Decoration Plateau borders in two-dimensional liquid foams: geometry and excess energy

Abstract

Of all Plateau borders (PBs) that may be realized in an equilibrium two-dimensional wet foam, we identify the class of ‘decoration PBs’ as those for which the (circular) film prolongations into the PB meet at the equilibrium angles at a single point. Any three-sided PB is a decoration PB, but this is, in general, not true of n-sided PBs with n > 3. For decoration PBs we define an excess energy as the difference between the energy of the PB surfaces and that of the film prolongations into the PB. We analyse arbitrary three-sided PBs and show that their excess energy ε ₃ is approximately proportional to their linear dimensions. We further investigate the ratios

and

of three-sided PBs, where P ₀₃ and P ₃ are, respectively, the total perimeter of the PB surfaces and the perimeter of their chords. These ratios are found to be nearly independent of PB shape. We also calculate the same quantities, now denoted ε ₄,

and

, for four-sided PBs with one or two mirror symmetry planes. Finally, we provide examples of PB decoration of periodic two-dimensional foams at constant bubble (gas) area, and obtain their energy as a function of liquid fraction.