Abstract

Co-authors: S. Ataei Talebi (Universit Grenoble 1), S. Cox (Aberystwyth University), F. Graner (Universit Paris Diderot), J. Kfer (Universit Lyon 1), C. Quilliet (Universit Grenoble 1)

Two-dimensional foams are characterised by their number of bubbles, $N_{}$, area distribution, $p(A)$, and number-of-sides distribution, $p(n)$. When the liquid fraction is very low (``dry’’ foams), their bubbles are polygonal, with shapes that are locally governed by the laws of Laplace and Plateau. Bubble size distribution and packing (or ``topology”) are crucial in determining extit{e.g.} rheological properties or coarsening rate. When a foam is shuffled (either mechanically or thermally), $N_{}$ and $p(A)$ remain fixed, but bubbles undergo ``T1’’ neighbour changes, which induce a random exploration of the foam configurations.

We explore the relation between the distributions of bubble number-of-sides (topology) and bubble areas (geometry). We develop a statistical model which takes into account physical ingredients and space-filling constraintes. The model predicts that the mean number of sides of a bubble with area $A$ within a foam sample with moderate size dispersity is given by: $$ar{n}(A) = 3left(1+dfrac{ qrt{A}}{langle qrt{A} angle} ight),$$ where $langle . angle$ denotes the average over all bubbles in the foam. The model also relates the extit{topological disorder} $ Delta n / langle n angle = qrt{langle n^{2 angle – langle n angle}2}/langle n angle$ to the (known) moments of the size distribution: $$left(dfrac{Delta n}{langle n angle} ight)^{2=rac{ 1 }{4}left(langle A}{1/2} angle langle A ^{angle+langle A angle langle A}{1/2} angle^{-2} -2 ight).$$ Extensive data sets arising from experiments and simulations all collapse surprisingly well on a straight line, even at extremely high values of geometrical disorder.

At the other extreme, when approaching the perfectly regular honeycomb pattern, we identify and quantitatively discuss a crystallisation mechanism whereby topological disorder vanishes.