The foam geometry "paradox"

Abstract

Plateau's rule states that bubble lamellae in a foam meet at equal angles. Attempts to rationalize this rule via a naive "force along a tangent line" argument employing vertex variables are shown to fail, since they do not properly account for bubble volume constraints. Indeed Plateau's rule appears to make a foam system overdetermined, in the sense that there seem to be more constraints than available variables. The resolution of this paradox is that the angle constraints of Plateau's rule cannot be regarded as all independent. This is explained in detail for the two-bubble system in two dimensions. By exploiting just pressure-curvature relations and geometry, it is shown that the lamella joining the two bubbles is obliged to subtend precisely the angle needed to satisfy Plateau's rule and minimize energy. Speculations are offered for a many bubble foam. PACS Nos.: 68.10Cr, 68.15+e