron, cube, and dodecahedron. We refer to these cells as regular Platonic foam cells.

Figure 4

Planar four-sided foam cell in a rectangular frame. (a) Quadrilateral cell suspended in a rectangle of size $L\times \left(\delta L\right)$. (b) Parametrization of cell deformations in terms of $\alpha$ yields analytical expressions for all relevant quantities. (c) The cell deformation $Q$, growth rate $\mathcal{G}$ relative to the growth rate ${\mathcal{G}}_0$ of the isotropic cell ($\alpha =\pi /12$), and radius of curvature $R$ of the cell edges, as functions of $\alpha$.

Figure 5

Illustration of the Platonic foam cell models. The interior (red) soap films bound a body of fixed volume suspended by (blue) films from a fixed wire frame. All of the soap films (facets) are free to move during surface area minimization.

Figure 6

Deformation of the wire frames and the corresponding deformed foam cells: (top) extension, (middle) shear, and (bottom) torsion for tetrahedral (left), cubic (center), and dodecahedral (right) foam cells.

Figure 7

Growth rate $\mathcal{G}$ as a function of $Q$ for the three Platonic foam cells. The left $y$ axis shows the relative growth rate $\mathcal{G}$/${\mathcal{G}}_0$; the right $y$ axis, the absolute growth rate $\mathcal{G}$. The value of ${\mathcal{G}}_0$ is approximately $-3.966$, $-2.850$, and $-0.4530$ for the tetrahedron, cube, and dodecahedron, respectively. The parameter $L$ is the wire-frame size. Under compression the tetrahedral cell experiences negligible distortion, so the results are omitted.

Figure 8

Surface area $S$ vs deformation $Q$ for a cubic foam cell under torsion (shown above). $S_0$ refers to the regular cube foam cell; i.e., the cube IPP.

Figure 9

Experimental and simulated images of soap bubbles suspended from initially cubic wire frames that are subjected to torsion (left), compression (center), and shear (right). Simulations indicate that the surface area of a cubic bubble decreases under torsion; see Fig. 8.

Figure 10

All of the growth rates in Fig. 7 plotted against the mean curvature in the sharp edges of the cells.