Figure 3: Simulations for reproducing the gathering effect.

(a) Pillars on the substrate are present in different arrangements. They divide the 2D space into hexagonal, dodecagonal and square cells, marked by dashed circles. The three kinds of cells have identical side lengths, a, as indicated. (b–d) Microscope observations of bubble deformation at the corresponding position marked in a. (e) Quantitative description of the radius of curvature varying with bubble area when bubbles grow and deform in square (n=4), hexagonal (n=6) and dodecagonal (n=12) cell. Bubble area (A/a²) and radius of curvature (r/a) are non-dimensionalized. Arrows above the curves denote that bubbles begin to deform from roundness and their radii of curvature decrease sharply. (f) The evolution process at different times in the simulation, reproducing the gathering effect (see Supplementary Movie 3 for complete simulation). (g,h) Area (g) and radius of curvature (h) variations of every bubble in the foams as a function of time. In the graphs, each curve denotes the area variation or radius of curvature for a bubble in the simulation in f. Insets in (g,h) are details with an enlarged scale of the curves marked by the red arrows. They show that, when bubbles fill the dodecagonal cells, a very small change in area can result in a significant change in the radius of curvature. Scales, a 50 μm, b 20 μm, c 50 μm, d 20 μm.